/* 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, 7, 11, -2, -7, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [5, 5, -2*w^5 + w^4 + 13*w^3 - 2*w^2 - 19*w - 5], [9, 3, 2*w^5 - w^4 - 14*w^3 + 2*w^2 + 23*w + 6], [11, 11, w - 1], [25, 5, w^3 + w^2 - 4*w - 3], [29, 29, w^5 - w^4 - 7*w^3 + 4*w^2 + 11*w], [41, 41, w^4 - w^3 - 5*w^2 + 3*w + 3], [49, 7, w^5 - w^4 - 7*w^3 + 4*w^2 + 11*w + 1], [59, 59, 2*w^5 - w^4 - 14*w^3 + 2*w^2 + 24*w + 7], [59, 59, -w^5 + 8*w^3 + 2*w^2 - 15*w - 8], [61, 61, w^5 - 7*w^3 - 2*w^2 + 12*w + 4], [61, 61, -w^5 + 7*w^3 - 11*w - 1], [64, 2, 2], [71, 71, w^3 + w^2 - 5*w - 3], [71, 71, -3*w^5 + w^4 + 21*w^3 - w^2 - 33*w - 9], [79, 79, -2*w^5 + w^4 + 13*w^3 - 3*w^2 - 19*w - 5], [81, 3, 2*w^5 - w^4 - 13*w^3 + w^2 + 19*w + 8], [89, 89, 2*w^5 - w^4 - 13*w^3 + 2*w^2 + 20*w + 7], [89, 89, -w^5 + 8*w^3 + w^2 - 16*w - 5], [89, 89, -3*w^5 + w^4 + 20*w^3 - 30*w - 11], [89, 89, -w^5 + 7*w^3 + 2*w^2 - 11*w - 4], [89, 89, -2*w^5 + 14*w^3 + 3*w^2 - 22*w - 8], [89, 89, 2*w^5 - w^4 - 14*w^3 + 3*w^2 + 22*w + 6], [101, 101, 3*w^5 - 2*w^4 - 18*w^3 + 6*w^2 + 22*w + 6], [101, 101, w^2 - w - 4], [101, 101, -3*w^5 + w^4 + 20*w^3 - 2*w^2 - 29*w - 7], [109, 109, w^4 - w^3 - 5*w^2 + 4*w + 5], [121, 11, w^5 - 7*w^3 - 2*w^2 + 10*w + 6], [131, 131, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 4*w - 3], [131, 131, -2*w^5 + w^4 + 13*w^3 - 3*w^2 - 20*w - 6], [131, 131, 3*w^5 - w^4 - 21*w^3 + w^2 + 35*w + 11], [131, 131, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w + 2], [139, 139, -w^5 + 6*w^3 + w^2 - 8*w - 1], [151, 151, -w^3 + w^2 + 4*w - 2], [179, 179, w^4 - 6*w^2 + 6], [181, 181, w^3 + w^2 - 3*w], [191, 191, 3*w^5 - w^4 - 20*w^3 + 2*w^2 + 28*w + 5], [199, 199, -2*w^5 + w^4 + 12*w^3 - 3*w^2 - 15*w - 2], [199, 199, 5*w^5 - w^4 - 34*w^3 - w^2 + 52*w + 15], [211, 211, 3*w^5 - w^4 - 19*w^3 + w^2 + 26*w + 9], [229, 229, -w^5 + w^4 + 7*w^3 - 3*w^2 - 11*w], [229, 229, -w^4 + 2*w^3 + 6*w^2 - 7*w - 6], [239, 239, -w^5 + w^4 + 7*w^3 - 3*w^2 - 13*w - 6], [239, 239, 3*w^5 - w^4 - 21*w^3 + w^2 + 35*w + 10], [239, 239, w^3 - w^2 - 4*w + 1], [239, 239, 5*w^5 - 2*w^4 - 34*w^3 + 2*w^2 + 53*w + 17], [241, 241, -w^5 + 7*w^3 + w^2 - 13*w - 4], [241, 