For an odd and squarefree level N, Kohnen proved that there is a canonically defined subspace [...] S κ + 1 2 n e w ( N ) ⊂ S κ + 1 2 ( N ) , and S κ + 1 2 n e w ( N ) and S 2 k n e w ( N ) $S_{\kappa +\frac{1}{2}}^{\mathrm{n}\mathrm{e}\mathrm{w}}\left(N\right)\subset S_{\kappa +\frac{1}{2}}\left(N\right),\text{and}{S}_{\kappa +\frac{1}{2}}^{\mathrm{n}\mathrm{e}\mathrm{w}}\left(N\right)\text{and}{S}_{2k}^{\mathrm{n}\mathrm{e}\mathrm{w}}\left(N\right)$ are isomorphic as modules over the Hecke algebra. Later he gave a formula for the product [...] a g ( m ) a g ( n ) ¯ $a_g\left(m\right)\overline{a_g\left(n\right)}$ of two arbitrary Fourier coefficients of a Hecke eigenform g of halfintegral weight and of level 4N in terms of certain cycle integrals of the corresponding form f of integral weight. To this...

Let $f={\sum }_{n=1}^{\infty }a\left(n\right)q^n\in S_{k+1/2}(N,{\chi }_0)$ be a nonzero cuspidal Hecke eigenform of weight $k+\frac{1}{2}$ and the trivial nebentypus ${\chi }_0$, where the Fourier coefficients $a\left(n\right)$ are real. Bruinier and Kohnen conjectured that the signs of $a\left(n\right)$ are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies ${\left\{a\left(t{n}^2\right)\right\}}_n$, where $t$ is a squarefree integer such that $a\left(t\right)\ne 0$. Let $q$ and $d$ be natural numbers such that $(d,q)=1$. In this work, we show that ${\left\{a\left(t{n}^2\right)\right\}}_n$ is equidistributed over any arithmetic progression $n\equiv dmodq$.

1. Introduction. Let Q be a positive definite n × n matrix with integral entries and even diagonal entries. It is well known that the theta function ${\vartheta }_Q\left(z\right):={\sum }_{g\in {\mathbb{Z}}^n}exp\pi i{}^{t}gQgz$, Im z > 0, is a modular form of weight n/2 on ${\Gamma }_0\left(N\right)$, where N is the level of Q, i.e. $N{Q}^{-1}$ is integral and $N{Q}^{-1}$ has even diagonal entries. This was proved by Schoeneberg [5] for even n and by Pfetzer [3] for odd n. Shimura [6] uses the Poisson summation formula to generalize their results for arbitrary n and he also computes the theta multiplier explicitly....

Higher-order non-holomorphic Eisenstein series associated to a Fuchsian group $\Gamma$ are defined by twisting the series expansion for classical non-holomorphic Eisenstein series by powers of modular symbols. Their functional identities include multiplicative and additive factors, making them distinct from classical Eisenstein series. In this article we prove the meromorphic continuation of these series and establish their functional equations which relate values at $s$ and $1-s$. In addition, we construct...