/* This code can be loaded, or copied and pasted, into Magma. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. At the *bottom* of the file, there is code to recreate the Hilbert modular form in Magma, by creating the HMF space and cutting out the corresponding Hecke irreducible subspace. From there, you can ask for more eigenvalues or modify as desired. It is commented out, as this computation may be lengthy. */ P := PolynomialRing(Rationals()); g := P![-1, -2, 4, 5, -4, -2, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [13, 13, -w^5 + 3*w^4 + 2*w^3 - 9*w^2 + w + 4], [13, 13, -w^2 + w + 2], [27, 3, 2*w^5 - 4*w^4 - 7*w^3 + 9*w^2 + 4*w - 2], [27, 3, -2*w^5 + 5*w^4 + 5*w^3 - 12*w^2 - w + 5], [29, 29, w^3 - 2*w^2 - 2*w + 3], [29, 29, 2*w^5 - 4*w^4 - 7*w^3 + 8*w^2 + 4*w - 2], [43, 43, -w^5 + 3*w^4 + w^3 - 6*w^2 + 3*w + 1], [43, 43, -w^4 + w^3 + 5*w^2 - 4], [49, 7, w^5 - 4*w^4 + 11*w^2 - 3*w - 4], [64, 2, -2], [71, 71, 2*w^5 - 6*w^4 - 4*w^3 + 17*w^2 - w - 6], [71, 71, 2*w^4 - 4*w^3 - 6*w^2 + 7*w + 2], [71, 71, -w^5 + 3*w^4 + 2*w^3 - 7*w^2 - w], [71, 71, -2*w^5 + 5*w^4 + 6*w^3 - 14*w^2 - 3*w + 5], [83, 83, -3*w^5 + 7*w^4 + 9*w^3 - 17*w^2 - 5*w + 5], [83, 83, 3*w^5 - 6*w^4 - 10*w^3 + 12*w^2 + 5*w - 2], [83, 83, -2*w^5 + 5*w^4 + 5*w^3 - 11*w^2 - 3*w + 3], [83, 83, 3*w^5 - 7*w^4 - 8*w^3 + 15*w^2 + 2*w - 4], [97, 97, -3*w^5 + 6*w^4 + 10*w^3 - 12*w^2 - 5*w + 3], [97, 97, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 5*w - 3], [97, 97, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 5*w - 1], [97, 97, -3*w^5 + 7*w^4 + 8*w^3 - 15*w^2 - 2*w + 3], [113, 113, -3*w^5 + 6*w^4 + 11*w^3 - 14*w^2 - 7*w + 3], [113, 113, 2*w^5 - 5*w^4 - 5*w^3 + 12*w^2 - 6], [113, 113, -w^5 + 2*w^4 + 3*w^3 - 4*w^2 - 1], [113, 113, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 2*w + 5], [125, 5, 3*w^5 - 6*w^4 - 10*w^3 + 11*w^2 + 6*w - 1], [125, 5, -2*w^5 + 4*w^4 + 6*w^3 - 6*w^2 - 3*w - 1], [127, 127, -2*w^5 + 6*w^4 + 3*w^3 - 15*w^2 + 2*w + 5], [127, 127, 2*w^5 - 3*w^4 - 9*w^3 + 6*w^2 + 7*w - 2], [127, 127, -3*w^5 + 7*w^4 + 9*w^3 - 17*w^2 - 4*w + 4], [127, 127, -w^5 + 2*w^4 + 3*w^3 - 4*w^2 - w + 3], [139, 139, 3*w^5 - 7*w^4 - 9*w^3 + 16*w^2 + 6*w - 4], [139, 139, -w^5 + w^4 + 5*w^3 - 6*w - 1], [167, 167, -2*w^5 + 5*w^4 + 6*w^3 - 12*w^2 - 5*w + 4], [167, 167, w^5 - 4*w^4 - w^3 + 13*w^2 - 2*w - 4], [169, 13, -w^5 + 4*w^4 - 11*w^2 + 3*w + 5], [169, 13, 2*w^5 - 5*w^4 - 6*w^3 + 13*w^2 + 3*w - 3], [181, 181, 2*w^5 - 4*w^4 - 6*w^3 + 6*w^2 + 2*w + 1], [197, 197, -w^4 + 3*w^3 + 2*w^2 - 6*w - 1], [197, 197, -3*w^5 + 6*w^4 + 11*w^3 - 14*w^2 - 6*w + 4], [211, 211, 2*w^5 - 5*w^4 - 6*w^3 + 12*w^2 + 6*w - 1], [211, 211, -2*w^5 + 4*w^4 + 8*w^3 - 9*w^2 - 8*w + 2], [211, 211, -w^5 + 2*w^4 + 3*w^3 - 3*w^2 - 3*w + 3], [211, 211, -2*w^4 + 5*w^3 + 5*w^2 - 9*w - 3], [223, 223, -w^5 + 8*w^3 + w^2 - 11*w + 1], [223, 223, 2*w^5 - 5*w^4 - 5*w^3 + 12*w^2 + w - 1], [239, 239, -2*w^5 + 5*w^4 + 6*w^3 - 13*w^2 - 2*w + 3], [239, 239, 2*w^5 - 5*w^4 - 4*w^3 + 9*w^2 + w], [239, 239, -3*w^5 + 7*w^4 + 10*w^3 - 18*w^2 - 7*w + 5], [239, 239, w^5 - w^4 - 6*w^3 + 2*w^2 + 5*w - 2], [251, 251, w^5 - w^4 - 6*w^3 + 3*w^2 + 6*w - 2], [251, 251, -w^5 + 4*w^4 - 12*w^2 + 3*w + 6], [281, 281, -5*w^5 + 11*w^4 + 17*w^3 - 27*w^2 - 10*w + 9], [281, 281, 2*w^5 - 5*w^4 - 5*w^3 + 10*w^2 + 3*w + 1], [293, 293, -2*w^5 + 4*w^4 + 7*w^3 - 7*w^2 - 7*w + 2], [293, 293, -w^4 + 2*w^3 + 2*w^2 - 3*w + 3], [293, 293, -2*w^5 + 7*w^4 + w^3 - 19*w^2 + 7*w + 9], [293, 293, -2*w^5 + 3*w^4 + 9*w^3 - 5*w^2 - 7*w + 2], [307, 307, 2*w^2 - 3*w - 4], [307, 307, -w^5 + 4*w^4 - 13*w^2 + 6*w + 6], [337, 337, 4*w^5 - 9*w^4 - 12*w^3 + 20*w^2 + 6*w - 5], [337, 337, -2*w^5 + 3*w^4 + 10*w^3 - 6*w^2 - 12*w + 2], [349, 349, 4*w^5 - 9*w^4 - 11*w^3 + 19*w^2 + 2*w - 4], [349, 349, 5*w^5 - 11*w^4 - 15*w^3 + 24*w^2 + 5*w - 7], [379, 379, -2*w^5 + 5*w^4 + 4*w^3 - 10*w^2 + w + 5], [379, 379, -2*w^5 + 5*w^4 + 6*w^3 - 12*w^2 - 5*w + 5], [379, 379, -4*w^5 + 8*w^4 + 14*w^3 - 17*w^2 - 10*w + 4], [379, 379, -5*w^5 + 12*w^4 + 13*w^3 - 27*w^2 - 2*w + 7], [421, 421, 2*w^5 - 6*w^4 - 4*w^3 + 15*w^2 + w - 3], [421, 421, 3*w^5 - 9*w^4 - 6*w^3 + 25*w^2 - w - 9], [433, 433, -5*w^5 + 10*w^4 + 18*w^3 - 21*w^2 - 13*w + 4], [433, 433, 5*w^5 - 12*w^4 - 15*w^3 + 30*w^2 + 7*w - 9], [433, 433, -3*w^5 + 8*w^4 + 8*w^3 - 22*w^2 - w + 8], [433, 433, 2*w^5 - 3*w^4 - 10*w^3 + 8*w^2 + 9*w - 3], [449, 449, 3*w^5 - 9*w^4 - 6*w^3 + 24*w^2 + w - 9], [449, 449, 3*w^5 - 6*w^4 - 11*w^3 + 15*w^2 + 6*w - 5], [449, 449, 2*w^5 - 3*w^4 - 8*w^3 + 3*w^2 + 