Full opinion text
WESLEY, Circuit Judge. Defendants Timothy J. Rigas and John J. Rigas (“Defendants”) appeal from a judgment of conviction following a jury trial in the United States District Court for the Southern District of New York (Sand, J.). Defendants were convicted of conspiracy to commit securities fraud, conspiracy to make and cause to be made false statements in filings with the SEC, and conspiracy to commit bank fraud under 18 U.S.C. § 371 (Count One); securities fraud under 15 U.S.C. §§ 78j(b) and 78ff, and 18 U.S.C. § 2 (Counts Two through Sixteen); and bank fraud under 18 U.S.C. § 1344 (Counts Twenty-Two and Twenty-Three). Defendants make four claims on appeal: (1) the government should have been required to present evidence that Defendants violated Generally Accepted Accounting Principles (“GAAP”) and to call an accounting expert; (2) government witness Robert DiBella improperly gave expert accounting testimony; (3) the bank fraud convictions should be vacated because the indictment was constructively amended or they should be reversed because there was insufficient evidence for the jury to find that any misrepresentations to the bank were “material”; and (4) Defendants were prejudiced by the improper admission of uncharged crime evidence, which also constituted a constructive amendment of the indictment. For the reasons set forth below, we affirm the judgments of conviction on all Counts except Count Twenty-Three. We reverse Defendants’ conviction on Count Twenty-Three, and we remand for an entry of a judgment of acquittal on this Count and for resentencing. BaCkground Adelphia Communications Company (“Adelphia”) announced its 2001 Fourth Quarter and Full-Year results in a March 27, 2002 press release. In a footnote on the final page of that press release, Adelp-hia, at the recommendation of its accounting firm, Deloitte & Touche, first disclosed publicly that it had approximately $2.2 billion in liabilities not previously reported on its balance sheet. On the day of disclosure, Adelphia’s stock price plummeted by about twenty-five percent to $20.39; by the time the stock was delisted in May 2002, the price per share was $1.16. The company filed for bankruptcy in June 2002, wiping out all shareholder value. A month later, John Rigas, his sons Michael and Timothy, and two other Adelphia employees were arrested and charged with looting the company. The Story of Adelphia Adelphia, one of the largest cable television providers in the country before its bankruptcy, had modest beginnings. In the early 1950s, John Rigas, the son of Greek immigrants, borrowed money from his family to buy a movie theater in Coud-ersport, a small town about twenty miles south of the New York-Pennsylvania state line. In 1952, he purchased the rights to wire the town for cable television. By the time John Rigas’s sons Michael and Timothy joined Adelphia in the mid-1980s, the privately owned company boasted hundreds of thousands of cable subscribers. In 1986, John Rigas took Adelphia public. Adelphia issued two classes of common stock: Class A, with one vote per share, and Class B, with 10 votes per share. The Rigas family owned almost all of the Class B shares, and, as a result, was able to maintain control of the company and the Board of Directors. Indeed, Ri-gas family members filled many of the top positions in Adelphia. John Rigas was Adelphia’s President, Chairman of the Board, and Chief Executive Officer until he resigned in May 2002. Timothy Rigas served as Board member, Executive Vice President, and Chief Financial Officer. Michael Rigas was also on Adelphia’s Board and was Executive Vice President for Operations. Another son, James, filled out the Rigases’ majority control of the seven-member Board of Directors. Peter Venetis, John Rigas’s son-in-law, was added to the Board when it expanded to nine members. Not all of the companies controlled by the Rigas family went public when Adelp-hia did. Rather, Adelphia managed some of the cable companies — the Rigas Managed Entities (“RMEs”) — that the family continued to own privately. Adelphia’s management of the RMEs was disclosed in public filings; however, Adelphia did not disclose the amount of the fees charged to, or paid by, the RMEs, or that cash generated from the RMEs was commingled with that generated by Adelphia. Certain transactions between Adelphia and the RMEs were at issue during the trial; the government argued that Defendants utilized the Adelphia-RMEs business arrangement to effect and conceal aspects of their frauds. Adelphia’s business during the time relevant to this case was “cash flow negative.” That is, it did not generate enough cash revenue from subscriber fees to pay for its capital expenditures, interest payments, and cost of operations. Adelphia’s capital expenditures included $1.5 to $2 billion per year to update its cable systems to higher bandwidth and two-way communication capabilities (the “Rebuild Plan”). Between 1998 and 2002, Adelphia paid approximately $5.2 billion in cash and issued more than 72 million new shares of Class A common stock to acquire other cable entities in an effort to lower costs as a result of operating efficiencies (the “Acquisition Plan”). Banks and holders of Adelphia stocks and bonds watched as Adelphia’s leverage ratios climbed. Indeed, as Moody’s Investors Service noted in August 2001, Adelp-hia was “one of the most highly leveraged companies in the cable sector.” Adelphia set about raising sufficient capital to offset its annual operating losses, to fund the Rebuild and Acquisition Plans, and to pay down increasing interest expenses. This new cash mainly came from $4.9 billion in public sales of newly issued common and preferred stock, $4.4 billion in public sales of notes and convertible debentures, and bank loans. Adelphia’s disclosed bank borrowings were $5.4 billion in September 2001, more than a six-fold increase from March 1998. Generally, each separate bank loan was entered into by a group of Adelphia subsidiaries that pledged their assets as collateral; the group was referred to as a “borrowing group.” Certain bank loans were set up through a “co-borrowing” arrangement (the “Co-Borrowing Arrangement”) between the RMEs and Adelphia subsidiaries. Timothy Rigas proposed the Co-Borrowing Arrangement to the Adelp-hia Board in 1999, and argued it would lower borrowing costs and prevent competition for bank financing between the RMEs and Adelphia entities. Under this Co-Borrowing Arrangement and at the Ri-gases’ direction, Adelphia entered into three separate “Co-Borrowing Agreements” — loans for which the RMEs and Adelphia subsidiaries were jointly and severally liable. These Co-Borrowing Agreements totaled about $5.5 billion. Adelp-hia’s accounting firm, Deloitte & Touche, reviewed and approved the manner in which Adelphia disclosed and accounted for the co-borrowed debt on its public financial statements. The Rigas family wished not only to expand Adelphia, but also to maintain control over the company, in part because Adelphia’s loan agreements provided that the Rigases’ loss of voting control would constitute default. To maintain family control, every sale of stock to the public required a concurrent sale of stock to the Rigases. Arguing that their stock purchases represented the family’s “public vote of confidence” in Adelphia “because in addition to selling shares to the public, they were buying new shares, that is, they were investing fresh money of