## Functiones et Approximatio Commentarii Mathematici

- Funct. Approx. Comment. Math.
- Volume 34 (2005), 27-146.

### Fourier coefficients of modular forms and eigenvalues of a Hecke operator

#### Abstract

We prove results analogous to certain theorems of Deshouillers and Iwaniec (Invent. Math. 70 (1982), 219-288]. Our proofs parallel theirs in the use made of the summation formulae of Bruggeman and Kuznetsov: where they require a lower bound on eigenvalues $\lambda_j = 1/4 + \kappa^2_j$ of the hyperbolic Laplacian operator (using that of Selberg) we need instead upper bounds on the moduli of the eigenvalues of a Hecke operator, obtaining these from recent work of Kim and Sarnak [J. Amer. Math. Soc. 16 (2003), 139-183]. Specifically, we give new bounds for sums $\sum_{Q/2\lt q\leqslant Q}\sum _{|\kappa_{j}|\leqslant K}|\sum_{N/2 \lt n \leqslant N}b_{n} \rho j(Dn)|^2$, where ($b_n$) is a complex sequence, and $j$ indexes the elements, $u_{j}(z)$, of a suitable orthonormal basis of the space spanned by the Maass cusp forms for the Hecke congruence subgroup $\Gamma_{0}(q)$, while $\rho_{j}(n)$ is the $n$-th Fourier coefficient at the cusp $\infty$ for $u_{j}(z)$, and $D$ is a large positive integer. Our bounds are strongest in cases where every prime factor of $D$ is a small power of $D$.

One application (briefly discussed in the paper) is a new mean-square bound for the modulus of a certain multiple sum involving Dirichlet characters modulo $D$. It is hoped this will be useful in the study of Carmichael numbers.

#### Article information

**Source**

Funct. Approx. Comment. Math., Volume 34 (2005), 27-146.

**Dates**

First available in Project Euclid: 29 September 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.facm/1538186585

**Digital Object Identifier**

doi:10.7169/facm/1538186585

**Mathematical Reviews number (MathSciNet)**

MR2269662

**Zentralblatt MATH identifier**

1247.11067

**Subjects**

Primary: 11F30: Fourier coefficients of automorphic forms

Secondary: 11A25: Arithmetic functions; related numbers; inversion formulas 11F11: Holomorphic modular forms of integral weight 11F25: Hecke-Petersson operators, differential operators (one variable) 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F72: Spectral theory; Selberg trace formula 11L05: Gauss and Kloosterman sums; generalizations 11L40: Estimates on character sums 11M06: $\zeta (s)$ and $L(s, \chi)$ 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$

**Keywords**

Maass cusp form Fourier coefficient Hecke operator eigenvalue mean value Kloosterman sum Dirichlet character

#### Citation

Watt, Nigel. Fourier coefficients of modular forms and eigenvalues of a Hecke operator. Funct. Approx. Comment. Math. 34 (2005), 27--146. doi:10.7169/facm/1538186585. https://projecteuclid.org/euclid.facm/1538186585