Spaces of Bianchi modular forms (awaiting review)

The Bianchi modular forms of fixed weight $k$ and level $\Gamma$ form a finite-dimensional vector space $\mathcal{M}_k(\Gamma)$ over the complex numbers $\C$. The forms $F$ which are cuspidal comprise the cuspidal subspace $\mathcal{S}_k(\Gamma)$ of $\mathcal{M}_k(\Gamma)$. The algebra $\mathbb{T}$ of Hecke operators acts on the space $\mathcal{M}_k(\Gamma)$, and preserves the cuspidal subspace $\mathcal{S}_k(\Gamma)$.

$\mathcal{S}_k(\Gamma)$ is an inner product spaces with respect to an inner product analogous to the classical Peterssen inner product on spaces of elliptic modular forms over $\Q$. For level $\Gamma_0(\mathcal{N})$ the Hecke operators are all self-adjoint, so have totally real eigenvalues.

The newspace $\mathcal{S}_k(\mathcal{N})^{\text{new}}$ at level $\mathcal{N}$ and weight $k$ is the orthogonal complement in $\mathcal{S}_k(\mathcal{N})$, with respect to this inner product, of all oldforms: these are forms coming from lower levels $\mathcal{M}\mid\mathcal{N}$ with $N(\mathcal{M})<N(\mathcal{N})$. The Hecke algebra preserves the newspace and acts semisimply on it, so that the newspace has a basis of simultaneous eigenforms for the Hecke algebra. A newform is an eigenform normalised so that the coefficient of $(1)$ in its Fourier expansion is $1$.

The complex vector space $\mathcal{S}_k(\mathcal{N})$ has a rational structure: the subset $\mathcal{S}_k(\mathcal{N})_{\Q}$ of forms with rational Fourier coefficients is a vector space over $\Q$ such that $\mathcal{S}_k(\mathcal{N}) = \mathcal{S}_k(\mathcal{N})_{\Q}\otimes\C$. There is also an integral structure, and the Hecke algebra $\mathbb{T}$ preserves both, so that the Hecke eigenvalues are all algebraic integers.

The irreducible components of $\mathcal{S}_k(\mathcal{N})_{\Q}$ under the action of $\mathbb{T}$ have dimension $d\ge1$, with a basis consisting of $d$ conjugate newforms whose Fourier coefficients are Galois conjugate elements of the associated Hecke field, a Galois extension of $\Q$ of degree $d$. The dimension of a newform is therefore the dimension of the $\Q$-irreducible component it lies in, and also the degree of its Hecke field.