241, -3*w^5 + w^4 + 21*w^3 - 35*w - 10], [241, 241, -w^5 + 7*w^3 - 11*w], [251, 251, 3*w^5 - w^4 - 21*w^3 + 35*w + 11], [251, 251, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 8*w - 3], [269, 269, -w^5 + 8*w^3 + 3*w^2 - 16*w - 8], [269, 269, -2*w^5 + w^4 + 13*w^3 - w^2 - 21*w - 9], [269, 269, 3*w^5 - 21*w^3 - 3*w^2 + 32*w + 12], [271, 271, -5*w^5 + 3*w^4 + 34*w^3 - 8*w^2 - 53*w - 14], [281, 281, -w^5 + 8*w^3 + 3*w^2 - 16*w - 11], [281, 281, 4*w^5 - w^4 - 29*w^3 + 49*w + 13], [281, 281, w^5 - w^4 - 7*w^3 + 3*w^2 + 11*w + 5], [281, 281, -5*w^5 + 3*w^4 + 33*w^3 - 8*w^2 - 50*w - 12], [311, 311, 2*w^5 - w^4 - 14*w^3 + w^2 + 21*w + 9], [311, 311, -3*w^5 + 20*w^3 + 3*w^2 - 30*w - 10], [311, 311, 5*w^5 - 2*w^4 - 35*w^3 + 2*w^2 + 57*w + 19], [331, 331, -w^5 - w^4 + 7*w^3 + 6*w^2 - 11*w - 7], [349, 349, -6*w^5 + 3*w^4 + 41*w^3 - 7*w^2 - 64*w - 17], [349, 349, 4*w^5 - w^4 - 29*w^3 - w^2 + 50*w + 17], [349, 349, 5*w^5 - w^4 - 34*w^3 - w^2 + 52*w + 17], [359, 359, w^5 - 6*w^3 - w^2 + 5*w + 3], [379, 379, -w^5 + w^4 + 8*w^3 - 3*w^2 - 17*w - 3], [379, 379, 2*w^5 - 14*w^3 - 4*w^2 + 22*w + 11], [379, 379, -5*w^5 + 3*w^4 + 34*w^3 - 8*w^2 - 53*w - 13], [379, 379, 4*w^5 - 2*w^4 - 27*w^3 + 3*w^2 + 43*w + 14], [389, 389, 4*w^5 - 2*w^4 - 28*w^3 + 4*w^2 + 44*w + 13], [401, 401, -2*w^5 + 2*w^4 + 13*w^3 - 6*w^2 - 20*w - 6], [409, 409, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w], [419, 419, -5*w^5 + w^4 + 34*w^3 + 2*w^2 - 53*w - 19], [419, 419, 2*w^5 - w^4 - 14*w^3 + 4*w^2 + 23*w + 3], [421, 421, -w^5 - w^4 + 8*w^3 + 6*w^2 - 13*w - 7], [421, 421, 2*w^5 - 2*w^4 - 13*w^3 + 8*w^2 + 18*w + 2], [421, 421, 4*w^5 - w^4 - 26*w^3 + 37*w + 11], [431, 431, 2*w^5 - w^4 - 13*w^3 + 2*w^2 + 20*w + 9], [431, 431, w^5 - 8*w^3 - w^2 + 14*w + 2], [449, 449, 4*w^5 - 2*w^4 - 26*w^3 + 4*w^2 + 38*w + 9], [449, 449, -3*w^5 + w^4 + 20*w^3 - 31*w - 14], [461, 461, 3*w^5 - 2*w^4 - 18*w^3 + 6*w^2 + 23*w + 4], [461, 461, -4*w^5 + 2*w^4 + 26*w^3 - 5*w^2 - 36*w - 9], [461, 461, -4*w^5 + 2*w^4 + 27*w^3 - 3*w^2 - 42*w - 15], [491, 491, -3*w^5 + w^4 + 20*w^3 - 29*w - 10], [491, 491, -5*w^5 + w^4 + 35*w^3 + 2*w^2 - 56*w - 18], [491, 491, 2*w^5 - 14*w^3 - 4*w^2 + 24*w + 13], [491, 