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For $q = e^{2\pi iz}$, let $f(z) = \sum_{n = 1}^{\infty} a(n)q^n\in S_k(N,\chi)$ be a cusp form of integral weight $k \geq 2$ and level $N$ with Dirichlet character $\chi\pmod N$. Let $\alpha_p,\beta_ …

Let $K$ a totaly real number field, $\mathcal{O}_K$ its ring of integre and $h$ the narrow class number of $K$. Let $\mathbf{f}$ a collection $(f_1, ..., f_h)$ of Hilbert cusp forms $f_\lambda$ $(\la …

Let $\Gamma$ be a discrete subgroup of $SL_2(\mathbb{R})$ which has a cusp at $\infty.$ suppose that $\mu(\Gamma\setminus\mathbb{H})<\infty,$ consider the Eisenstein series :$$E(z,s,\Gamma)=\sum_{\gam …

Let $f(z)=\sum_{n\ge 1}a(n)n^{(k-1/2)/2}e(nz)\in S_{k+1/2}(\Gamma_0(4N),\chi)$ be a cuspidal Hecke eigenform. Let $$M(s)=\sum_{p\ge 2, \text{prime}}\frac{|a(p)|^2}{p^s}$$ and $$R_f(s)=\sum_{n\ge 1}\f …

I am trying to understand some open steps in the following article The Sign of Fourier coefficients of Half-integral Weight Cusp Form by Hulse, Kiral, Kuan, and Lim, I find the following : Let $f\i …