Bianchi newform (awaiting review)

A Bianchi newform $F$ is a Bianchi modular form lying in the new subspace $\mathcal{S}_k(\Gamma_0(\mathcal{N}))^{\text{new}}$ of the space $\mathcal{S}_k(\Gamma_0(\mathcal{N}))$ of Bianchi cusp forms of weight $k$ and level $\Gamma_0(\mathcal{N})$ which is an normalised eigenform for the Hecke algebra $\mathbb{T}$.

For each prime $\frak{p}$ of $K$ not dividing the level $\mathcal{N}$, the eigenvalue $a_{\frak{p}}$ of $T_{\frak{p}}$ on $F$ is an algebraic integer lying in the Hecke field of $F$. This is a totally real Galois extension of $\Q$ whose degree $d$ is the dimension of the irreducible component of the rational newspace $\mathcal{S}_k(\Gamma_0(\mathcal{N}))^{\text{new}}_{\Q}$ containing $F$. The integer $d$ is the dimension of the newform $F$. The Hecke eigenvalue $a_{\frak{p}}$ is also the coefficient of index $\frak{p}$ in the Fourier expansion of $F$.