their own into the company,” Timothy Rigas persuaded the Adelphia board to sell Class B shares to the Rigas family with each new offering to the public. During the relevant time, family members purchased $1.6 billion in new shares. The Charged Conduct Timothy Rigas and John Rigas were charged with conspiracy to commit securities fraud, to commit wire fraud, to make and cause to be made false statements in filings with the SEC, to falsify the books and records of a public corporation, and to commit bank fraud under 18 U.S.C. § 371 (Count One); securities fraud under 15 U.S.C. §§ 78¡j(b) and 78ff, and 18 U.S.C. § 2 (Counts Two through Sixteen); wire fraud under 18 U.S.C. § 1341 (Counts Seventeen through Twenty-One); and bank fraud under 18 U.S.C. § 1344 (Counts Twenty-Two and Twenty-Three). The conduct underlying these charges, as set forth in the Superseding Indictment and the government’s case at trial, is summarized below. I. The Rigas Family’s Fraudulent Stock Purchases The Rigases did not have enough cash to provide the promised “fresh money” for the shares they purchased to maintain control over Adelphia. The steps they took to purchase these shares constitute several of the charged frauds. The purchase agreements for the stocks required that at the closing date, the Rigases “shall deliver to the Company the purchase price for the Shares in immediately available funds”; Adelphia’s public filings and press releases suggested to investors and analysts that the Rigases had paid cash for the stocks. However, this was not the case. Instead, for the earlier purchases, Defendants borrowed funds to pay Adelphia, but then caused Adelphia to use that cash to pay off other family debts. For the later purchases, Defendants caused Adelphia to “move” debt it owed under the Co-Borrowing Agreements from its books to the books of one of the RMEs. The process of moving debt from Adelphia’s financial statement to one of the RMEs’ financial statements was called “reclassification,” and the debt, itself, was referred to as having been “reclassified.” As the government argued at trial, even if the RMEs had assumed Adelphia’s debts, Adelphia was worse off than if the Rigases had paid cash and Adelphia paid down its existing borrowings. When the Rigases assumed debt from Adelphia under one of the Co-Borrowing Agreements, Adelphia’s capital funding strategy was adversely affected in two ways: first, Adelp-hia would still be liable for those debts because the Co-Borrowing Agreements provided for joint and several liability, and second, had the Rigases paid cash, those funds could have been used to pay down the debts on the Co-Borrowing Agreements, thus freeing up the credit available for Adelphia. Most importantly, the Ri-gases misrepresented that they paid cash for the stocks, raising the necessary funds from margin loans, from leveraging their private cable properties, and from outside investors, and that this cash would be used to pay down debts. The government introduced evidence supporting its allegations that Defendants engaged in fraudulent securities purchases through, inter alia, the testimony of former members of Adelphia’s Board, the stock purchase agreements, bank records, general ledger journal entries relating to the sales, and borrowing and paydown notices for the bank creditors. II. The Transfers of the Co-Borrowing Debt The government also alleged that Defendants masked other debts that the Rigas family, the RMEs, and the RNCEs owed to Adelphia. Defendants accomplished this by reporting all the amounts the RMEs and RNCEs owed Adelphia as a single “related party receivable,” which they reported on a net basis — that is, Adelphia’s financial statements did not itemize the amounts owed by each of the RFEs, but instead listed a single figure which “netted” all the payables and receivables related to the RFEs on a combined basis. The reclassification scheme used to effectuate the stock sales described above also contributed to this concealment. By reporting the amounts owed as a related party receivable, Defendants masked both the actual amount of cash advanced to the RMEs and the RNCEs and the fact that the cash was advanced to RNCEs that Adelphia did not manage. Once this net related party receivable reached $200 million, Vice President of Finance James Brown and Timothy Rigas discussed masking the size of the receivables by moving debt from Adelphia’s books to the RMEs’ books. As an example, Adelphia might move $20 million of debt owed to the banks under the Co-Borrowing Agreements to an RME’s books; Adelphia would then credit the RME with the $20 million assumption of debt, thus decreasing the amount the RME owed Adelphia by $20 million. Brown testified that this arrangement provided no benefit to the Adelphia shareholders, but merely avoided disclosing on Adelphia’s books the high net receivable balance from the RMEs. After the first reclassification of over $200 million, additional debt was reclassified on a quarterly basis. In total, the Rigases reclassified over $2.8 billion dollars worth of debt, including the stock purchase reclassifica-tions, from the first quarter of 2000 until the end of the conspiracy. These reclassifications were memorialized only in general ledger journal entries; neither Adelphia nor the RMEs executed formal assumption agreements. As the reclassified funds had been borrowed under the Co-Borrowing Agreements, Adelp-hia would still be liable for the full amount due if the RMEs were unable to pay the debt. The government argued, and the jury apparently agreed, that these ledger entries were fraudulent and intended to mislead stockholders and analysts about the debt the Rigas family and the RFEs owed to Adelphia. III. Fraud Regarding Adelphia’s Operating Performance The government also alleged that Defendants misrepresented three key indices of Adelphia’s performance: (1) its basic cable subscriber growth; (2) its success in rebuilding its cable systems; and (3) its pro forma earnings, measured in terms of “Earnings Before Interest, Taxes, Depreciation, and Amortization” (“EBITDA”). These misrepresentations allowed Adelp-hia to appease investors and comply with covenants under its bond indentures, and they affected indices used to set interest rates under its various bank loans. They were disseminated to the public through Adelphia’s SEC filings and quarterly press releases, and through conference calls, conferences, and “road shows” with investors. Given Adelphia’s rapid expansion, and the associated cash flow deficits, investors were paying particularly close attention to the indices Adelphia manipulated. a. Misleading Cable Subscriber Growth The government provided proof that Adelphia distributed materially misleading cable subscriber growth numbers to the public from 2000 to 2002. Timothy Rigas directed or approved fraudulent quarterly earnings press releases, and John Rigas knew of and approved them. Karen Chrosniak, Adelphia’s Director of Investor Relations, testified that Timothy Rigas and others directed her to add subscribers to the earnings releases to artificially increase Adelphia’s reported basic subscriber growth rate. The government argued that Timothy Rigas directed the fraudulent inflation of Adelphia’s basic subscriber number and basic subscriber growth rate by adding, in 2000, subscribers from companies in Brazil and Venezuela in which Adelphia owned an interest. The government contended that including the subscribers