## Hilbert Modular Forms and Modular Elliptic Curves

Information: In these tables, we tabulate some Hilbert modular forms for a number of totally real fields up to degree 6. For information on how this data was computed, see the paper A database of Hilbert modular forms (with Steve Donnelly).

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• COEFFS is a minimal polynomial for F with the convention:

[a[1],a[2],...,a[n+1]] corresponds to a[n+1]*x^n + ... + a[2]*x + a[1];

• n is the degree [F:Q];
• d is the discriminant of F;
• PRIMES is the list of primes (up to at least norm 1000), with the convention:

[N, n, alpha] corresponds to the ideal frakN of norm N generated by n and alpha, where n is the smallest positive integer in frakN;

• NEWFORMS is the list of newforms f, specified by the data
N, label, eigenvalues    or    N, label, g, eigenvalues
where:
• N is the level of f;
• label is an (arbitrary) label ("a", "b", ..., "z", "aa", ...);
• g is the minimal polynomial for the field H_f of Hecke eigenvalues, generated by e, absent if H_f = Q;
• eigenvalues is a list of eigenvalues with entries in H_f, each specified by a polynomial in e.

Only one representative of a level up to automorphisms of F appears. In particular, if F is Galois, then only one level up to the action of Galois is listed. The conjugate levels, by a theorem of Shimura, are obtained by applying the permutation of the list of primes induced by an automorphism to the list of eigenvalues.

The tables are cumulative and complete up to a given level norm, so if no entry appears in a given level then it has dimension zero. There is also a maximal level norm computed for each field.

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