Hilbert modular forms of half-integral weight

The files below are the result of using the main result from

Nicolás Sirolli and Gonzalo Tornaría, Effective construction of Hilbert modular forms of half-integral weight,

for computing the central values twisted L-series attached to Hilbert modular forms. They extend the data available here, obtained using results from our previous articles.

Each of the files corresponds to a Hilbert modular form g, and is named after the label of g in the LMFDB database.

The files contain a header with commented information, including the quaternion algebra and the order R used in each case, along with the following lines:

The first line contains a list with information about the quaternionic modular form φ in correspondence with g.
The elements of this list are tuples (c_I,L_I) corresponding to representatives I for the right ideal classes for R.
Here L_I is a basis over the rational integers for the left order of I and c_I is the I-th coefficient of φ, divided by the order of the torsion group of L_I.
The second line contains a list of tuples (p,pv), where p ranges over the prime divisors of the level of g, and pv is the auxiliary vector used for defining the weight function at p.
The prime p is described by giving a basis for it over the rational integers.
The subsequent lines correspond to the different admissible functions gamma; they are preceeded by a commented line describing the function in question.
Each line contains a list [l,lvec,cvs].
Here l is the auxiliary parameter, and lvec the auxilary vector for computing the weight function at l.
Finally, cvs is the list containing central values and Fourier coefficients, which we describe with detail below.

The main object of these computations are the lists cvs.

Each of these lists is made out of tuples (N,cvsN), which appear in the list sorted increasingly according to the absolute value of N.
Here cvsN is a list of tuples which have entries (D,λ(D),L_D).
These entries include every integral D of type γ such that (D,O) is a fundamental discriminant having norm equal to N, and such that -lD belongs to a certain Shintani cone which serves as fundamental domain for the action of the group (O^x)² acting on F⁺. Here the absolute value of N goes up to the precision N_max indicated in the header.

Finally,

λ(D) = λ(D,O;f)

L_D = L(1/2,g⊗χ^D)

are respectively the (D,O)-th Fourier coefficient of the theta series f corresponding to g and the central value of the L-series twisted by χ^D. They satisfy the central values formula

L(1/2,g⊗χ^D) = 2^ω(D,N) · <g,g> · c_g,γ / |D|^1/2 · |λ(D,O;f)|² / <f,f>.

See Theorem A from our article for notation and details.

Citations

Please reference this data as

Nicolás Sirolli and Gonzalo Tornaría, Hilbert modular forms of half-integral weight, Computational Number Theory, 2021. http://www.cmat.edu.uy/cnt/