artificially increased Adelphia’s reported pro forma basic subscriber growth rate. The third quarter 2001 report was also increased, again at Timothy Rigas’s instruction, to include 60,000 home security system subscribers, even though the home security subscribers were tallied separately from the cable subscribers and those home security subscribers who also had cable would, in effect, be double counted. Finally, after he learned that his projections to analysts had fallen short, Timothy Rigas instructed Chrosniak to inflate the 2001 year-end number of subscribers to the Powerlink internet service by including 7,000 “pending installs” — subscribers who had signed up for service but not yet started making payments to Adelphia as the service had not yet been installed — as actual subscribers. As a result of the fraudulent increases in subscriber growth rates, the year-end 2000 subscriber growth rate was reported as 1.3 percent and the year-end 2001 figure was reported as 0.5 percent. The actual figures were 0.5 percent and negative 1.2 percent. b. Misrepresentations about the Rebuild Program Adelphia expended between $1.5 and $2 billion annually to rebuild its cable system to provide digital cable and high speed internet access to its subscribers. As this enhanced technology was critical to the company’s long-term health and the annual expenditures on it were substantial, investors closely followed the status of the Rebuild Program. But they were misled by Timothy Rigas, who, during road shows, investor conferences, and shareholders’ meetings, fraudulently overstated the percentage of Adelphia’s systems that had been upgraded to higher bandwidth and two-way communication capabilities. Adelphia also gave these inflated numbers to its bank lenders. c. Adelphia’s Inflated EBITDA Several witnesses testified that investors commonly use EBITDA to assess the earnings from operations of cable companies. Brown testified that he told John Rigas Adelphia’s real EBITDA and how it compared with its competitors’ results. Brown also told John Rigas what would happen if Adelphia reported the actual EBITDA: Adelphia would default on some of its public debt, its stock price could decline, and its interest expenses and the cost of borrowing from banks would increase. While John Rigas told Brown that Adelphia “needed to get away” from using what Brown described as “accounting magic” to manipulate the numbers, he never told Brown to stop manipulating the numbers. The “accounting magic” used to manipulate EBITDA comprised two schemes: (1) fraudulent allocations of management fees that the RMEs owed to Adelphia and (2) “wash transactions” with Adelphia’s suppliers. Brown explained that he would arbitrarily inflate the management fees that an RME owed to Adelphia, and then record a corresponding interest expense that Adelp-hia “owed” the RME. The interest expense would ensure that there was no real cost to the RME as a result of the scheme but, because it was interest, it would not be included in Adelphia’s EBITDA. As a result, then, this scheme — which Timothy Ri-gas “went along with” — artificially inflated Adelphia’s EBITDA. In his testimony about the wash transactions, Brown indicated that Timothy Ri-gas discussed, and then implemented, schemes with two separate equipment suppliers, Motorola and Scientific Atlanta. The effect of the wash transaction schemes was to increase Adelphia’s EBIT-DA by $87.1 million. In these schemes, Adelphia increased the price it paid to Motorola and Scientific Atlanta for digital converter boxes, and Motorola and Scientific Atlanta agreed to pay Adelphia the amount of the increase for advertising and market support. Because the payments to the equipment suppliers were booked as capital expenses, and the payments from the suppliers were booked as revenue, this scheme artificially inflated the EBITDA. According to Brown, Timothy Rigas instructed him to book nearly $20 million in increased advertising revenue even before the two equipment suppliers agreed to the wash transaction scheme; Adelphia never provided any advertising services for these suppliers. IV. The Scheme to Defraud Adelphia’s Bank Lenders The jury convicted Timothy and John Rigas of conspiracy to commit bank fraud and two substantive counts of bank fraud related to two of the three Co-Borrowing Agreements. The Co-Borrowing Agreements required minimum leverage ratios of debt to EBITDA and tied interest rates to this leverage ratio. The government argued to the jury that the EBITDA manipulations resulted in lower interest payments to the banks than if the EBIT-DA had been accurately reported. The EBITDA manipulations were carried out at the level of the Adelphia parent company as described above. In addition, when Brown, Timothy Rigas, and Michael Mul-cahey (a co-defendant of the Rigases who was Adelphia’s Assistant Treasurer) determined that the EBITDA of particular borrowing groups (the Adelphia and RME entities in each Co-Borrowing Agreement) was not high enough, expenses would be moved between the subsidiaries and affiliate or interest income would be transferred from one internal company to another. Y. Looting from Adelphia’s Cash Management System The evidence at trial showed that throughout the period of the conspiracy, Defendants took over $200 million dollars from Adelphia’s Cash Management System for personal expenses ranging from $200 to purchase 100 pairs of bedroom slippers for Timothy Rigas, to over $3 million to produce a film by Ellen Rigas, to $200 million to pay off Rigas family margin loans. The missing money was obscured by the commingling of cash between Adelphia and the RMEs and the RNCEs. Cash transfers for the benefit of the Rigas family needed only to be approved by a member of the Rigas family or James Brown. No promissory notes were ever signed in favor of Adelphia, and, in some instances, personal expenses were falsely recorded as Adelphia’s expenses. Timothy Rigas also unilaterally changed the price allocation approved by Adelphia’s Board of Directors regarding the co-purchase of certain cable systems; he shifted an extra $50 million of the purchase price from the RFEs to Adelphia without informing Adelphia’s independent directors. The cash transfers to the Rigas family were not reported as compensation or loans, as required by the SEC, or disclosed to investors as related party transactions. Adelphia’s financial statements and annual reports did little to apprise shareholders of what the Rigas family owed Adelphia. All related party transactions between Adelphia and the Rigas family and the RNCEs were combined and “netted out” against transactions with the RMEs, which obscured what the Rigas family actually owed Adelphia. DiBella’s Testimony Robert DiBella reviewed and analyzed Adelphia’s accounting records from December 31, 1988 through April 30, 2002 and testified extensively about a summary chart, Government Exhibit 101, prepared with data retrieved from Adelphia’s general ledger, journal entries, and other supporting documents to “summarize the affiliate receivable transactions between Adelphia and certain of the Rigas entities.” He totaled the cash that flowed into the Cash Management System from the RMEs and the RNCEs and then deducted the payments made on behalf of the RMEs and the RNCEs. The result was that there were net receivables due to Adelphia from the Rigas entities of $54.9 million, $164.7 million, $10.5 million, $39.9 million, and $386 million for the