7*w], [449, 449, 4*w^5 - 8*w^4 - 14*w^3 + 17*w^2 + 8*w - 2], [449, 449, -3*w^4 + 5*w^3 + 12*w^2 - 9*w - 6], [449, 449, -3*w^5 + 8*w^4 + 7*w^3 - 20*w^2 - 2*w + 6], [461, 461, 2*w^4 - 4*w^3 - 7*w^2 + 7*w + 2], [461, 461, -4*w^5 + 10*w^4 + 11*w^3 - 24*w^2 - 6*w + 8], [463, 463, 4*w^5 - 8*w^4 - 14*w^3 + 18*w^2 + 6*w - 5], [463, 463, w^5 - w^4 - 5*w^3 + w^2 + 4*w - 3], [491, 491, -3*w^5 + 7*w^4 + 8*w^3 - 15*w^2 + 5], [491, 491, -2*w^5 + 6*w^4 + 3*w^3 - 14*w^2 + 3], [547, 547, 2*w^5 - 3*w^4 - 10*w^3 + 8*w^2 + 10*w - 5], [547, 547, w^5 - 4*w^4 + 12*w^2 - 2*w - 5], [547, 547, -3*w^5 + 5*w^4 + 13*w^3 - 11*w^2 - 11*w + 3], [547, 547, -2*w^5 + 6*w^4 + 3*w^3 - 15*w^2 + w + 6], [547, 547, -2*w^5 + 6*w^4 + 4*w^3 - 15*w^2 - w + 4], [547, 547, 4*w^5 - 8*w^4 - 15*w^3 + 18*w^2 + 12*w - 6], [587, 587, -5*w^5 + 12*w^4 + 14*w^3 - 28*w^2 - 6*w + 7], [587, 587, 5*w^5 - 12*w^4 - 14*w^3 + 27*w^2 + 6*w - 6], [601, 601, -3*w^5 + 9*w^4 + 5*w^3 - 24*w^2 + 5*w + 10], [601, 601, -3*w^5 + 8*w^4 + 8*w^3 - 23*w^2 - w + 9], [601, 601, -w^5 + 8*w^3 + 2*w^2 - 11*w - 1], [601, 601, 2*w^5 - 3*w^4 - 10*w^3 + 8*w^2 + 10*w - 4], [617, 617, w^5 - w^4 - 7*w^3 + 4*w^2 + 10*w - 3], [617, 617, -5*w^5 + 11*w^4 + 16*w^3 - 26*w^2 - 7*w + 8], [617, 617, 3*w^5 - 7*w^4 - 9*w^3 + 15*w^2 + 7*w - 2], [617, 617, -2*w^5 + 2*w^4 + 11*w^3 - w^2 - 13*w - 2], [631, 631, 2*w^4 - 4*w^3 - 7*w^2 + 8*w + 2], [631, 631, w^5 - w^4 - 5*w^3 - w^2 + 4*w + 5], [631, 631, 4*w^5 - 9*w^4 - 11*w^3 + 19*w^2 + 2*w - 6], [631, 631, -3*w^5 + 7*w^4 + 9*w^3 - 15*w^2 - 7*w + 3], [643, 643, -2*w^5 + 3*w^4 + 9*w^3 - 6*w^2 - 8*w + 4], [643, 643, 3*w^5 - 8*w^4 - 7*w^3 + 20*w^2 + 2*w - 5], [659, 659, 3*w^5 - 9*w^4 - 5*w^3 + 23*w^2 - 3*w - 10], [659, 659, -2*w^4 + 3*w^3 + 7*w^2 - 3*w - 5], [673, 673, 3*w^5 - 6*w^4 - 11*w^3 + 14*w^2 + 8*w - 2], [673, 673, 3*w^5 - 8*w^4 - 6*w^3 + 19*w^2 - 3*w - 8], [673, 673, 3*w^5 - 6*w^4 - 10*w^3 + 13*w^2 + 3*w - 2], [673, 673, 2*w^5 - 3*w^4 - 10*w^3 + 6*w^2 + 11*w - 2], [701, 701, 2*w^5 - 5*w^4 - 6*w^3 + 13*w^2 + 6*w - 4], [701, 701, 2*w^5 - 5*w^4 - 5*w^3 + 11*w^2 + 2*w - 6], [701, 701, -w^5 + w^4 + 5*w^3 - 5*w - 5], [701, 701, -w^5 + w^4 + 6*w^3 - 2*w^2 - 9*w + 2], [727, 727, -w^5 + 5*w^4 - w^3 - 16*w^2 + 4*w + 8], [727, 727, w^5 - w^4 - 4*w^3 - 2*w^2 + 