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 $T_1,T_2,\dots ,T_D$ act linearly independently on the winding elements inside the space of weight $2k$ cuspidal modular symbol ${\mathbb{S}}_{2k}({\mathrm{\Gamma }}_0(p))$ 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 $l≠p$.