years 1998 through 2002. But, the government argued, even these numbers underrepresented — by over $2.8 billion — the actual debt that the Rigas family owed Adelphia because of the debt reclassification scheme described above. The reclassification scheme was included on Government Exhibit 101, and DiBella explained to the jury that, while the net receivable to Adelphia with the debt reclassifications was $386 million, it would have been around $3.2 billion without the reclassifications. The Defense Case Timothy Rigas called no witness, and John Rigas called a character witness and two lawyers who testified that a government witness had made a prior statement that was inconsistent with his trial testimony. The Verdict After a four and a half month trial, the testimony of twenty witnesses, and the submission of hundreds of exhibits, the jury found John and Timothy Rigas guilty of conspiracy to commit securities fraud, conspiracy to make and cause to be made false statements in filings with the SEC, and conspiracy to commit bank fraud under 18 U.S.C. § 371; securities fraud under 15 U.S.C. §§ 78j(b) and 78ff, and 18 U.S.C. § 2; and bank fraud under 18 U.S.C. § 1344. John Rigas, Timothy Ri-gas, and Michael Rigas were acquitted of wire fraud and conspiracy to commit wire fraud. The jury acquitted Michael Mulca-hey of all charges and acquitted Michael Rigas of the conspiracy and wire fraud counts; the jury was undecided as to the remaining counts against Michael Rigas. John Rigas and Timothy Rigas remain free on bail. Discussion The Government Was Not Required to Prove Defendants Violated GAAP or to Call an Accounting Expert Defendants challenge their convictions for conspiracy under 18 U.S.C. § 371 (Count One) and securities fraud under 15 U.S.C. §§ 783(b) and 78ff; 17 C.F.R. § 240.10b-5; and 18 U.S.C. § 2 (Counts Two through Sixteen), on the grounds that the prosecution should have been required to call an accounting expert to familiarize the jury with GAAP. Specifically, they contend that Financial Accounting Statement (“FAS”) Number 5 (“FAS 5”) by the Financial Accounting Standards Board (“FASB”) applies to the Co-Borrowing Agreements that formed the basis for the securities fraud conviction, and that the government was required to introduce FAS 5 and an accounting expert to explain it. We conclude that the government was not required to present this evidence. Defendants wisely do not argue that the prosecution was required to prove that they violated GAAP to establish that they committed securities fraud. It has been the long-held view in this Circuit that GAAP neither establishes nor shields guilt in a securities fraud case. United States v. Simon, 425 F.2d 796, 805-06 (2d Cir. 1969) (Friendly, J.). Making GAAP compliance determinative of securities fraud charges would require jurors to “accept the accountants’ evaluation whether a given fact was material to overall fair presentation” — a proposition this Court rejected in Simon. Id. at 806. Instead, compliance with GAAP is relevant only as evidence of whether a defendant acted in good faith. Id. at 805. Simon was recently, and unequivocally, reaffirmed by this Court in United States v. Ebbers, 458 F.3d 110 (2d Cir.2006). In Ebbers, we held that “GAAP may have relevance in that a defendant’s good faith attempt to comply with GAAP or reliance upon an accountant’s advice regarding GAAP may negate the government’s claim of an intent to deceive,” id. at 125 (citing Simon, 425 F.2d at 805), but that even when “improper accounting is alleged,” we look to the statute to determine what the government must prove. Id. Defendants argue that Simon should apply only to cases where no specific accounting provision speaks to the alleged accounting malfeasance. They base this argument on language from Simon that notes accountants’ evaluations do not bind a jury, “at least not when the accountants’ testimony was not based on specific rules or prohibitions to which they could point. ...” Simon, 425 F.2d at 806. They contend that because FAS 5 applies to their situation, the district court should have required the prosecution to prove noncompliance, or, at the very least, offer expert testimony on the subject. Defendants are wrong. The government was not required to present expert testimony about GAAP’s requirements because these requirements are not essential to the securities fraud alleged here. See Ebbers, 458 F.3d at 125. A single reference to GAAP in the Superseding Indictment does not change that conclusion, and the district court properly instructed the jury on the elements of securities fraud and conspiracy to commit securities fraud. See United States v. Miller, 471 U.S. 130, 144, 105 S.Ct. 1811, 85 L.Ed.2d 99 (1985) (holding that courts may ignore “independent and unnecessary allegations in the indictments”). The jury heard testimony that the debt reclassifications were specifically designed to mislead investors about the amount of money the Rigas family and their other companies owed Adelp-hia, and it could have reasonably found that Defendants committed fraud. Even if Defendants complied with GAAP, a jury could have found, as the jury did here, that Defendants intentionally misled investors. Defendants reclassified debt owed under the Co-Borrowing Agreements — for which Adelphia remained jointly and severally liable — rather than paying for the securities they purchased from Adelphia in “immediately available funds.” This reclassified debt also reduced the amount of money that Adelphia could borrow under the Co-Borrowing Agreements. As a result, the jury could find that investors were misled into believing that Adelphia had been infused with more cash, when, in reality, debt for which Adelphia remained jointly and severally liable was moved onto the RMEs’ books. Whether the reclassification was permitted under GAAP was not the issue. In Ebbers, we also foreclosed Defendants’ argument that the court should have required the prosecution to call expert witnesses to testify regarding GAAP and, specifically, FAS 5: “The government is not required in addition to prevail in a battle of expert witnesses over the application of individual GAAP rules.” 458 F.3d at 125-26. While Defendants are correct that the district court opined that an expert might be helpful, the prosecution apparently thought it could explain the alleged fraud through the testimony of other witnesses — including James Brown, Adelp-hia’s former Vice President of Finance, and James Helms, an accountant/manager in Adelphia’s treasury department — with sufficient clarity to garner a conviction. The district court did not err by not requiring the prosecution to call accounting experts. Finally, in a letter submitted pursuant to Federal Rule of Appellate Procedure 28(j), Defendants contend that United States v. Lake, 472 F.3d 1247 (10th Cir.2007), supports their argument. The defendants in Lake were indicted for, inter alia, filing false 10K reports with the SEC because those reports failed to disclose the value of their personal use of corporate aircraft. Id. at 1253-54. “Highly pertinent” to the jury’s assessment of whether the Lake defendants acted with wrongful intent in failing to disclose their use of the company planes was “whether the personal use had to be reported to the SEC.” Id. at 1253. The SEC required disclosure only if the “aggregate incremental cost” exceeded a certain threshold. Id. Because the government did not show that the SEC required disclosure