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 \(T_1, T_2, \ldots , T_D\) $T_1,T_2,\dots ,T_D$ act linearly independently on the winding elements inside the space of weight 2k cuspidal modular symbol \(\mathbb {S}_{2k}(\Gamma _0(p))\) ${\mathbb{S}}_{2k}({\mathrm{\Gamma }}_0(p))$ with \(k\ge 1\) $k\ge 1$ for \(D^2\ll p\) $D^2\ll p$ . 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 \(l\not =p\) $l\ne p$ .