4*w + 4], [727, 727, -3*w^5 + 9*w^4 + 5*w^3 - 22*w^2 + 2*w + 7], [727, 727, -w^5 + 2*w^4 + 3*w^3 - 5*w^2 + 5], [727, 727, -5*w^5 + 11*w^4 + 16*w^3 - 24*w^2 - 10*w + 6], [727, 727, 4*w^5 - 8*w^4 - 14*w^3 + 17*w^2 + 7*w - 4], [743, 743, 5*w^5 - 11*w^4 - 16*w^3 + 26*w^2 + 6*w - 8], [743, 743, -4*w^5 + 7*w^4 + 16*w^3 - 14*w^2 - 12*w + 1], [743, 743, -4*w^5 + 9*w^4 + 14*w^3 - 23*w^2 - 10*w + 7], [743, 743, w^5 - 3*w^4 + 6*w^2 - 6*w - 2], [757, 757, -w^5 + 7*w^3 + 3*w^2 - 7*w - 6], [757, 757, 5*w^5 - 11*w^4 - 16*w^3 + 24*w^2 + 11*w - 5], [769, 769, -3*w^5 + 4*w^4 + 14*w^3 - 6*w^2 - 12*w + 1], [769, 769, 4*w^5 - 7*w^4 - 17*w^3 + 16*w^2 + 14*w - 5], [797, 797, -5*w^5 + 12*w^4 + 14*w^3 - 27*w^2 - 7*w + 7], [797, 797, 2*w^5 - 7*w^4 - w^3 + 17*w^2 - 4*w - 7], [811, 811, 5*w^5 - 13*w^4 - 13*w^3 + 33*w^2 + 3*w - 10], [811, 811, -4*w^5 + 10*w^4 + 9*w^3 - 22*w^2 + w + 6], [841, 29, 3*w^5 - 6*w^4 - 9*w^3 + 10*w^2 + 4*w - 1], [841, 29, -w^5 + w^4 + 4*w^3 + 3*w^2 - 3*w - 7], [853, 853, -4*w^5 + 9*w^4 + 12*w^3 - 21*w^2 - 4*w + 5], [853, 853, -3*w^5 + 6*w^4 + 10*w^3 - 13*w^2 - 4*w + 2], [853, 853, -w^5 + w^4 + 5*w^3 + w^2 - 4*w - 6], [853, 853, -3*w^5 + 7*w^4 + 8*w^3 - 16*w^2 - w + 7], [853, 853, -4*w^5 + 10*w^4 + 10*w^3 - 24*w^2 - w + 10], [853, 853, -w^4 + 3*w^3 + 3*w^2 - 8*w - 2], [883, 883, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - 4*w + 6], [883, 883, -w^5 + 5*w^4 - 3*w^3 - 12*w^2 + 9*w + 3], [883, 883, 5*w^5 - 12*w^4 - 14*w^3 + 27*w^2 + 7*w - 6], [883, 883, 2*w^5 - 6*w^4 - 3*w^3 + 16*w^2 - 3*w - 9], [937, 937, 5*w^5 - 11*w^4 - 16*w^3 + 26*w^2 + 8*w - 7], [937, 937, -w^5 + 2*w^4 + 4*w^3 - 3*w^2 - 6*w], [937, 937, 2*w^5 - 6*w^4 - 4*w^3 + 18*w^2 - 3*w - 10], [937, 937, 2*w^5 - 7*w^4 - 3*w^3 + 22*w^2 - 2*w - 10], [953, 953, 4*w^5 - 7*w^4 - 16*w^3 + 14*w^2 + 13*w - 3], [953, 953, 5*w^5 - 11*w^4 - 15*w^3 + 23*w^2 + 7*w - 3], [953, 953, w^5 + w^4 - 9*w^3 - 6*w^2 + 12*w + 3], [953, 953, -3*w^5 + 7*w^4 + 10*w^3 - 17*w^2 - 9*w + 5], [953, 953, w^5 - 4*w^4 + w^3 + 8*w^2 - 5*w + 1], [953, 953, -2*w^5 + 6*w^4 + 3*w^3 - 15*w^2 + 4*w + 7], [967, 967, 4*w^5 - 10*w^4 - 11*w^3 + 26*w^2 + w - 9], [967, 967, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 3*w - 8]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [4, -2, -1, 