of the aircraft use, there was no “evidence from which the jury could infer beyond a reasonable doubt that any of the reports wired to the SEC was false, fraudulent, or even misleading.” Id. at 1258,1260. Defendants argue that Lake’s, endorsement of the SEC standards for disclosure compels us to find that the government should have provided GAAP disclosure standards here. GAAP rules do not govern whether Adelphia’s disclosures regarding the Co-Borrowing Agreements were false and fraudulent, and a violation of GAAP is not an element of the offenses charged. Because Defendants’ guilt does not turn on whether Adelphia’s accounting statements complied with GAAP, Lake is inapposite. DiBella Did Not Present Expert Opinion Testimony Defendants assert that government witness Robert DiBella improperly offered expert opinion testimony. In our view, DiBella’s testimony was properly admitted. A district court’s decision to admit evidence is reviewed for abuse of discretion. See, e.g., Old Chief v. United States, 519 U.S. 172, 174 n. 1, 117 S.Ct. 644, 136 L.Ed.2d 574 (1997); United States v. Garcia, 413 F.3d 201, 210 (2d Cir.2005). Even if evidence is improperly admitted, reversal is warranted only if an error affects a “substantial right,” Fed.R.Evid. 103(a)-that is, if the error had a “substantial and injurious effect or influence” on the jury’s verdict. United States v. Dukagjini, 326 F.3d 45, 62 (2d Cir.2003) (internal quotation marks omitted); see also United States v. Grinage, 390 F.3d 746, 751 (2d Cir.2004); Bank of China, New York Branch, v. NBM LLC, 359 F.3d 171, 183 (2d Cir.2004). “Where the erroneously admitted evidence goes to the heart of the case against the defendant, and the other evidence against the defendant is weak, we cannot conclude that the evidence was unimportant or was not a substantial factor in the jury’s verdict.” Grinage, 390 F.3d at 751 (citing Wray v. Johnson, 202 F.3d 515, 524-30 (2d Cir.2000); United States v. Forrester, 60 F.3d 52, 64-65 (2d Cir.1995)). The government did not present DiBella as an expert witness. Instead, the government informed the district court that DiBella would be testifying only to Adelp-hia’s accounting records and not regarding “the appropriateness of [the] accounting treatment.” Noting the “overwhelming complexity of the case,” the court asked counsel, “[h]ave you ever seen a case in which a summing up was more appropriate than this one?” Over Defendants’ objection, the court accepted the government’s representation of DiBella’s testimony as that of “someone who has gone through the books and records and will testify to what the books and records reflect,” and permitted DiBella to testify as a fact witness. The court added that it would “revisit” its decision “if the government’s representation!,] inadvertently or otherwise!,] is not what the testimony of this witness will be.... ” Defendants later objected that several lines of questioning impermis-sibly invoked expert testimony; the district court allowed DiBella to continue. It is undisputed that DiBella had personal knowledge of Adelphia’s books. Tatum Partners, the company for which Di-Bella worked, was retained by Adelphia in August 2002, after Defendants were indicted, “to assist in the restatement or correction of Adelphia financial statements.” DiBella began working as a full-time Adelphia employee in September 2002. In the course of nearly twenty months at Adelphia, DiBella developed what he characterized as “fairly extensive knowledge of the debt area of Adelphia” by reviewing the Co-Borrowing Agreements, and other documents within the company, focusing on “several of the areas of related-party transactions with the Rigas family, including security purchases, margin loans, other transactions.” He also familiarized himself with Adelphia’s accounting system and the software used to generate reports. Using data collected by Adelphia’s accounting system, DiBella created Government Exhibit 101, a chart that summarized the affiliate receivable transactions between Adelphia and certain Rigas entities from 1999 through April 2002. DiBella testified, using Government Exhibit 101, about co-borrowing debt transferred from Adelphia’s books to the ledgers of the RFEs. Brown had already testified that the purpose of the reclassi-fications was to mask the amount of money that the RFEs owed Adelphia. DiBel-la explained that these reclassifications involved (1) the reduction of debt in Adelphia’s balance sheets; (2) a corresponding reduction in the amount owed to Adelphia by an RFE; and (3) the creation of a payable to the RFE from an Adelphia subsidiary. DiBella testified that Adelphia’s net related-party receivable balance would have been $2.8 billion higher without the debt reclassifications, for a total of around $3.2 billion. Defendants’ cross-examination attempted to show that the reclassifications were legitimate, and that the RFEs owed the reclassified $2.8 billion to the banks — not to Adelphia. On redirect, DiBella noted that the debt reclassification “really shouldn’t have occurred [because] Adelp-hia’s still responsible for that debt.” In its final redirect question, the government asked DiBella how much, “[b]ased on [his] review of the records and the analysis,” the RFEs owed Adelphia. DiBella’s answer — “$3.2 billion.” Defendants contend that DiBella gave expert opinion testimony about what Adelphia’s books should have shown. They argue that the government “concealed from the court, the defense, and ... the jury” that this was opinion, not fact, testimony. In support of this argument, Defendants point to DiBella’s deposition testimony in a subsequent Adelphia-relat-ed civil case. In mid-February 2005, Di-Bella testified that the receivable balance on Adelphia’s books did not include the $2.8 billion of reclassified debt, but indicated that, based on a “review of accounting literature,” it was “quite clear that Adelp-hia had no basis to relieve the debt from its balance sheet.” When asked to name the accounting literature he had reviewed, DiBella said that he considered FAS 140. Defendants also argue that the prosecutor “misrepresent[ed]” to the district court that DiBella would merely be a summary witness; this “deceit,” Defendants opine, was tantamount to a “foul blow that violated the prosecution’s fundamental obligation to see that justice is done.” See Berger v. United States, 295 U.S. 78, 88, 55 S.Ct. 629, 79 L.Ed. 1314 (1935). The government contends that DiBella did not offer expert testimony because he merely “d[id] the math” to explain how the reclassifications that Brown indicated were fraudulent affected Adelphia’s ledger. The government also argues that DiBella’s subsequent deposition testimony that the debt reclassification entries were improper under the relevant accounting literature does not transform his testimony in the Rigas trial into the product of accounting analysis. Did DiBella offer impermissible expert testimony? If his testimony “result[ed] from a process of reasoning familiar in everyday life,” it was permissible lay opinion testimony under Rule 701. Fed. R.Evid. 701, advisory committee’s note to 2000 amend, (quoting State v. Brown, 836 S.W.2d 530, 549 (Tenn.1992)). A witness’s specialized knowledge, or the fact that he was chosen to carry out an investigation because of this knowledge, does not render his testimony “expert” as long as it was based on his “investigation and reflected his investigatory findings and conclusions, and