/* 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, 0, 8, 4, -6, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [5, 5, w - 1], [11, 11, w^3 - 3*w], [19, 19, -w^5 + w^4 + 5*w^3 - 4*w^2 - 5*w], [25, 5, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 8*w], [29, 29, -w^4 + 4*w^2 + 1], [31, 31, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 3*w - 4], [31, 31, -w^2 + 2], [59, 59, -w^5 + w^4 + 5*w^3 - 3*w^2 - 4*w - 2], [59, 59, w^4 - w^3 - 5*w^2 + 3*w + 4], [59, 59, -w^4 - w^3 + 4*w^2 + 5*w], [61, 61, w^5 - w^4 - 4*w^3 + 5*w^2 - 4], [64, 2, 2], [71, 71, w^5 - w^4 - 6*w^3 + 4*w^2 + 6*w], [71, 71, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 + w + 4], [71, 71, 2*w^5 - 3*w^4 - 9*w^3 + 13*w^2 + 4*w - 4], [71, 71, w^3 - 5*w], [79, 79, w^5 - w^4 - 4*w^3 + 4*w^2 + w - 2], [89, 89, w^3 - w^2 - 4*w + 1], [101, 101, -2*w^5 + 3*w^4 + 11*w^3 - 13*w^2 - 12*w + 2], [101, 101, 2*w^5 - 2*w^4 - 13*w^3 + 7*w^2 + 20*w + 4], [101, 101, w^5 - w^4 - 5*w^3 + 5*w^2 + 2*w - 3], [101, 101, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 8*w - 1], [109, 109, -w^5 + w^4 + 7*w^3 - 4*w^2 - 11*w - 1], [121, 11, -w^5 + w^4 + 4*w^3 - 5*w^2 + w + 4], [131, 131, w^4 - 4*w^2 + w - 1], [131, 131, -2*w^5 + 2*w^4 + 10*w^3 - 9*w^2 - 7*w + 2], [139, 139, w^4 + w^3 - 3*w^2 - 4*w - 4], [149, 149, -w^5 + 3*w^4 + 6*w^3 - 13*w^2 - 8*w + 2], [149, 149, 2*w^5 - 11*w^3 - 2*w^2 + 11*w + 6], [151, 151, 3*w^5 - 5*w^4 - 16*w^3 + 21*w^2 + 16*w - 6], [169, 13, w^5 - 3*w^4 - 5*w^3 + 13*w^2 + 6*w - 3], [179, 179, w^5 - 2*w^4 - 7*w^3 + 9*w^2 + 12*w - 1], [179, 179, 2*w^5 - 4*w^4 - 9*w^3 + 18*w^2 + 4*w - 7], [179, 179, -3*w^5 + 3*w^4 + 15*w^3 - 13*w^2 - 12*w + 3], [179, 179, -w^5 + 3*w^4 + 5*w^3 - 14*w^2 - 5*w + 6], [181, 181, w^3 + w^2 - 4*w - 6], [191, 191, -w^5 + w^4 + 4*w^3 - 5*w^2 + w + 3], [191, 191, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w + 3], [191, 191, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 - w + 3], [191, 191, -w^5 + w^4 + 7*w^3 - 4*w^2 - 12*w - 2], [199, 199, w^5 - w^4 - 6*w^3 + 4*w^2 + 9*w - 1], [199, 199, w^4 + w^3 - 5*w^2 - 3*w + 3], [199, 199, -w^5 + w^4 + 3*w^3 - 4*w^2 + 3*w + 2], [199, 199, w^2 + w - 3], [211, 