5, -6, 9, 7, 10, 2, 7, -6, -15, -6, 12, -9, 0, 9, -6, 10, 14, 8, -5, -3, 6, -6, 0, 9, -18, 7, -8, -8, -8, -16, 5, 18, -24, -7, -19, -11, 6, 18, 20, 5, 1, -20, 5, -16, -6, -9, -12, 9, -12, 18, -9, 6, -18, -12, -9, -27, -8, 25, 14, 26, 13, -26, -34, 29, -5, 16, -14, -2, -16, 2, -16, -28, 27, -24, -12, 30, -15, -24, 3, 18, -4, 23, -42, -18, -38, -17, 40, -11, -7, -22, 6, -36, 44, 16, 26, 46, -12, 6, -9, 15, -20, 7, -2, 13, 23, -16, 48, -18, -8, 2, 44, 4, 24, -6, 3, -6, 8, 29, -16, -2, -10, 13, 36, -24, 15, 6, 2, -28, -40, -28, 33, 12, -2, -8, 10, -8, 38, 43, 46, -37, -35, 37, -14, -11, 32, 2, -2, 2, 2, 43, 36, -30, 18, 12, -27, -15, 58, -29]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := 1; // EXAMPLE: // pp := Factorization(2*ZF)[1][1]; // heckeEigenvalues[pp]; print "To reconstruct the Hilbert newform f, type f, iso := Explode(make_newform());"; function make_newform(); M := HilbertCuspForms(F, NN); S := NewSubspace(M); // SetVerbose("ModFrmHil", 1); NFD := NewformDecomposition(S); newforms := [* Eigenform(U) : U in NFD *]; if #newforms eq 0 then; print "No Hilbert newforms at this level"; return 0; end if; print "Testing ", #newforms, " possible newforms"; newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *]; print #newforms, " newforms have the correct Hecke field"; if #newforms eq 0 then; print "No Hilbert newform found with the correct Hecke field"; return 0; end if; autos := Automorphisms(K); xnewforms := [* *]; for f in newforms do; if K eq RationalField() then; Append(~xnewforms, [* f, autos[1] *]); else; flag, iso := IsIsomorphic(K,BaseField(f)); for a in autos do; Append(~xnewforms, [* f, a*iso *]); end for; end if; end for; newforms := xnewforms; for P in primes do; xnewforms := [* *]; for f_iso in newforms do; f, iso := Explode(f_iso); if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then; Append(~xnewforms, f_iso); end if; end for; newforms := xnewforms; if #newforms eq 0 then; print "No Hilbert newform found which matches the Hecke eigenvalues"; return 0; else if #newforms eq 1 then; print "success: unique match"; return newforms[1]; end if; end if; end for; print #newforms, "Hilbert newforms found which match the Hecke eigenvalues"; return newforms[1]; end function;