Completeness of Bianchi Modular Form data (reviewed)

The database contains information about Bianchi modular forms over several imaginary quadratic fields including all nine fields of class number $1$, for a range of levels.

Newform data

Bianchi Newform data is available for levels of the form $\Gamma_0(\mathfrak{n})\le \GL_2(\mathcal{O}_K)$ and not for the larger spaces forms of level $\Gamma_0(\mathfrak{n})\cap \SL_2(\mathcal{O}_K)$.

For the five Euclidean imaginary quadratic fields we have a complete set of Bianchi newforms of dimension 1 (that is, with rational coefficients) for levels of norm up to a bound that depends on the field. The current bounds are 150000, 100000, 50000, 50000, 50000 for the imaginary quadratic fields of discriminant -3, -4, -7, -8, -11, respectively.

Over $\mathbb{Q}(\sqrt{-1})$ only, for levels of norm up to 5000, we also have all newforms of dimension 2.

For each of the newforms of dimension 1, the database contains all Atkin-Lehner eigenvalues and Hecke eigenvalues $a_{\mathfrak{p}}$ for primes $\mathfrak{p}$ of norm less than $50$. For the newforms of dimension 2 it contains the first fifteen $a_{\mathfrak{p}}$, but no Atkin-Lehner eigenvalues.

Dimension data

Dimension data for full cuspidal and new spaces for a range of weights and $\SL_2$-levels over the imaginary quadratic fields of discriminant -8, -11, -19, -43, -67, -163 are included. Dimension data for the three remaining imaginary quadratic fields of class number one (discriminants -3, -4, -7) is in preparation.

Over the five Euclidean imaginary quadratic fields we have the dimensions of the full cuspidal space and the new subspace at each $\GL_2$-level, for weight 2 forms of level norm up to a bound depending on the field (listed above).

Over all nine class number one fields and also over $\mathbb{Q}(\sqrt{-5})$, we have the cuspidal and new dimensions for a range of $\SL_2$-levels, and for a range of weights.