/* 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, -5, 2, 8, -4, -2, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [9, 3, -w^2 + 2], [19, 19, -w^3 + 4*w], [19, 19, w^3 - 3*w - 1], [29, 29, w^5 - w^4 - 6*w^3 + 4*w^2 + 8*w - 3], [29, 29, w^4 - w^3 - 4*w^2 + 2*w + 2], [31, 31, -w^4 + 4*w^2 + w - 3], [41, 41, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 11*w - 4], [49, 7, -2*w^5 + 3*w^4 + 10*w^3 - 12*w^2 - 11*w + 6], [59, 59, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 7*w - 5], [59, 59, w^5 - w^4 - 4*w^3 + 3*w^2 + 3*w - 3], [59, 59, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 9*w - 4], [61, 61, -w^5 + w^4 + 5*w^3 - 3*w^2 - 7*w + 1], [64, 2, -2], [71, 71, -2*w^5 + 2*w^4 + 10*w^3 - 8*w^2 - 11*w + 6], [79, 79, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 9*w + 5], [79, 79, -w^4 - w^3 + 5*w^2 + 4*w - 3], [79, 79, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 8*w + 4], [81, 3, 3*w^5 - 5*w^4 - 14*w^3 + 19*w^2 + 13*w - 8], [89, 89, 2*w^5 - 2*w^4 - 9*w^3 + 6*w^2 + 8*w - 1], [101, 101, -w^4 - w^3 + 5*w^2 + 3*w - 3], [101, 101, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w + 4], [109, 109, -2*w^5 + 4*w^4 + 8*w^3 - 14*w^2 - 6*w + 5], [109, 109, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 10*w + 7], [125, 5, w^5 - w^4 - 5*w^3 + 4*w^2 + 5*w], [131, 131, -w^2 - w + 3], [131, 131, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 3*w - 4], [131, 131, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 5], [131, 131, -2*w^5 + 4*w^4 + 8*w^3 - 15*w^2 - 6*w + 7], [139, 139, w^4 - 3*w^3 - 3*w^2 + 9*w], [149, 149, 2*w^5 - 3*w^4 - 9*w^3 + 12*w^2 + 7*w - 5], [149, 149, w^5 - 2*w^4 - 4*w^3 + 8*w^2 + 2*w - 2], [151, 151, 3*w^5 - 4*w^4 - 13*w^3 + 14*w^2 + 10*w - 5], [151, 151, -3*w^5 + 4*w^4 + 14*w^3 - 15*w^2 - 12*w + 6], [151, 151, 4*w^5 - 5*w^4 - 18*w^3 + 18*w^2 + 15*w - 9], [151, 151, 2*w^5 - 2*w^4 - 9*w^3 + 7*w^2 + 8*w - 5], [169, 13, -2*w^5 + w^4 + 11*w^3 - 3*w^2 - 13*w + 1], [179, 179, w^3 - w^2 - 2*w + 3], [181, 181, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 10*w - 3], [181, 181, -2*w^4 + w^3 + 8*w^2 - 3*w - 2], [191, 191, w^5 - w^4 - 6*w^3 + 5*w^2 + 9*w - 4], [191, 