Hecke algebra (reviewed)

Let $K$ be an imaginary quadratic field and $\mathcal{M}_k(\Gamma)$ the space of Bianchi modular forms of weight $k$ and level $\Gamma=\Gamma_0(\mathcal{N})$ for some integral ideal $\mathcal{N}$ of $\mathcal{O}_K$.

The (level $\mathcal{N}$) Hecke algebra $\mathbb{T}$ is a commutative algebra of linear Hecke operators acting on $\mathcal{M}_k(\Gamma)$. It preserves the cuspidal subspace $\mathcal{S}_k(\mathcal{N})$ of Bianchi cusp forms and the new subspace $\mathcal{S}_k(\mathcal{N})^{\text{new}}$, and also preserves the rational and integral structures on these.

The Hecke algebra is generated by operators $T_\frak{p}$ for primes $\frak{p}$ of $K$. Each $T_\frak{p}$ is self-adjoint on $\mathcal{S}_k(\mathcal{N})$ and hence has eigenvalues which are totally real algebraic integers. The new space $\mathcal{S}_k(\mathcal{N})^{\text{new}}$ has a basis consisting of forms which are eigenforms for all Hecke operators.