/* 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![-4, -7, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -w - 2], [2, 2, -w - 1], [5, 5, -w^2 + w + 7], [5, 5, -w^2 + w + 5], [17, 17, -w^2 + 3*w + 1], [23, 23, -w^2 + w + 3], [27, 3, 3], [29, 29, -w^2 + 3*w - 1], [37, 37, w^2 + w + 1], [41, 41, w^2 - w - 9], [43, 43, -3*w^2 + 3*w + 17], [47, 47, 2*w^2 - 2*w - 13], [47, 47, -2*w + 5], [53, 53, 3*w^2 - w - 19], [59, 59, 2*w - 1], [67, 67, w^2 + w - 5], [71, 71, -3*w^2 + w + 21], [79, 79, 3*w^2 - w - 23], [89, 89, 2*w^2 - 4*w - 7], [89, 89, -2*w^2 - 2*w + 5], [89, 89, 2*w - 3], [97, 97, 2*w^2 - 9], [103, 103, -2*w^2 + 2*w + 3], [103, 103, -5*w^2 + 5*w + 27], [103, 103, -2*w^2 + 6*w + 3], [107, 107, w^2 - 3*w - 5], [109, 109, -w^2 + w - 1], [113, 113, -4*w^2 + 8*w + 9], [113, 113, 4*w^2 - 2*w - 29], [113, 113, -2*w^2 + 15], [127, 127, -w^2 - w + 9], [131, 131, 2*w^2 - 2*w - 1], [149, 149, 3*w^2 - 3*w - 19], [149, 149, -4*w^2 + 4*w + 21], [149, 149, -3*w^2 + 5*w + 9], [157, 157, -2*w^2 - 2*w + 1], [163, 163, 2*w^2 - 2*w - 7], [167, 167, 2*w^2 - 4*w - 1], [173, 173, 5*w^2 - 13*w - 5], [179, 179, -4*w^2 + 6*w + 17], [179, 179, 4*w + 5], [179, 179, -2*w - 7], [181, 181, w^2 - 3*w - 7], [191, 191, w^2 - w - 11], [197, 197, -2*w^2 - 4*w + 3], [199, 199, w^2 - 5*w + 5], [199, 199, -4*w^2 + 33], [199, 199, w^2 - 3*w - 9], [211, 211, w^2 - 3*w + 3], [211, 211, -5*w^2 + 9*w + 15], [211, 211, w^2 - 5*w - 5], [223, 223, w^2 + 7*w + 5], [227, 227, 4*w^2 - 6*w - 15], [229, 229, -w^2 - w + 13], [239, 239, 2*w^2 - 4*w - 9], [251, 251, -4*w^2 + 2*w + 23], [271, 271, -2*w^2 + 8*w - 7], [277, 277, 2*w^2 - 5], [281, 281, -8*w^2 + 6*w + 51], [289, 17, 3*w^2 + w - 15], [331, 331, -5*w^2 + 5*w + 29], [337, 337, 3*w^2 - 3*w - 13], [343, 7, -7], [347, 347, 7*w^2 - 3*w - 51], [347, 347, -4*w^2 - 4*w + 11], [347, 347, -2*w^2 + 19], [349, 349, 2*w^2 - 6*w - 7], [353, 353, 2*w^2 + 1], [367, 367, 4*w - 1], [373, 373, 3*w^2 - 9*w - 1], [373, 373, 4*w^2 - 4*w - 27], [373, 373, -3*w^2 - 5*w + 1], [379, 379, -6*w^2 + 4*w + 43], [389, 389, w^2 - 5*w + 3], [397, 397, 2*w^2 - 4*w - 11], [409, 409, -6*w^2 + 2*w + 43], [421, 421, -w^2 + 5*w + 