was not rooted exclusively in his expertise .... ” Bank of China, 359 F.3d at 181. If, however, the witness’s testimony was “not a product of his investigation, but rather reflected [his] specialized knowledge,” then it was impermissible expert testimony. Id. at 182. In particular, Rule 701(e), which prohibits testimony from a lay witness that is “based on scientific, technical, or other specialized knowledge,” is intended “to eliminate the risk that the reliability requirements set forth in Rule 702 will be evaded through the simple expedient of proffering an expert in lay witness clothing.” Fed.R.Evid. 701 advisory committee’s note to 2000 amend.; see also Bank of China, 359 F.3d at 181. The district court did not abuse its discretion in permitting DiBella to testify under Rule 701 about the effects of the disputed reclassifications. Garcia, 413 F.3d at 210. First, DiBella’s testimony was based upon his observations during his twenty months as an Adelphia employee. Fed.R.Evid. 701(a). DiBella was responsible for correcting Adelphia’s financial statements and was well-acquainted with the records of Adelphia and the RFEs. While Defendants argue that Di-Bella’s opinion was based on what Adelphia’s records should have shown, DiBella’s testimony was based upon Adelphia’s and the RFEs’ records, and addressed the aggregate of what the RFEs would actually owe Adelphia if the debt reclassifications, which Brown and others testified were fraudulent, had not occurred. Second, DiBella’s opinion about the effects of the reclassifications was “helpful to ... the determination of a fact in issue-” Fed.R.Evid. 701(b). As the district court noted, testimony that summed up the government’s allegations was quite “appropriate” in this complicated case. DiBella’s testimony about both the undisputed $386 million and the reclassified $2.8 billion helped explain how the allegedly improper reclassification affected what the RFEs owed Adelphia. Third, DiBella’s opinion about the re-classifications was “not based on ... specialized knowledge,” because he presumed that the reclassifications were shams, as Brown and others testified, and then explained how the reclassifications affected the amount the RFEs owed Adelphia. Fed.R.Evid. 701(c). Whether these reclas-sifications should have been carried on Adelphia’s books, as a matter of appropriate accounting techniques, was a separate issue. While DiBella did testify briefly on redirect that moving the reclassifications to the RFEs’ books was improper, the remainder of his testimony regarding the reclassifications related to how the reclas-sifications affected the amount the RFEs actually owed Adelphia. DiBella’s deposition testimony in a later case that FAS 140 required the debt reclassifications to be recorded on Adelphia’s books does not compel the conclusion that his testimony here was impermissible expert opinion. Finally, even if portions of DiBella’s redirect testimony were admitted in error, this error was harmless. Defendants have not shown that this testimony had a “substantial and injurious effect or influence” on the jury’s verdict. Dukagjini, 326 F.3d at 62; see also Bank of China, 359 F.3d at 183. We are confident that, given the importance of any wrongly admitted testimony and the overall strength of the government’s case, “the error did not influence the jury, or had but very slight effect.” Dukagjini, 326 F.3d at 62 (citation omitted). Bank Fraud Convictions: The Indictment Was Not Constructively Amended, But the Conviction on Count Twenty-Three Must Be Reversed on Sufficiency Grounds Defendants challenge their bank fraud convictions (Counts Twenty-Two and Twenty-Three) on two grounds: first, that the bank fraud charges were constructively amended, and second, that the evidence submitted at trial was insufficient to prove either the charged bank fraud or the constructively amended bank fraud. We conclude that the Superseding Indictment was not constructively amended, but that the government proffered insufficient evidence to prove that the misrepresentations alleged in Count Twenty-Three were material. We affirm Defendants’ convictions on Count Twenty-Two, but reverse their convictions on Count Twenty-Three on sufficiency grounds and instruct the district court to enter a judgment of acquittal on that Count. I. Constructive Amendment Defendants argue that the government’s proof at trial constituted a constructive amendment of the indictment. An indictment has been constructively amended “[wjhen the trial evidence or the jury charge operates to ‘broaden[] the possible bases for conviction from that which appeared in the indictment.’ ” United States v. Milstein, 401 F.3d 53, 65 (2d Cir.2005) (second alteration in original) (quoting United States v. Miller, 471 U.S. 130, 138, 105 S.Ct. 1811, 85 L.Ed.2d 99 (1985)); see also United States v. Kaplan, 490 F.3d 110, 128-29, No. 05-5531-cr, 2007 WL 1087270, at *16 (2d Cir. Apr.11, 2007). We exercise de novo review of a constructive amendment challenge, United States v. Wallace, 59 F.3d 333, 336 (2d Cir.1995), which is a per se violation of the Grand Jury Clause of the Fifth Amendment requiring reversal. United States v. Roshko, 969 F.2d 1, 5 (2d Cir.1992) (explaining that, where constructive amendment “affects an essential element of the offense,” it “destroyjs] the defendant’s substantial right to be tried only on charges presented in an indictment returned by a grand jury” (alteration in original) (internal quotation marks omitted)); see also Milstein, 401 F.3d at 65. Alternatively, “‘[a] variance occurs when the charging terms of the indictment are left unaltered, but the evidence offered at trial proves facts materially different from those alleged in the indictment.’ ” United States v. Salmonese, 352 F.3d 608, 621 (2d Cir.2003) (quoting United States v. Frank, 156 F.3d 332, 337 n. 5 (2d Cir.1998)). A defendant alleging variance must show “substantial prejudice” to warrant reversal. United States v. McDermott, 918 F.2d 319, 326 (2d Cir.1990); see also Fed.R.Crim.P. 52(a); United States v. Dupre, 462 F.3d 131, 140 (2d Cir.2006). Section IV of the Superseding Indictment explained, at paragraphs 159 and 161, that Co-Borrowing Agreements required “quarterly reports to ... lenders regarding each borrowing group’s compliance with the conditions of the credit facilities, and, in particular, [the borrowing group’s] ratio of cash flow to indebtedness.” The indictment further alleged, at paragraph 160, that Timothy Rigas and Mulcahey, along with other Adelphia employees, “prepared and submitted to lenders loan compliance reports that fraudulently misrepresented, among other things, the cash flow of the reporting entities.” If a borrowing group was not in compliance with its loan covenants, or if it could obtain a better interest rate by reporting a more favorable ratio of cash flow to indebtedness, paragraph 162 alleged, Timothy Ri-gas and Mulcahey, together with other Adelphia employees, “routinely made one or more fraudulent adjustments to the financial information disclosed in the required loan compliance documents.” Finally, paragraph 163 stated: Such fraudulent adjustments to financial information submitted to the banks took a number of forms. Often, TIMOTHY J. RIGAS and MICHAEL C. MULCA-HEY, together with Brown, would record revenue due from affiliates, without any factual basis, and direct Adelphia employees to credit such revenue to a particular borrowing group so that it would