Vector Valued Modular Forms of Dimension $3$

Abstract

In this thesis we describe and relate various representations of $3$-dimensional vector valued modular forms. In particular, we give algebraic formulas for families of $3$-dimensional vector valued modular forms on $\Gamma_0(2)$, a subgroup of the modular group $\Gamma = SL_2(\mathbb{Z})$. These formulas enable us to compute CM values of the $3$-dimensional vector valued modular forms at CM points in the upper half plane.

We also define families of Eisenstein series corresponding to one-dimensional representation, $\chi$, on $\Gamma_0(2)$. This gives a different description of the algebraic family discussed in the preceding paragraph. For Eisenstein series of weight $4$ and $6$, we evaluate their Fourier series expansion and compute their Fourier coefficients. The constant term in the Fourier series expansion of Eisenstein series of weight $4$ and $6$ is then expressed using Bessel function of the first kind and Kloosterman sums.