491, -3*w^5 + w^4 + 21*w^3 - 2*w^2 - 33*w - 10], [499, 499, 6*w^5 - 3*w^4 - 40*w^3 + 6*w^2 + 61*w + 18], [499, 499, -4*w^5 + w^4 + 29*w^3 + w^2 - 49*w - 17], [499, 499, -w^5 - w^4 + 8*w^3 + 5*w^2 - 14*w - 4], [499, 499, -3*w^5 + w^4 + 19*w^3 - 2*w^2 - 26*w - 8], [509, 509, -4*w^5 + 2*w^4 + 27*w^3 - 5*w^2 - 43*w - 11], [509, 509, -2*w^5 + 16*w^3 + 2*w^2 - 30*w - 9], [521, 521, 3*w^5 - 21*w^3 - 3*w^2 + 34*w + 10], [521, 521, -3*w^5 + w^4 + 19*w^3 - 26*w - 9], [521, 521, 4*w^5 - w^4 - 27*w^3 + 40*w + 15], [521, 521, -3*w^5 + 22*w^3 + 3*w^2 - 37*w - 11], [541, 541, -4*w^5 + w^4 + 29*w^3 + w^2 - 49*w - 18], [569, 569, -6*w^5 + 2*w^4 + 40*w^3 - 2*w^2 - 60*w - 19], [569, 569, -w^5 + w^4 + 7*w^3 - 2*w^2 - 12*w - 6], [571, 571, 4*w^5 - w^4 - 28*w^3 + 43*w + 14], [571, 571, -4*w^5 + 2*w^4 + 28*w^3 - 4*w^2 - 45*w - 14], [599, 599, -w^5 + w^4 + 7*w^3 - 2*w^2 - 13*w - 7], [601, 601, 4*w^5 - w^4 - 26*w^3 + 38*w + 13], [601, 601, 4*w^5 - 2*w^4 - 28*w^3 + 5*w^2 + 44*w + 11], [601, 601, 3*w^5 - w^4 - 21*w^3 + 2*w^2 + 35*w + 9], [619, 619, 4*w^5 - 2*w^4 - 27*w^3 + 4*w^2 + 40*w + 13], [619, 619, -4*w^5 + w^4 + 26*w^3 - 36*w - 12], [619, 619, w^4 - w^3 - 6*w^2 + 4*w + 4], [631, 631, -3*w^5 + w^4 + 20*w^3 - 30*w - 9], [631, 631, -5*w^5 + 2*w^4 + 35*w^3 - 3*w^2 - 57*w - 15], [641, 641, 5*w^5 - w^4 - 34*w^3 - w^2 + 52*w + 16], [641, 641, -5*w^5 + 2*w^4 + 34*w^3 - 2*w^2 - 52*w - 18], [659, 659, 3*w^5 - w^4 - 20*w^3 + w^2 + 32*w + 9], [659, 659, -4*w^5 + w^4 + 27*w^3 - w^2 - 40*w - 11], [661, 661, 4*w^5 - w^4 - 26*w^3 + 38*w + 12], [661, 661, -w^5 + 6*w^3 - 8*w + 1], [691, 691, 3*w^5 - 20*w^3 - 3*w^2 + 29*w + 9], [691, 691, w^5 + w^4 - 8*w^3 - 7*w^2 + 14*w + 9], [691, 691, -w^5 + w^4 + 6*w^3 - 3*w^2 - 7*w + 2], [701, 701, -2*w^5 + w^4 + 13*w^3 - 2*w^2 - 19*w - 9], [701, 701, 5*w^5 - 2*w^4 - 33*w^3 + 3*w^2 + 48*w + 14], [701, 701, w^4 + 3*w^3 - 3*w^2 - 10*w - 1], [701, 701, -2*w^5 + w^4 + 14*w^3 - 3*w^2 - 24*w - 3], [709, 709, -2*w^5 + 2*w^4 + 14*w^3 - 7*w^2 - 22*w - 5], [709, 709, -3*w^5 + 21*w^3 + 4*w^2 - 34*w - 11], [719, 719, 5*w^5 - 2*w^4 - 35*w^3 + 3*w^2 + 58*w + 17], [719, 719, 4*w^5 - 2*w^4 - 26*w^3 + 4*w^2 + 37*w + 11], [719, 719, 4*w^5 - 2*w^4 - 27*w^3 + 4*w^2 + 