191, -w^3 + 2*w^2 + 3*w - 4], [191, 191, -2*w^5 + w^4 + 11*w^3 - 3*w^2 - 12*w], [191, 191, w^4 - 2*w^3 - 4*w^2 + 7*w], [199, 199, -3*w^5 + 4*w^4 + 15*w^3 - 14*w^2 - 16*w + 6], [199, 199, -2*w^5 + 3*w^4 + 10*w^3 - 12*w^2 - 12*w + 7], [211, 211, -2*w^5 + 4*w^4 + 9*w^3 - 15*w^2 - 8*w + 5], [211, 211, -3*w^5 + 5*w^4 + 13*w^3 - 18*w^2 - 10*w + 8], [229, 229, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 10*w + 6], [229, 229, w^4 - 4*w^2 - 1], [229, 229, w^5 - w^4 - 3*w^3 + 3*w^2 - 3*w - 2], [239, 239, -2*w^5 + 3*w^4 + 9*w^3 - 11*w^2 - 8*w + 8], [241, 241, w^5 - 3*w^4 - 4*w^3 + 10*w^2 + 3*w - 2], [241, 241, w^5 - w^4 - 5*w^3 + 3*w^2 + 4*w + 1], [251, 251, -3*w^5 + 3*w^4 + 15*w^3 - 10*w^2 - 17*w + 3], [271, 271, -2*w^5 + 4*w^4 + 9*w^3 - 14*w^2 - 8*w + 2], [271, 271, -2*w^5 + 3*w^4 + 8*w^3 - 9*w^2 - 4*w], [271, 271, w^5 - 2*w^4 - 3*w^3 + 7*w^2 - 2*w - 4], [289, 17, 4*w^5 - 6*w^4 - 17*w^3 + 22*w^2 + 12*w - 9], [311, 311, -2*w^5 + 3*w^4 + 7*w^3 - 9*w^2 - 3*w + 2], [331, 331, 3*w^5 - 5*w^4 - 14*w^3 + 19*w^2 + 14*w - 12], [349, 349, -4*w^5 + 6*w^4 + 18*w^3 - 21*w^2 - 17*w + 7], [349, 349, -w^4 + w^3 + 5*w^2 - 3*w - 3], [359, 359, -2*w^5 + 4*w^4 + 9*w^3 - 15*w^2 - 10*w + 9], [361, 19, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 6*w - 4], [361, 19, -3*w^5 + 5*w^4 + 13*w^3 - 18*w^2 - 10*w + 9], [379, 379, -4*w^5 + 5*w^4 + 18*w^3 - 17*w^2 - 14*w + 7], [379, 379, 2*w^5 - 4*w^4 - 10*w^3 + 15*w^2 + 11*w - 5], [389, 389, -5*w^5 + 7*w^4 + 22*w^3 - 25*w^2 - 17*w + 11], [389, 389, w^5 - 3*w^4 - 3*w^3 + 12*w^2 - 2*w - 5], [401, 401, -4*w^5 + 6*w^4 + 18*w^3 - 23*w^2 - 15*w + 10], [401, 401, -w^5 + 3*w^4 + 4*w^3 - 12*w^2 - w + 4], [401, 401, 4*w^5 - 5*w^4 - 20*w^3 + 19*w^2 + 21*w - 11], [401, 401, -3*w^5 + 3*w^4 + 16*w^3 - 12*w^2 - 18*w + 6], [409, 409, -2*w^4 + 2*w^3 + 7*w^2 - 6*w - 2], [409, 409, -w^5 + 3*w^4 + 5*w^3 - 13*w^2 - 5*w + 9], [419, 419, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 10*w - 6], [419, 419, -w^5 + 7*w^3 - w^2 - 10*w + 4], [419, 419, w^5 - 2*w^4 - 5*w^3 + 10*w^2 + 5*w - 9], [421, 421, 3*w^5 - 3*w^4 - 16*w^3 + 12*w^2 + 18*w - 7], [431, 431, 4*w^5 - 7*w^4 - 17*w^3 + 25*w^2 + 13*w - 10], [431, 431, -2*w^5 + 3*w^4 + 7*w^3 - 10*w^2 - w + 5], [431, 431, 4*w^5 - 6*w^4 - 19*w^3 + 23*w^2 + 17*w - 10], [439, 439, -2*w^5 + 4*w^4 + 10*w^3 - 15*w^2 - 11*w + 4], [461, 461, w^5 - 2*w^4 - 3*w^3 + 6*w^2 + 2], [461, 461, w^5 - w^4 - 5*w^3 + 5*w^2 + 6*w - 7], [479, 479, -2*w^5 + w^4 + 9*w^3 - 2*w^2 - 8*w], [479, 479, 2*w^5 - 2*w^4 - 10*w^3 + 8*w^2 + 12*w - 7], [491, 491, -3*w^5 + 3*w^4 + 14*w^3 - 12*w^2 - 12*w + 7], [491, 491, -3*w^5 + 3*w^4 + 15*w^3 - 12*w^2 - 14*w + 8], [491, 491, -2*w^2 + w + 6], [499, 499, -3*w^5 + 3*w^4 + 14*w^3 - 10*w^2 - 13*w + 6], [509, 509, -2*w^5 + 2*w^4 + 8*w^3 - 6*w^2 - 5*w], [521, 521, 3*w^5 - 4*w^4 - 14*w^3 + 15*w^2 + 15*w - 9], [521, 521, -3*w^5 + 5*w^4 + 14*w^3 - 19*w^2 - 15*w + 9], [521, 521, -4*w^5 + 5*w^4 + 18*w^3 - 17*w^2 - 16*w + 6], [529, 23, w^5 - 3*w^4 - 5*w^3 + 11*w^2 + 7*w - 5], [541, 541, -3*w^5 + 4*w^4 + 14*w^3 - 15*w^2 - 14*w + 10], [569, 569, -w^5 - w^4 + 6*w^3 + 4*w^2 - 6*w - 1], [571, 571, -2*w^4 + w^3 + 9*w^2 - 3*w - 4], [599, 599, -3*w^5 + 3*w^4 + 14*w^3 - 11*w^2 - 11*w + 4], [601, 601, w^4 - 2*w^2 - w - 2], [619, 619, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 10*w - 5], [619, 619, w^5 - 3*w^4 - 4*w^3 + 12*w^2 + 3*w - 3], [619, 619, -3*w^5 + 5*w^4 + 13*w^3 - 20*w^2 - 10*w + 10], [619, 619, w^5 - 2*w^4 - 5*w^3 + 10*w^2 + 5*w - 8], [619, 619, -4*w^5 + 6*w^4 + 19*w^3 - 23*w^2 - 17*w + 11], [619, 619, -4*w^5 + 6*w^4 + 17*w^3 - 22*w^2 - 13*w + 10], [631, 631, w^5 - 3*w^4 - 5*w^3 + 13*w^2 + 5*w - 6], [641, 641, -w^3 + w^2 + 3*w - 5], [641, 641, -w^5 + 3*w^4 + 2*w^3 - 10*w^2 + 3*w + 4], [659, 659, -w^5 + 3*w^4 + 5*w^3 - 12*w^2 - 5*w + 5], [661, 661, -2*w^5 + 3*w^4 + 8*w^3 - 12*w^2 - 3*w + 8], [661, 661, 2*w^5 - 3*w^4 - 8*w^3 + 10*w^2 + 6*w - 1], [691, 691, -w^3 - w^2 + 5*w + 1], [691, 691, -3*w^5 + 4*w^4 + 13*w^3 - 13*w^2 - 10*w + 6], [691, 691, 3*w^5 - 4*w^4 - 15*w^3 + 15*w^2 + 14*w - 6], [701, 701, 3*w^5 - 6*w^4 - 13*w^3 + 23*w^2 + 10*w - 10], [701, 701, -3*w^5 + 6*w^4 + 13*w^3 - 23*w^2 - 11*w + 9], [709, 709, 3*w^5 - 6*w^4 - 13*w^3 + 22*w^2 + 11*w - 10], [709, 709, -w^5 + 5*w^3 - 4*w - 2], [709, 709, -w^5 + w^4 + 7*w^3 - 6*w^2 - 11*w + 5], [719, 719, -2*w^5 + 3*w^4 + 9*w^3 - 12*w^2 - 7*w + 4], [751, 751, w^4 + w^3 - 5*w^2 - 6*w + 4], [769, 769, 4*w^5 - 5*w^4 - 19*w^3 + 18*w^2 + 18*w - 10], [769, 769, 4*w^5 - 6*w^4 - 17*w^3 + 22*w^2 + 13*w - 11], [809, 809, w^5 - w^4 - 4*w^3 + 5*w^2 + w - 7], [809, 809, 2*w^5 - 5*w^4 - 9*w^3 + 19*w^2 + 10*w - 7], [809, 809, -w^5 + 5*w^3 + w^2 - 7*w], [809, 809, -3*w^5 + 4*w^4 + 15*w^3 - 16*w^2 - 15*w + 11], [811, 811, -2*w^5 + 4*w^4 + 8*w^3 - 13*w^2 - 6*w + 5], [811, 811, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - w], [811, 811, 2*w^5 - 4*w^4 - 9*w^3 + 16*w^2 + 10*w - 9], [811, 811, 2*w^5 - 4*w^4 - 10*w^3 + 16*w^2 + 10*w - 7], [821, 821, -w^5 + 2*w^4 + 5*w^3 - 6*w^2 - 5*w - 2], [829, 829, 3*w^5 - 5*w^4 - 13*w^3 + 19*w^2 + 12*w - 9], [839, 839, 2*w^5 - 4*w^4 - 7*w^3 + 14*w^2 + w - 6], [839, 839, 4*w^5 - 5*w^4 - 18*w^3 + 19*w^2 + 14*w - 10], [841, 29, -2*w^5 + 4*w^4 + 10*w^3 - 15*w^2 - 10*w + 5], [841, 29, -w^5 + 3*w^4 + 6*w^3 - 12*w^2 - 10*w + 6], [859, 859, 3*w^5 - 3*w^4 - 15*w^3 + 10*w^2 + 14*w - 2], [881, 881, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 10*w - 4], [911, 911, -4*w^5 + 5*w^4 + 19*w^3 - 20*w^2 - 17*w + 11], [919, 919, 2*w^5 - 3*w^4 - 10*w^3 + 13*w^2 + 9*w - 10], [929, 929, -w^5 + w^4 + 5*w^3 - 2*w^2 - 8*w - 1], [941, 941, -5*w^5 + 7*w^4 + 23*w^3 - 26*w^2 - 20*w + 11], [961, 31, -3*w^5 + 4*w^4 + 16*w^3 - 16*w^2 - 20*w + 8], [971, 971, w^5 - 5*w^3 + 5*w - 4], [991, 991, 3*w^5 - 5*w^4 - 12*w^3 + 18*w^2 + 8*w - 6], [991, 991, 3*w^5 - 5*w^4 - 14*w^3 + 18*w^2 + 12*w - 7]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [5, -5, -5, 0, 0, 7, -3, -1, 0, 0, 15, 7, 5, -3, 10, -10, 10, -13, -15, 12, -3, 10, -10, 6, 12, -3, 12, 12, 5, -15, 0, 2, -23, -13, -8, -10, 15, 7, 2, -3, -3, 12, 12, -5, 20, 22, -23, 10, 25, 5, 0, 17, -23, 27, -28, -28, -13, 20, -3, 7, 10, 10, 0, 7, -38, -10, -20, -15, -30, -3, 12, 27, 27, -10, -40, 0, 0, 0, 32, -3, 27, 12, -10, -33, -18, 0, -30, -33, 27, 12, -10, 0, -18, -3, -3, 35, 7, -15, 32, -30, -8, 20, 20, 25, 25, 5, 35, -28, -18, 12, 45, 17, -43, 37, 22, 2, 27, -18, -10, 10, 10, -30, -8, 40, 10, -30, 30, 30, 15, 52, -28, 2, -23, -18, -50, 30, 0, 22, 22, 25, -18, 12, 25, 30, -33, -38, -48, 2, 7]; 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;