15], [431, 431, 8*w^2 - 6*w - 53], [439, 439, -9*w^2 + 19*w + 19], [443, 443, -3*w^2 - w + 11], [449, 449, 4*w^2 - 23], [457, 457, 8*w^2 - 4*w - 57], [461, 461, w^2 - 5*w - 7], [463, 463, -6*w^2 + 12*w + 17], [467, 467, -4*w^2 + 8*w + 13], [479, 479, 2*w^2 - 6*w + 3], [491, 491, 4*w^2 - 6*w - 7], [499, 499, 2*w^2 - 8*w + 5], [499, 499, -4*w^2 + 2*w + 31], [499, 499, 4*w^2 - 4*w - 19], [503, 503, -2*w^2 - 2*w + 21], [509, 509, -2*w - 9], [529, 23, 3*w^2 - w - 13], [547, 547, -2*w^2 - 6*w + 1], [557, 557, 6*w + 7], [557, 557, 6*w^2 - 16*w - 3], [557, 557, -5*w^2 + 9*w + 13], [563, 563, -8*w^2 + 20*w + 11], [569, 569, 4*w^2 - 2*w - 21], [571, 571, 2*w^2 + 4*w - 5], [577, 577, 3*w^2 - 5*w - 15], [577, 577, 6*w + 11], [577, 577, -w^2 + 3*w - 5], [587, 587, -5*w^2 + w + 37], [607, 607, 3*w^2 - 3*w - 7], [613, 613, -6*w - 1], [617, 617, -w^2 - w - 5], [619, 619, -2*w^2 + 8*w + 3], [641, 641, -7*w^2 + w + 55], [643, 643, -6*w^2 + 2*w + 41], [643, 643, 6*w^2 - 4*w - 35], [643, 643, 3*w^2 + w - 17], [653, 653, -w^2 + 7*w + 7], [659, 659, -3*w^2 - w + 3], [673, 673, 2*w^2 + 2*w - 11], [673, 673, w^2 - 7*w + 13], [673, 673, -7*w^2 + 7*w + 37], [683, 683, w^2 - 5*w - 13], [683, 683, 2*w^2 - 6*w - 9], [683, 683, w^2 + w - 15], [691, 691, -6*w^2 + 6*w + 31], [691, 691, 4*w - 5], [691, 691, -4*w^2 + 35], [701, 701, 2*w^2 + 4*w + 5], [719, 719, -w^2 - 3*w - 7], [727, 727, 4*w^2 - 6*w - 11], [733, 733, w^2 + 3*w - 7], [739, 739, 3*w^2 - w - 11], [743, 743, 5*w^2 - 5*w - 33], [773, 773, -6*w^2 + 10*w + 23], [787, 787, -3*w^2 + 5*w + 1], [809, 809, 4*w^2 - 12*w - 1], [811, 811, -5*w^2 + 11*w + 5], [821, 821, -7*w^2 + 5*w + 41], [829, 829, w^2 + 7*w + 1], [841, 29, 6*w^2 - 16*w - 9], [853, 853, -11*w^2 + 9*w + 65], [857, 857, -9*w^2 + 17*w + 25], [857, 857, 4*w^2 - 4*w - 17], [857, 857, 9*w^2 - 7*w - 59], [859, 859, w^2 + 3*w - 13], [859, 859, -3*w^2 - w + 25], [859, 859, 6*w^2 - 8*w - 29], [877, 877, 3*w^2 - w - 9], [877, 877, 5*w^2 + 9*w - 7], [877, 877, -10*w^2 + 20*w + 27], [881, 881, -3*w^2 - w + 5], [887, 887, 3*w^2 + w - 7], [911, 911, 2*w^2 + 2*w - 13], [937, 937, 2*w^2 - 8*w - 1], [937, 937, -w^2 + 7*w + 1], [937, 937, 2*w^2 - 10*w + 11], [953, 953, -6*w^2 + 8*w + 25], [953, 953, -5*w^2 + w + 41], [953, 953, w^2 + 3*w - 11], [977, 977, 4*w^2 + 2*w - 19], [983, 983, 3*w^2 + 11*w + 3], [991, 991, 2*w - 11], [997, 997, 4*w^2 - 27], [997, 997, -w^2 - 9*w - 17], [997, 997, -4*w - 13]]; primes := [ideal : I in primesArray]; heckePol := x^4 - 8*x^2 + 2*x + 11; K := NumberField(heckePol); heckeEigenvaluesArray := [e, 1, -1, e^3 + 2*e^2 - 5*e - 6, -2*e + 2, -2*e^2 + 10, 2*e^2 + 2*e - 8, -2*e^3 + 10*e - 2, 2*e^3 + 2*e^2 - 12*e - 6, -2, e^3 - 3*e, -e^3 - 4*e^2 + 3*e + 20, -3*e^3 - 2*e^2 + 13*e + 2, -e^3 - 2*e^2 + 7*e + 4, -2*e^3 - 8*e^2 + 8*e + 28, e^3 - 5*e + 6, -2*e^3 - 6*e^2 + 8*e + 24, 2*e^3 + 2*e^2 - 8*e, -2*e^2 + 20, -4*e^3 - 6*e^2 + 14*e + 22, -e^3 + 2*e^2 + 5*e - 10, -2*e^3 - 2*e^2 + 8*e + 12, -2*e^2 - 6*e + 10, -3*e^3 - 4*e^2 + 15*e + 14, -4*e^3 - 8*e^2 + 20*e + 24, 2*e^2 + 6*e - 10, 3*e^3 + 6*e^2 - 13*e - 24, 4*e^3 + 2*e^2 - 20*e - 4, e^3 + 2*e^2 - 7*e - 12, 4*e^3 + 4*e^2 - 18*e - 6, 4*e^3 + 2*e^2 - 22*e - 6, e^3 + 2*e^2 - 5*e - 4, 4*e^3 + 8*e^2 - 20*e - 22, -e^3 - 4*e^2 + e + 16, 2*e^2 - 2*e - 2, -2*e^3 - 2*e^2 + 6*e + 16, e^3 + 6*e^2 - 5*e - 20, 2*e^3 + 4*e^2 - 4*e - 8, -2*e^2 + 2*e + 2, 4*e^3 + 2*e^2 - 20*e - 2, 4*e, 2*e^3 + 8*e^2 - 10*e - 32, 2*e^3 + 8*e^2 - 6*e - 34, 2*e^3 - 4*e^2 - 14*e + 16, 4*e + 2, -3*e^3 - 8*e^2 + 13*e + 36, -2*e^3 + 2*e^2 + 10*e - 6, e^3 - e - 6, 2*e^3 + 2*e^2 - 14*e - 10, -5*e^3 - 4*e^2 + 29*e + 10, 8*e^2 + 4*e - 28, -6*e^3 - 12*e^2 + 24*e + 44, -2*e^3 - 2*e^2 + 6*e - 6, -3*e^3 - 10*e^2 + 11*e + 30, e^3 + 6*e^2 - 5*e - 20, -4*e - 8, -4*e^3 - 6*e^2 + 24*e + 10, 3*e^3 + 2*e^2 - 15*e - 6, -e^3 + 13*e + 8, -4*e^3 - 14*e^2 + 18*e + 50, -3*e^3 + 23*e + 2, -6*e^2 - 2*e + 34, -5*e^3 + 2*e^2 + 31*e - 10, 6*e^3 + 10*e^2 - 22*e - 34, -4*e^3 - 6*e^2 + 14*e + 4, -8*e^3 - 8*e^2 + 36*e + 28, 4*e^3 + 4*e^2 - 16*e - 6, 8*e^3 + 10*e^2 - 32*e - 36, 4*e^3 + 8*e^2 - 24*e - 32, 4*e^3 + 12*e^2 - 12*e - 50, 6*e^3 + 6*e^2 - 24*e - 22, 2*e^3 + 4*e^2 - 10*e - 14, -e^3 - 2*e^2 - 5*e + 6, 2*e^2 - 4*e - 12, -4*e^3 - 10*e^2 + 14*e + 26, 4*e^3 + 8*e^2 - 16*e - 14, 3*e^3 - 4*e^2 - 11*e + 32, 2*e^3 + 4*e^2 - 14*e - 28, -10*e^3 - 16*e^2 + 