Modular Galois representations data
On this page, you can download data and tables describing the Galois representations modulo ℓ attached to eigenforms of level 1 and weight k<ℓ which have residual degree 1 and whose image contains SL2(Fℓ), for ℓ up to 31, ℓ≠17, as well as to a few forms which admit a companion form mod ℓ and which are specified by their LMFDB label.
These data were computed by the algorithm described in my article Computing modular Galois representations, and certified by the method described in my article Certification of Galois representations.
For each Galois representation, the data available for download below here are formed of
- an irreducible polynomial f(x) in Z[x],
- an ordered list of the roots of f(x) in the field Fp[x]/f(x), where p is a prime such that f(x) is irreducible mod p,
- and of a list of resolvents which can be used to compute the image of Frobenius elements by the representation, along with the polynomial h(x)∈Z[x] used to compute them (cf. my articles for details).
The splitting field of f(x) is the smallest number field through which the representation modulo S factors, where S is the largest subgroup of Fℓ* which does not contain -1. The roots of f(x) correspond via the Galois representation to the points of (Fℓ2 -{0})/S, and are ordered as follows:
(0,1), (0,2), ..., (0,ℓ-1), (1,0), ε(0,1), ε(0,2), ..., ε(0,ℓ-1), ε(1,0), ε2(0,1), ε2(0,2), ..., ε2(0,ℓ-1), ε2(1,0),...
where ε is the smallest positive integer which generates the multiplicative group of Fℓ.
The resolvents each come with a tag which indicates the conjugacy class of GL2(Fℓ)/S it corresponds to:
| Tag | Conjugacy class |
|---|---|
| "Scal",a | Scalar matrix with eigenvalue a |
| "Diag",[a,b] | Split semisimple matrices with eigenvalues a and b |
| "Irr",[t,d] | Nonsplit semisimple matrices with trace t and determinant d |
| "Nil",a | Nonsemisimple matrices with eigenvalue a. |
The format the data is given in is:
[f(x),[root1,root2,...],h(x),[[resolvent1,tag1],[resolvent2,tag2],...]].
On the third- and second-to-last lines, f24=q+24(22+α)q2+O(q3) is the eigenform of level 1 and weight 24, where α2-α-36042=0.
Here is a PDF file containing tables giving the image by the above representations of the Frobenius elements at the first 40 primes above 101000.
Finally, here is a polynomial and an orderng of its roots describing the representation attached to a companion form 3.22.1.b mod 41 mentioned in this article. These data, unlike the above, are not certified at present.