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/s406870180155z] 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 JonesRoberts 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 ruling 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].
 Review status: beta
 Last edited by Andrew Sutherland on 20190422 20:55:40