be in compliance. At other times, TIMOTHY J. RIGAS and MULCAHEY would direct Adelphia employees either to lower the borrowing group’s actual costs or increase its extra revenues, again with no factual basis. Such fraudulent adjustments had the effect of increasing the cash flow for a particular borrowing group so as to bring it into compliance with its loan agreements. The charging paragraphs for Counts Twenty-Two and Twenty-Three — paragraphs 210-11 — incorporated by reference the allegations contained in paragraphs 1-197 and 204-05, and alleged that John Rigas, Timothy Rigas, Michael Rigas, and Mulcahey committed bank fraud by “falsely representing] that the borrowers on [two] credit agreements ... were in compliance with certain material terms of those credit agreements.” The Superseding Indictment briefly described, and set forth the approximate dates of, the two Co-Borrowing Agreements. Defendants contend the only bank fraud theory properly set forth in the Superseding Indictment was that “post-closing adjustments” to financial information resulted in bank fraud. They argue that their convictions were based on an entirely different theory, referenced only in Section II of the indictment, that related to the EBITDA manipulations from marketing support contracts with Motorola and Scientific Atlanta. They argue that the jury should not have been permitted to consider any conduct or scheme other than the one specifically alleged in Section IV of the Superseding Indictment. The government argues that the Superseding Indictment was sufficiently broad for the jury to consider whether the fraudulent EBITDA manipulations from the marketing support contracts “trickled down” to affect the leverage ratios reported in compliance reports to the banks. The government contends the indictment did not limit it to proving only that post-closing adjustments and management fee forgiveness affected the leverage ratios that were submitted to the banks. The government also notes that the Superseding Indictment alleged that the Defendants “prepared and submitted to lenders loan compliance reports that fraudulently misrepresented, among other things, the cash flow of the reporting entities” and “falsely represented that the borrowers on the credit agreements set forth below were in compliance with certain material terms of those credit agreements.” Moreover, the Superseding Indictment alleged that Defendants “caused Adelphia to engage in sham transactions with affiliates for the purpose of substantiating Adelphia’s false and fraudulent loan compliance reports” and “caused Adelphia to record false and misleading entries in its books and records for the purpose of substantiating Adelp-hia’s false and fraudulent loan compliance reports.” The Superseding Indictment also alleged that Defendants “caused Adelphia to submit false and misleading compliance reports, and to make other false and misleading statements, to banks and holders of Adelphia’s corporate debt.” Because the Superseding Indictment was sufficiently broad, the government argues, the sham marketing support transactions with Motorola and Scientific Atlanta, along with journal entries which booked nonexistent fee income from certain RMEs and RNCEs, permissibly demonstrated the means by which Defendants caused Adelphia to engage in sham transactions “for the purpose of substantiating the fraudulent loan compliance reports.” To establish a constructive amendment, the Rigases must show that trial evidence or the jury instructions “so altered an essential element of the charge that, upon review, it is uncertain whether the defendant was convicted of conduct that was the subject of the grand jury’s indictment.” Salmonese, 352 F.3d at 620 (internal quotation marks omitted). “[W]here a generally framed indictment encompasses the specific legal theory or evidence used at trial,” there is no constructive amendment. Milstein, 401 F.3d at 65 (quoting Salmonese, 352 F.3d at 620). As a result, “an indictment drawn in more general terms may support a conviction on alternate bases, even though an indictment with specific charging terms will not.” United States v. Zingaro, 858 F.2d 94, 99 (2d Cir.1988). Our constructive amendment jurisprudence has resulted in what we recently characterized as apparently “divergent results.” Milstein, 401 F.3d at 65 (collecting cases). One constant, however, is that we have “consistently permitted significant flexibility in proof, provided that the defendant was given notice of the core of criminality to be proven at trial.” United States v. Patino, 962 F.2d 263, 266 (2d Cir.1992) (emphasis added) (internal quotation marks omitted). “[Pjroof at trial need not, indeed cannot, be a precise replica of the charges contained in an indictment.” United States v. Heimann, 705 F.2d 662, 666 (2d Cir.1983). However, “even an amendment or a variance that does not alter an essential element may still deprive a defendant of an opportunity to meet the prosecutor’s case.” United States v. Helmsley, 941 F.2d 71, 90 (2d Cir.1991). The Supreme Court recently reiterated the “two constitutional requirements for an indictment: first,. that it contains the elements of the offense charged and fairly informs a defendant of the charge against which he must defend, and, second, that it enables him to plead an acquittal or conviction in bar of future prosecutions for the same offense.” United States v. Resendiz-Ponce, — U.S. -, 127 S.Ct. 782, 788, 166 L.Ed.2d 591 (2007) (internal alterations and quotation marks omitted). The issue in determining whether an indictment has been constructively amended, then, is whether the deviation between the facts alleged in the indictment and the proof adduced at trial undercuts these constitutional requirements. If the indictment notifies the defendant of the “core of criminality,” Patino, 962 F.2d at 265-66, and the government’s proof at trial does not “modify essential elements of the offense charged to the point that there is a substantial likelihood that the defendant may have been convicted of an offense other than the one charged by the grand jury,” United States v. Clemente, 22 F.3d 477, 482 (2d Cir.1994), then he has sufficient notice “of the charge against which he must defend,” Resendiz-Ponce, 127 S.Ct. at 788. We recently affirmed a conviction for wire fraud where the only wire transfer actually alleged in the indictment was not proven. Dupre, 462 F.3d at 140-141. There was no constructive amendment, we held, “because the evidence at trial concerned the same elaborate scheme to defraud investors as was described in the indictment,” even though none of the wire transfers presented in the trial had been alleged in the indictment. Id. The indictment and the evidence at trial contained the same starting and ending dates of the conspiracy, and the prosecution demonstrated the same overall scheme — that defendants misled investors into believing that they would eventually be able to obtain certain funds belonging to family members of former Philippine president Ferdinand Marcos. Id. at 141. The discrepancy between the wire transfer alleged in the indictment and the transfers proven at trial constituted a non-prejudicial variance; we affirmed the conviction. Id. at 141-42. In Milstein, the indictment alleged that pharmaceuticals were “misbranded” because the “[florgery or falsification of any part of the packaging material, including the instructional inserts, lot numbers or expiration dates, renders