50*e + 44, -e^3 - 4*e^2 + 7*e + 12, 8*e^3 + 12*e^2 - 34*e - 30, 3*e^3 - 19*e + 12, 2*e^3 + 8*e^2 - 2*e - 22, -8*e^3 - 12*e^2 + 40*e + 20, -4*e^3 - 4*e^2 + 20*e, -4*e - 8, e^3 + 10*e^2 + 7*e - 44, -4*e^2 - 12*e + 28, -8*e^3 - 14*e^2 + 34*e + 46, -2*e^3 + 8*e + 24, -2*e^3 - 6*e^2 + 20*e + 24, -3*e^3 + 4*e^2 + 25*e - 10, 4*e^3 + 2*e^2 - 22*e + 10, 2*e^3 - 2*e^2 - 12*e - 16, -4*e^3 - 4*e^2 + 12*e + 6, -2*e^3 - 2*e^2 + 2*e, -12*e^2 + 46, -4*e^3 - 4*e^2 + 20*e, -2*e^3 - 6*e^2 + 18*e + 32, -2*e^3 - 2*e - 16, 4*e^2 + 12*e - 26, 10*e^2 + 12*e - 28, -6*e^3 - 6*e^2 + 28*e + 6, 4*e^3 + 8*e^2 - 24*e - 20, 10*e^3 + 12*e^2 - 50*e - 28, 6*e^3 + 14*e^2 - 18*e - 48, 4*e^3 + 6*e^2 - 8*e - 12, -8*e^3 - 12*e^2 + 36*e + 24, 7*e^3 + 16*e^2 - 25*e - 62, -e^3 - 4*e^2 + 7*e - 16, 2*e^2 + 4*e - 10, -4*e^2 + 32, 8*e^3 - 40*e + 6, 6*e^3 + 12*e^2 - 26*e - 48, -11*e^3 - 28*e^2 + 51*e + 88, -5*e^3 - 6*e^2 + 27*e - 16, 2*e^3 - 2*e^2 - 20*e + 30, -2*e^3 - 4*e^2 + 2*e, 6*e^3 + 16*e^2 - 22*e - 52, 4*e^3 + 14*e^2 - 12*e - 58, -2*e^3 - 6*e^2 + 22*e + 38, -4*e^3 - 6*e^2 + 34*e + 10, 2*e^3 - 2*e^2 + 2*e + 26, 2*e^3 - 2*e^2 - 18*e + 32, 4*e^3 - 6*e^2 - 26*e + 20, 2*e^3 + 4*e^2 - 6*e, -2*e^3 + 10*e - 6, 8*e^3 + 12*e^2 - 32*e - 24, -4*e^3 - 12*e^2 + 12*e + 52, 4*e^3 - 16*e + 18, -6*e^3 - 12*e^2 + 38*e + 36, -5*e^3 - 14*e^2 + 19*e + 64, -4*e^3 + 2*e^2 + 14*e - 16, -e^3 + 15*e + 6, 8*e^2 + 8*e - 18, 4*e^2 + 10*e - 10, 12*e^3 + 12*e^2 - 48*e - 30, -2*e^3 + 8*e^2 + 10*e - 34, 4*e^3 + 8*e^2 - 20*e - 30, -12*e^3 - 24*e^2 + 52*e + 74, 10*e^2 + 10*e - 22, -2*e^3 + 2*e^2 - 4*e - 32, 2*e^3 - 8*e^2 - 26*e + 24, -2*e^3 + 4*e^2 + 22*e - 14, 4*e^3 + 8*e^2 - 32*e - 30, -8*e^3 - 20*e^2 + 26*e + 90, -6*e^3 - 18*e^2 + 30*e + 36, -2*e^3 + 6*e^2 + 20*e - 24, 4*e^3 + 14*e^2 - 18*e - 60, -e^3 - 6*e^2 + 7*e + 32, 5*e^3 + 2*e^2 - 43*e - 12, 2*e^3 + 16*e^2 - 14*e - 78, e^3 + 10*e^2 - 5*e - 30, -e^3 - 12*e^2 + e + 12, -6*e^3 - 8*e^2 + 30*e + 6, -4*e^3 + 4*e^2 + 24*e - 6, 4*e + 8, -3*e^3 + 4*e^2 + 23*e - 10, e^3 + 12*e^2 + 5*e - 54, 4*e^2 - 8*e - 18, 4*e^3 - 28*e + 14]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := -1; 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;