Petersson products of bases of spaces of cusp forms and estimates for Fourier coefficients

Abstract

We prove a bound for the Fourier coefficients of a cusp form of integral weight which is not a newform by computing an explicit orthogonal basis for the space of cusp forms of given integral weight and level.

Integers represented by positive-definite quadratic forms and Petersson inner products

“…In this section we prove Theorem 1. First, we rely on the preprint [23] of Rainer Schulze-Pillot and Abdullah Yenirce to translate the bound on C, C given above into a bound on the Fourier coefficient |a C (n)|. The weight 2 case of Theorem 10 of [23] gives that…”

“…It is well-known that S 2 (Γ 0 (N), χ) is spanned by newforms f of level M|N with charater χ and their "translates", that is, the images under the operator V In general, the space of "translates" of the newform f is isometric to the tensor product of the spaces W p i (f ) for the p i |M/N with the isometry mapping f |V (p r 1 1 ) ⊗f |V (p r 2 2 ) ⊗ · · · ⊗f |V (p rz z ) →f |V (p r 1 1 p r 2 2 · · · p rz z ). (This is given on the bottom of page 4 and the top of page 5 of [23].) As a consequence, there is an orthonormal basis for W p (f ) with the property that if gcd(n, N) = 1, then the nth Fourier coefficient of every basis element is ≪ d(n) √ n f,f (1 + 1/p 1/2 ), because if gcd(n, N) = 1, no contribution is made to the nth coefficient from f |V (p i ) if i ≥ 1.…”

Universal Sums of m-Gonal Numbers

“…In this section, we will first study this implied constant and then we will use the above bound to obtain an upper bound on the coefficients of the cuspidal part. We use an explicit argument of Schulze-Pillot and Yenirce [23] in order to obtain a bound on the Fourier coefficients in terms of the Petersson norm. (1) For every h ∈ N…”

“…(2) The result in Lemma 5.1 may essentially be read off from [23,Theorem 11]. However, they use the normalization f instead of f Γ , and hence we provide a full argument for the reader in order to avoid confusion.…”

“…For g 1 = g 2 ∈ H new k (ℓ, χψ) it is well known that g 1 , g 2 = 0. Moreover, for each g ∈ H new k (ℓ, χψ), Schulze-Pillot and Yenirce [23] constructed an explicit orthogonal basis of the eigenspace W g spanned by g|V d with d | M N ℓ 2 . They then proved (see the proof of [23,Theorem 11]) that there exists an orthonormal (with respect to ·, · , not ·, · Γ ) basis {F g,d :…”