the drug mis-branded under federal law.” 401 F.3d at 64 (alteration in original). We found that, by charging him with misbranding because he had “re-packaged drugs as if they were the original product from the licensed manufacturers,” the government had “not necessarily place[d] Milstein on notice” that it would also attempt to prove that the drugs were unsterile. Id. at 65. Thus, we were persuaded that the indictment was constructively amended and reversed on that count. Id. Defendants’ case lies somewhere between Dupre and Milstein. Here we must determine whether permitting the jury to consider the trickle-down effects of the marketing support agreements with Motorola and Scientific Atlanta constituted a constructive amendment of the indictment. The issue, then, is whether the Superseding Indictment put Defendants on notice that the jury might consider these EBIT-DA manipulations. See, e.g., Resendiz-Ponce, 127 S.Ct. at 788. While Paragraph 163 appears to limit the manner in which the government planned to prove bank fraud, it is not the only paragraph in the indictment that addresses bank fraud. The government’s argument that there was no constructive amendment finds support in other paragraphs that suggest that the specific allegations of bank fraud are merely exemplary. Furthermore, the charging paragraphs for bank fraud incorporate by reference Paragraph 204, which alleges broadly that “[Djefendants and their co-conspirators caused Adelphia to record false and misleading entries in its books and records for the purpose of substantiating Adelphia’s false and fraudulent loan compliance reports.” When the crime charged involves making false statements, “the ‘core of criminality’ is not the substance of the false statements but rather that knowing falsehoods were submitted.... ” United States v. Sindona, 636 F.2d 792, 797 (2d Cir.1980) (citing United States v. Bernstein, 533 F.2d 775 (2d Cir.1976)). In our opinion, Defendants were notified of the “core of criminality” the government intended to prove. Patino, 962 F.2d at 265-66. Furthermore, we must read an indictment “to include facts which are necessarily implied by the specific allegations made.” United States v. LaSpina, 299 F.3d 165, 177 (2d Cir.2002) (internal quotation marks omitted). The Superseding Indictment explained that the sham transactions “g[a]ve the false appearance of revenue to Adelphia,” and this sham increase in revenue artificially inflated Adelphia’s EBITDA. Adelphia was merely a holding company— any borrowing was done through its subsidiaries or, as through the Co-Borrowing Agreements, combinations of its subsidiaries and certain RFEs. The leverage ratios reported to the banks under the Co-Borrowing Agreements were, roughly, debt divided by cash flow. An increase in revenue from the sham transactions increased the subsidiaries’ cash flow, artificially decreasing the leverage ratios they reported to the banks. The Co-Borrowing Agreements linked interest rates to the leverage ratios and provided that leverage ratios above a certain level were an event of default; manipulating the leverage ratios could, therefore, artificially lower interest rates or present the false appearance that the subsidiaries complied with the conditions of the Agreements. Defendants therefore had notice that the government would seek to prove that they “caused Adelphia to record false and misleading entries in its books and records for the purpose of substantiating Adelphia’s false and fraudulent loan compliance reports” and that the government would introduce evidence about the sham marketing support agreements that resulted in an artificial increase in revenue. That this increase in revenue would contribute to the false and fraudulent loan compliance report is “necessarily implied by the specific allegations made.” LaSpina, 299 F.3d at 177; see Dupre, 462 F.3d at 140-141. Our holding also comports with Sindo-na. In Sindona, we held that where, in response to a request for a bill of particulars, the government referred defense counsel to certain counts of an indictment — counts that had, incidentally, been dismissed — the defendant had “notice that the core of the crime charged was the concealment of the source of the funds and not the illegality of the fiduciary system” used to conceal those funds. 636 F.2d at 797. We found that there was no constructive amendment, and that the defendant’s claim on appeal that he was “surprised by the ‘shift’ of the [g]overnment late in the trial” was, if anything, nonprejudicial variance. Id. at 797-98. Here, likewise, the fact that the jury was permitted to consider proof of the trickle-down EBITDA manipulation in determining whether Defendants were guilty of bank fraud would, at most, constitute a variance. While Defendants’ brief contains, in a footnote, a cursory allegation of prejudice, they have not shown the “substantial prejudice” required to warrant reversal on variance grounds. McDermott, 918 F.2d at 326; Fed.R.Crim.P. 52(a). Defendants’ claim that evidence presented at trial constituted a constructive amendment, or prejudicial variance, of the Superseding Indictment thus fails. II. Sufficiency A defendant challenging the sufficiency of the evidence “bears a heavy burden.” United States v. Jackson, 335 F.3d 170, 180 (2d Cir.2003) (quoting United States v. Finley, 245 F.3d 199, 202 (2d Cir.2001)). Sufficiency analysis requires a court to review the separate “[pjieces of evidence ... not in isolation but in conjunction,” United States v. Miller, 116 F.3d 641, 676 (2d Cir.1997), and to draw all reasonable inferences in the light most favorable to both the jury’s verdict, United States v. Stavroulakis, 952 F.2d 686, 695 (2d Cir. 1992), and the government, United States v. Moore, 208 F.3d 411, 413 (2d Cir.2000). If “any rational trier of fact could have found the essential elements of the crime beyond a reasonable doubt,” we must affirm the conviction. Jackson v. Virginia, 443 U.S. 307, 319, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979). The federal bank fraud statute criminalizes: knowingly executing], or attempting] to execute, a scheme or artifice— (1) to defraud a financial institution; or (2) to obtain any of the moneys, funds, credits, assets, securities, or other property owned by, or under the custody or control of, a financial institution, by means of false or fraudulent pretenses, representations, or promises.... 18 U.S.C. § 1344. “[T]he ‘scheme to defraud’ clause ... requires that the defendant engage in ... a pattern or course of conduct designed to deceive a federally chartered or insured financial institution into releasing property, with the intent to victimize the institution by exposing it to actual or potential loss.” Stavroulakis, 952 F.2d at 694. First, the government must prove that the defendant engaged in a deceptive course of conduct by making material misrepresentations. Neder v. United States, 527 U.S. 1, 16, 119 S.Ct. 1827, 144 L.Ed.2d 35 (1999); United States v. Rodriguez, 140 F.3d 163, 167-68 (2d Cir.1998). “A false statement is material if it has a ‘natural tendency to influence, or is capable of influencing, the decision of the decisionmaking body to which it was addressed.’ ” United States v. Whab, 355 F.3d 155, 163 (2d Cir.2004) (quoting Neder, 527 U.S. at 16, 119 S.Ct. 1827). We have also held that “[t]o be material, the information withheld either must be of some independent value or must bear on the ultimate value of the tra