Modular forms with non-vanishing central values and linear independence of Fourier coefficients

In this article, we are interested in modular forms with non-vanishing central critical values and linear independence of Fourier coefficients of modular forms. The main ingredient is a generalization of a theorem due to VanderKam to modular symbols of higher weights. We prove that for sufficiently large primes p, Hecke operators \(T_1, T_2, \ldots , T_D\) act linearly independently on the winding elements inside the space of weight 2k cuspidal modular symbol \(\mathbb {S}_{2k}(\Gamma _0(p))\) with \(k\ge 1\) for \(D^2\ll p\). This gives a bound on the number of newforms with non-vanishing arithmetic L-functions at their central critical points and linear independence on the reductions of these modular forms for prime modulo \(l\not =p\).