Data related to computing quinary orthogonal modular forms
This data was generated by the code in the repositories https://github.com/assaferan/omf5 and https://gitlab.fing.edu.uy/grama/quinary.
The data consists of the following
- Single rows of Hecke matrices
- Hecke matrices
- Hecke eigensystems
In folder hecke_rows, there are files named hecke_row_{N}_{p}.dat, for N < 1000 and p < 100 a prime not dividing N.
Each file contains a dictionary, whose keys are the possible conductors d for the spinor norm twist, corresponding to the different Atkin-Lehner spaces.
The value at d is the first row of the matrix representing the Hecke Operator T(p,1) on the space of paramodular forms of weight (3,0), level N and coresponding Atkin-Lehner signs.
In folder hecke_mats, there are files named hecke_mat_{N}_{p}.dat, for N < 1000 and p < 10 a prime not dividing N.
Each file contains a dictionary, whose keys are the possible conductors d for the spinor norm twist, corresponding to the different Atkin-Lehner spaces.
The value at d is the matrix representing the Hecke Operator T(p,1) on the space of paramodular forms of weight (3,0), level N and coresponding Atkin-Lehner signs.
Folder hecke_evs_3_0 contains three subfolders - data, data_nl_200 and logs. The subfolder data contains files named hecke_ev_3_0_{N}.dat. Each file contains an array of dictionaries, each dictionary corresponds to a modular form in the space of orthogonal modular forms of trivial weight and level N, corresponding to Siegel modular forms. The fields for each modular form are as follows.
- 'aut_rep_type' : 'F', 'P', 'Y', 'G' or 'O', the automorphic type of the associated automorphic representation (F/P/Y/G) or 'O' if this is an oldform. (appears at lower level)
- 'field_poly' : The coefficients of the defining polynomial of the Hecke field (the field over which the Hecke eigenvalues are defined), given as [a_0,...,a_n] where f(x) = a_0 + ... + a_n x^n.
- 'hecke_ring_power_basis' : True if the eigenvalues are represented by the coordinates in the power basis [1, a, a^2, ..., a^{n-1}] where a is a root of the field polynomial f(x).
- 'hecke_ring_numerators' : If hecke_ring_power_basis if False, an array of arrays, each representing an element in the field in the above power basis, corresponding to numerators of generators for the Hecke ring.
- 'hecke_ring_denominators': If hecke_ring_power_basis if False, an array representing an element in the field in the above power basis, corresponding to a denominator of generators for the Hecke ring. That is, if a_1,...,a_n are the numerators and b is the denominator, then a_1/b,...,a_n/b are generators for the Hecke ring.
- 'hecke_ring_inverse_numerators': Same as above, but for the inverse generators (writing the power basis as a linear combination of the generators).
- 'hecke_ring_inverse_denominators': Same as above, but for the inverse generators (writing the power basis as a linear combination of the generators).
- 'lambda_p' : an array of arrays, representing the eigenvalues of the form with respect to the Hecke operator T(p,1) in the supplied basis of the Hecke ring for prime p < 100 not dividing N.
- 'lambda_p_square' : an array of arrays, representing the eigenvalues of the form with respect to the Hecke operator T(p,2) in the supplied basis of the Hecke ring for prime p < 100 not dividing N.
- 'trace_lambda_p' : an array of the traces of the elements of 'lambda_p'.
- 'trace_lambda_p_square' : an array of the traces of the elements of 'lambda_p_square'.
- 'atkin_lehner_eigenvals' : an array of pairs, each consists of a divisor d of N, and the eigenvalue of the form under the Atkin-Lehner operator W_d.
- 'atkin_lehner_string' : a string of '+' or '-', corresponding to the Atkin-Lehner eigenvalues at the different primes in ascending order.
- 'hecke_ring_index' : The index of the Hecke ring inside the ring of integer of the number field.
- 'hecke_ring_generator_nbound' : The maximal index in the array of eigenvalues one needs in order that they generate the entire Hecke ring.
The subfolder data_nl_200 contains files named hecke_ev_3_0_nl_200_{N}.dat. Each file contains an array of modular forms, as above, but only for the newforms which are nonlifts (automorphic type is 'G'), and the array of eigenvalues is for primes p < 200.
The subfolder logs contains files named hecke_ev_3_0_{N}.log. Each file contains the log data of the run that generated the corresponding data file.
In the main folder there are several more files.
The file qf5db.sage is a sage script. When loaded it creates the variable qf5, which is an array of arrays of pairs of matrices. qf5[N] represents a set of genus representatives for the chosen genus of discriminant N, each pair of matrices consists of a gram matrix of the lattice, and an isometry from the first lattice to this one. All the data was generate with respect to this basis for the space of orthognoal modular forms, and using these isometries for the spinor norm.
The file qf5.db is an array of arrays, containing only the gram matrices of the genus representatives in an abbreviated format (lower triangular part).
The file hecke_3_0.dat contains an array of dictionaries, indexed by the level N, whose Nth entry is a dictionary of the matrices of the Hecke operators T(p,1) where p is the smallest prime not dividing N for each of the Atkin-Lehner spaces (keys are the conductors of the spinor norm). The matrices are with respect to the basis described by qf5db.sage, and were generated by the PARI/GP code in https://gitlab.fing.edu.uy/grama/quinary.
The file hecke_all.dat contains an array of dictionaries, indexed by the level N, whose Nth entry is a dictionary of the matrices of the Hecke operators T(p,1) where p is the smallest prime not dividing N for each of the Atkin-Lehner spaces (keys are the conductors of the spinor norm). The matrices are with respect to the basis described by qf5db.sage, and were generated by the PARI/GP code in https://github.com/assaferan/omf5.
The file verify.sage verifies that the data in both files agrees by checking that the characteristic polynomials of the matrices are the same. (Therefore would work even if a different basis is being used).
The file discrepancies.sage specifies the matrices which are not identical, due to a different choice of basis. This occurs in the spaces with non-trivial conductor, as there is a choice to make in the order of the basis elements.
The file newforms30_dims.txt contains an array, indexed by the level, such that at index N one has an array containing the dimensions of the irreducible Hecke modules corresponding to newforms which are nonlifts (i.e. the degrees of the number fields over which their Hecke eigenvalues are defined).