A short note on inadmissible coefficients of weight 2 and 2k+1 newforms

Malik Amir,Andreas Hatziiliou

doi:10.1007/s40316-021-00168-4

Abstract

Let f(z)=q+sum _{nge 2}a(n)q^n be a weight k normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in Amir and Hong (On L-functions of modular elliptic curves and certain K3 surfaces, Ramanujan J, 2021) for k=2 by ruling out or locating all odd prime values |ell |<100 of their Fourier coefficients a(n) when n satisfies some congruences. We also study the case of odd weights kge 1 newforms where the nebentypus is given by a quadratic Dirichlet character.

Highlights

Introduction and statement of the resultsIn an article entitled “On certain arithmetical functions” [17], Ramanujan introduced the τ -function in 1916, known as the Fourier coefficients of the weight 12 modular form ∞(z) = q (1 − qn)24:= τ (n)qn = q − 24q2 + 252q3 − 1472q4 + 4830q5 − ... n=1 n=1M
We extend slightly the results of [2] on inadmissible coefficients for Lfunctions of modular elliptic curves and give a procedure to rule out odd prime values, positive or negative, as coefficients of any normalized newform of odd weight k ≥ 1 with integer coefficients having trivial residual mod 2 Galois representation and a quadratic Dirichlet character
Theorem 1.1 Suppose f (z) = q+ n≥2 a(n)qn ∈ S2new( 0(N ))∩Z[[q]] has trivial residual mod 2 Galois representation, namely, E/Q is an elliptic curve of conductor N with a rational 2-torsion point

Summary

Introduction and statement of the results

In an article entitled “On certain arithmetical functions” [17], Ramanujan introduced the τ -function in 1916, known as the Fourier coefficients of the weight 12 modular form. One may be interested in studying odd values taken by τ (n) or, more generally, the coefficients of any newform. This is the question we consider as a variation of Lehmer’s original speculation. We extend slightly the results of [2] on inadmissible coefficients for Lfunctions of modular elliptic curves and give a procedure to rule out odd prime values , positive or negative, as coefficients of any normalized newform of odd weight k ≥ 1 with integer coefficients having trivial residual mod 2 Galois representation and a quadratic Dirichlet character. Theorem 1.4 Let E/Q be an elliptic curve with conductor N and f the corresponding newform with Fourier coefficients a(n).

Lucas sequences and their primitive prime divisors

Modular forms and their Galois representation

Integer points on Thue equations

Some congruences