Source of Classical Modular Forms Data (reviewed)

Details regarding the construction of the database of classical modular forms can be found in the paper Computing Classical Modular Forms [arXiv:2002.04717] by Alex J. Best, Jonathan Bober, Andrew R. Booker, Edgar Costa, John Cremona, Maarten Derickx, David Lowry-Duda, Min Lee, David Roe, Andrew V. Sutherland, and John Voight. In particular, the classical modular forms data in the LMFDB comes from three sources:

• For newforms of weight $k > 1$ and level $N\le 1000$ satisying $Nk^2\le 4000$ complex embedding data was computed by Jonathan Bober using software available at [github.com/jwbober/mflib] based on the trace formula described in [MR:1111555]. The software library [arb] was used to rigorously control precision. This data was also used to compute trace forms.

• For newforms of weight $k > 1$ and dimension $d\le 20$ exact algebraic eigenvalue data was computed using the modular symbols package implemented in [Magma]. For newforms of weight $k>1$ and dimension $d>20$ with $Nk^2>4000$ Magma was also used to compute complex embedding data.

• For newforms of weight $k=1$ the modular forms package [10.1007/s40687-018-0155-z] in [Pari/GP] was used to compute exact algebraic eigenvalue data.

For newforms of weight $k=1$ the projective fields cut out by the kernel of the projective Galois representation associated to the newform come from four sources:

• In cases where the associated Artin representation was already present in the LMFDB, the projective field was computed directly from the Artin field.

• In cases where the projective image is one of $D_2$, $D_3$, $D_4$, $A_4$, $S_4$, $D_5$, $A_5$ , the projective fields were obtained via an exhaustive enumeration of all number fields with compatible Galois group and ramification data by determining a unique candidate with compatible Frobenius elements. The enumeration of these fields used the LMFDB database of number fields together with additional information provided by the Jones-Roberts database [NFDB], along with a list of quartic fields enumerated using the algorithms in [MR:MR1954977], and a list of A5 number fields enumerated by John Jones using a targeted Hunter search [MR:MR1726089, 10.1007/BFb0054880].

• In case where the projective image type is $D_n$ with $n>5$ and the distinguished quadratic subfield $K$ is real, the projective image was computed using class field theoretic techniques to enumerate all cyclic $n$-extensions of $K$ with compatible conductors and determining a unique candidate with compatible Frobenius elements.

• In all remaining cases the projective fields were computed as fixed fields of uniquely determined subgroups of class groups of imaginary quadratic orders using the CRT method to directly construct a defining polynomial for the quotient field via class invariants as described in [MR:MR2970725, 10.1112/S1461157012001015].