40*w + 12], [719, 719, 2*w^5 - 16*w^3 - 3*w^2 + 30*w + 10], [739, 739, -5*w^5 + w^4 + 34*w^3 + w^2 - 51*w - 17], [751, 751, -w^5 - w^4 + 9*w^3 + 7*w^2 - 18*w - 10], [761, 761, 4*w^5 - 2*w^4 - 28*w^3 + 3*w^2 + 46*w + 16], [761, 761, -4*w^5 + 2*w^4 + 27*w^3 - 5*w^2 - 43*w - 12], [769, 769, -4*w^5 + w^4 + 28*w^3 - 44*w - 12], [769, 769, -w^4 + w^3 + 3*w^2 - 3*w + 1], [809, 809, 2*w^5 - 2*w^4 - 13*w^3 + 7*w^2 + 20*w + 1], [811, 811, -2*w^5 + 15*w^3 + 2*w^2 - 24*w - 8], [811, 811, -w^5 + 9*w^3 + w^2 - 18*w - 6], [821, 821, -w^5 + 8*w^3 + 3*w^2 - 17*w - 9], [821, 821, w^5 + w^4 - 7*w^3 - 6*w^2 + 11*w + 8], [821, 821, 4*w^5 - 2*w^4 - 28*w^3 + 4*w^2 + 46*w + 11], [821, 821, 5*w^5 - 2*w^4 - 34*w^3 + 4*w^2 + 51*w + 13], [829, 829, -2*w^5 + w^4 + 15*w^3 - w^2 - 28*w - 10], [829, 829, -6*w^5 + 3*w^4 + 42*w^3 - 7*w^2 - 68*w - 21], [829, 829, 4*w^5 - 2*w^4 - 26*w^3 + 5*w^2 + 38*w + 12], [829, 829, w^5 - 7*w^3 - 3*w^2 + 11*w + 6], [839, 839, 3*w^5 - 2*w^4 - 20*w^3 + 5*w^2 + 32*w + 11], [839, 839, -w^4 + w^3 + 6*w^2 - 3*w - 4], [841, 29, -3*w^5 + w^4 + 22*w^3 - 38*w - 11], [911, 911, 3*w^5 - w^4 - 21*w^3 + 36*w + 12], [919, 919, -w^5 - w^4 + 7*w^3 + 6*w^2 - 12*w - 5], [941, 941, -w^5 + w^4 + 7*w^3 - 5*w^2 - 11*w], [941, 941, 4*w^5 - w^4 - 29*w^3 + 48*w + 15], [971, 971, -4*w^5 + w^4 + 27*w^3 + w^2 - 41*w - 13], [991, 991, w^5 - w^4 - 8*w^3 + 3*w^2 + 16*w + 2], [991, 991, -4*w^5 + 3*w^4 + 27*w^3 - 9*w^2 - 42*w - 10]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [-1, -1, -2, 9, 10, -2, -10, 5, 5, 3, -13, -5, 3, 3, 0, 8, 5, 15, 0, 5, 0, -15, 2, -2, -2, 10, -7, 7, 2, 8, 8, 5, -13, -5, 12, -7, -15, 5, -2, -20, 10, 15, 0, 15, 15, -12, -23, -17, 28, 27, -20, -10, 10, 22, 3, 2, -8, -18, -12, 18, -13, 8, 10, -15, 0, -20, -20, 30, -25, 25, 10, 32, 10, 30, 25, -2, 38, -8, -3, -33, -35, -25, 18, -13, 38, -17, -17, -18, -2, 15, 20, 20, -20, 15, -30, -2, -23, 22, 22, -17, 0, 0, 23, 22, 40, 28, -2, 2, 25, -10, -40, -32, 2, 42, -3, -20, -15, -22, 12, -12, 8, -3, 7, 27, 47, -47, 15, -20, -30, 0, 30, 15, -20, 2, -13, 22, -30, -20, -40, 2, 28, -2, 38, 37, 13, 30, 30, -15, 30, -30, -40, 57, 12, 45, 23, -18, -33, 22, 13]; 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;