On this page, you can download data and tables describing all of 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(F) for ℓ up to 31, as well as a few others, including three representations attached to forms which admit a companion form mod ℓ. These forms are specified by their LMFDB label.

These data were computed by the algorithm described in my article Computing modular Galois representations, and (when ℓ≤31) certified by the method described in my article Certification of Galois representations. The three representations attached to companion forms were studied in my article Companion forms and explicit computation of PGL2 number fields with very little ramification.

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 F* which does not contain -1. The roots of f(x) correspond via the Galois representation to the points of (F2 -{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 F.

The resolvents each come with a tag which indicates the conjugacy class of GL2(F)/S it corresponds to:

Tag Conjugacy class
"Scal",a Scalar matrix with double 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 double eigenvalue a.

The data are provided in the following format:

[f(x), [root1,root2,...], h(x), [[resolvent1,tag1],[resolvent2,tag2],...]].

They have been rigorously certified unless explicit mention of the contrary.

• ℓ=11
• ℓ=13
• ℓ=17
• Δ (uncertified)
• E4Δ (uncertified)
• ℓ=19
• ℓ=23
• ℓ=29
• ℓ=31
• ℓ=37
• E8Δ (uncertified)
• ℓ=41
• ℓ=43
Finally, 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.