211, 3*w^5 - 4*w^4 - 16*w^3 + 17*w^2 + 16*w - 3], [239, 239, -3*w^5 + 3*w^4 + 16*w^3 - 12*w^2 - 16*w + 3], [239, 239, 4*w^5 - 6*w^4 - 21*w^3 + 25*w^2 + 19*w - 5], [239, 239, -w^5 + w^4 + 7*w^3 - 5*w^2 - 10*w + 2], [241, 241, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 4*w + 2], [251, 251, 3*w^5 - 4*w^4 - 16*w^3 + 18*w^2 + 15*w - 5], [251, 251, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 2], [251, 251, -3*w^5 + 4*w^4 + 16*w^3 - 17*w^2 - 15*w + 2], [251, 251, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 - w + 4], [269, 269, w^5 - w^4 - 6*w^3 + 6*w^2 + 7*w - 5], [271, 271, 3*w^5 - 3*w^4 - 15*w^3 + 12*w^2 + 11*w - 1], [271, 271, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 2*w], [281, 281, w^2 - w - 4], [281, 281, -3*w^5 + 4*w^4 + 16*w^3 - 18*w^2 - 14*w + 4], [281, 281, w^5 - w^4 - 6*w^3 + 4*w^2 + 9*w], [281, 281, -3*w^5 + 3*w^4 + 14*w^3 - 13*w^2 - 8*w + 4], [289, 17, 2*w^5 - 2*w^4 - 11*w^3 + 8*w^2 + 10*w - 1], [311, 311, -w^5 - w^4 + 6*w^3 + 6*w^2 - 8*w - 5], [311, 311, -2*w^5 + 3*w^4 + 10*w^3 - 13*w^2 - 8*w + 7], [331, 331, 2*w^5 - 3*w^4 - 11*w^3 + 12*w^2 + 12*w - 4], [331, 331, -3*w^5 + 4*w^4 + 15*w^3 - 16*w^2 - 12*w + 3], [331, 331, -w^5 + w^4 + 4*w^3 - 3*w^2 + w - 1], [331, 331, -2*w^5 + 5*w^4 + 10*w^3 - 21*w^2 - 8*w + 5], [359, 359, 2*w^4 - 2*w^3 - 9*w^2 + 7*w + 3], [361, 19, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2], [379, 379, 3*w^5 - 2*w^4 - 15*w^3 + 8*w^2 + 11*w - 1], [379, 379, 2*w^5 - 2*w^4 - 11*w^3 + 8*w^2 + 13*w - 3], [389, 389, w^5 + w^4 - 5*w^3 - 5*w^2 + 3*w + 4], [389, 389, 3*w^5 - 5*w^4 - 16*w^3 + 21*w^2 + 16*w - 5], [389, 389, -w^5 + 4*w^4 + 5*w^3 - 17*w^2 - 4*w + 4], [389, 389, 3*w^5 - 4*w^4 - 15*w^3 + 18*w^2 + 10*w - 6], [401, 401, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 8*w + 3], [401, 401, -2*w^5 + 4*w^4 + 11*w^3 - 17*w^2 - 13*w + 3], [409, 409, w^3 + w^2 - 3*w - 5], [419, 419, -w^5 + w^4 + 4*w^3 - 6*w^2 + 4], [419, 419, -w^4 + 2*w^3 + 5*w^2 - 8*w - 6], [421, 421, w^5 - w^4 - 5*w^3 + 5*w^2 + 3*w - 6], [431, 431, 2*w^3 + w^2 - 9*w - 3], [439, 439, -w^5 + 6*w^3 + w^2 - 6*w - 4], [439, 439, w^5 - 2*w^4 - 4*w^3 + 9*w^2 - 7], [439, 439, 3*w^5 - 5*w^4 - 16*w^3 + 21*w^2 + 14*w - 4], [439, 439, -2*w^5 + 3*w^4 + 11*w^3 - 11*w^2 - 13*w - 1], [439, 439, w^4 - w^3 - 3*w^2 + 3*w - 3], [439, 439, -3*w^5 + 4*w^4 + 16*w^3 - 16*w^2 - 17*w + 2], [449, 449, -w^3 - 2*w^2 + 5*w + 6], [461, 461, w^5 - w^4 - 6*w^3 + 3*w^2 + 10*w + 1], [479, 479, 3*w^5 - 5*w^4 - 16*w^3 + 21*w^2 + 15*w - 2], [491, 491, w^5 - w^4 - 6*w^3 + 3*w^2 + 10*w + 2], [499, 499, w^5 - 5*w^3 - 2*w^2 + 4*w + 4], [509, 509, -w^5 + 2*w^4 + 5*w^3 - 10*w^2 - 4*w + 4], [509, 509, -w^5 + w^4 + 6*w^3 - 6*w^2 - 7*w + 4], [521, 521, -3*w^5 + 5*w^4 + 16*w^3 - 22*w^2 - 14*w + 5], [541, 541, -w^4 + w^3 + 5*w^2 - 2*w - 5], [541, 541, 2*w^5 - 3*w^4 - 9*w^3 + 14*w^2 + 4*w - 5], [541, 541, -2*w^5 + 3*w^4 + 9*w^3 - 13*w^2 - 2*w + 4], [569, 569, 2*w^5 - w^4 - 11*w^3 + 5*w^2 + 11*w - 3], [571, 571, w^5 - w^4 - 6*w^3 + 2*w^2 + 8*w + 4], [571, 571, 3*w^5 - 5*w^4 - 15*w^3 + 22*w^2 + 10*w - 7], [599, 599, -w^4 + w^3 + 3*w^2 - 4*w], [599, 599, -w^4 + 6*w^2 + w - 4], [601, 601, -2*w^5 + 3*w^4 + 10*w^3 - 12*w^2 - 7*w + 4], [619, 619, -3*w^5 + 4*w^4 + 15*w^3 - 16*w^2 - 11*w + 2], [631, 631, w^4 + 2*w^3 - 5*w^2 - 8*w + 2], [631, 631, -w^5 + 4*w^4 + 5*w^3 - 18*w^2 - 4*w + 6], [631, 631, w^4 + 2*w^3 - 4*w^2 - 8*w + 1], [641, 641, -w^5 - w^4 + 5*w^3 + 5*w^2 - 5*w - 2], [641, 641, -w^4 + 6*w^2 + w - 5], [641, 641, w^5 - 2*w^4 - 4*w^3 + 9*w^2 + 2*w - 5], [659, 659, -w^4 - 2*w^3 + 5*w^2 + 8*w - 1], [661, 661, 2*w^5 - 2*w^4 - 9*w^3 + 8*w^2 + 3*w + 1], [661, 661, 2*w^5 - 3*w^4 - 11*w^3 + 12*w^2 + 10*w - 1], [661, 661, w^5 - 3*w^4 - 6*w^3 + 12*w^2 + 9*w], [661, 661, -2*w^5 + 3*w^4 + 12*w^3 - 13*w^2 - 14*w + 1], [691, 691, w^5 - 6*w^3 - w^2 + 9*w + 1], [691, 691, 3*w^5 - 5*w^4 - 15*w^3 + 21*w^2 + 13*w - 5], [691, 691, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 6*w + 2], [691, 691, 2*w^5 - 3*w^4 - 12*w^3 + 12*w^2 + 16*w - 2], [701, 701, -w^5 - w^4 + 6*w^3 + 4*w^2 - 8*w - 2], [709, 709, -3*w^5 + 3*w^4 + 14*w^3 - 12*w^2 - 8*w + 3], [709, 709, w^5 + w^4 - 6*w^3 - 4*w^2 + 8*w + 1], [709, 709, 2*w^5 - 4*w^4 - 10*w^3 + 16*w^2 + 9*w - 2], [709, 709, 2*w^5 - 2*w^4 - 12*w^3 + 9*w^2 + 14*w - 3], [719, 719, w^4 - w^3 - 6*w^2 + 3*w + 4], [729, 3, -3], [739, 739, -w^5 + w^4 + 6*w^3 - 3*w^2 - 9*w - 5], [751, 751, -2*w^5 + 3*w^4 + 11*w^3 - 13*w^2 - 12*w + 1], [751, 751, -w^5 + w^4 + 4*w^3 - 4*w^2 - 2*w - 1], [761, 761, -2*w^5 + 2*w^4 + 12*w^3 - 9*w^2 - 15*w], [769, 769, -3*w^5 + 5*w^4 + 17*w^3 - 21*w^2 - 19*w + 3], [769, 769, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 5*w - 1], [769, 769, -2*w^5 + 4*w^4 + 10*w^3 - 18*w^2 - 9*w + 7], [769, 769, 2*w^5 - w^4 - 12*w^3 + 3*w^2 + 14*w + 2], [811, 811, -3*w^5 + 2*w^4 + 16*w^3 - 9*w^2 - 15*w + 2], [821, 821, 2*w^3 - w^2 - 8*w + 1], [829, 829, w^5 - w^4 - 7*w^3 + 3*w^2 + 11*w + 1], [829, 829, 2*w^5 - 2*w^4 - 10*w^3 + 9*w^2 + 8*w - 6], [839, 839, -w^4 - 2*w^3 + 5*w^2 + 6*w - 1], [841, 29, -2*w^5 + 2*w^4 + 12*w^3 - 9*w^2 - 16*w + 2], [859, 859, -w^5 + 2*w^4 + 4*w^3 - 10*w^2 - w + 4], [859, 859, 2*w^5 - 3*w^4 - 10*w^3 + 11*w^2 + 7*w + 1], [881, 881, -2*w^5 + 3*w^4 + 9*w^3 - 14*w^2 - 4*w + 4], [881, 881, 2*w^5 - 2*w^4 - 10*w^3 + 10*w^2 + 6*w - 5], [911, 911, w^5 - w^4 - 4*w^3 + 5*w^2 - 2*w - 5], [911, 911, w^3 - 4*w - 4], [929, 929, -w^5 + 3*w^4 + 6*w^3 - 12*w^2 - 9*w + 1], [941, 941, 2*w^4 - 10*w^2 + w + 6], [941, 941, 3*w^5 - 3*w^4 - 15*w^3 + 14*w^2 + 10*w - 5], [941, 941, -w^5 + 5*w^3 + w^2 - 2*w - 4], [961, 31, -w^4 - w^3 + 6*w^2 + 4*w - 4], [961, 31, 2*w^5 - 3*w^4 - 12*w^3 + 12*w^2 + 15*w - 3], [971, 971, -2*w^5 + 3*w^4 + 12*w^3 - 13*w^2 - 14*w], [991, 991, w^5 - 3*w^4 - 4*w^3 + 14*w^2 + w - 6]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [0, 2, 2, -4, 1, -8, -6, 8, -12, -8, 0, -9, -8, -12, 6, -8, 10, -2, 10, 10, 6, 0, -10, 16, -12, -20, -8, 18, 18, -2, -14, -12, -12, 4, 4, -10, -10, -16, -16, 0, 16, 0, 4, -22, -6, -12, 12, 18, -2, 18, -30, 12, -20, -6, 16, 2, -6, 20, 6, -22, -30, 28, 4, -8, 16, 32, 28, -22, -4, 4, -16, -16, -18, -22, -12, 30, 0, 30, 12, -14, 6, -12, 22, 0, 8, 38, 0, 26, -18, 18, 38, -20, -16, 14, 8, -10, -10, 10, -18, 30, 20, -10, -40, -20, 26, 20, 28, 8, -44, -48, -28, -30, 40, -18, -2, 40, 12, -34, -42, -36, -20, 34, -34, 6, -6, -20, 48, -10, -20, 24, -20, -22, 18, -6, -6, 36, -32, 6, 54, -14, 0, 14, 32, -4, -50, 6, -34, -16, -18, -14, 48, -2, -44, -30, 32, -32]; 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;