/* 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![-2, 1, 8, -1, -6, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -w], [13, 13, -w^2 + 3], [17, 17, -w^5 + w^4 + 5*w^3 - 3*w^2 - 5*w + 1], [23, 23, w^4 - w^3 - 4*w^2 + 2*w + 1], [31, 31, w^5 - w^4 - 6*w^3 + 5*w^2 + 7*w - 5], [32, 2, w^5 - 6*w^3 - w^2 + 8*w + 1], [43, 43, w^4 - 5*w^2 + 3], [47, 47, -w^4 + w^3 + 5*w^2 - 2*w - 5], [49, 7, -w^5 + w^4 + 5*w^3 - 3*w^2 - 4*w - 1], [53, 53, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 5*w - 7], [59, 59, w^5 - w^4 - 6*w^3 + 4*w^2 + 7*w - 3], [71, 71, w^5 - w^4 - 5*w^3 + 4*w^2 + 4*w - 5], [79, 79, w^3 - w^2 - 4*w + 1], [83, 83, w^5 - 6*w^3 - w^2 + 7*w - 1], [83, 83, 2*w^5 - w^4 - 11*w^3 + 2*w^2 + 12*w + 1], [89, 89, -w^5 + w^4 + 4*w^3 - 2*w^2 - w - 1], [89, 89, -w^5 + w^4 + 5*w^3 - 3*w^2 - 4*w + 3], [89, 89, w^5 + w^4 - 6*w^3 - 6*w^2 + 6*w + 3], [89, 89, 2*w^4 - w^3 - 9*w^2 + w + 5], [101, 101, w^5 - 5*w^3 - w^2 + 5*w - 1], [107, 107, w^4 - w^3 - 3*w^2 + 2*w - 1], [107, 107, -w^5 + w^4 + 6*w^3 - 3*w^2 - 7*w + 1], [107, 107, -w^5 + 4*w^3 + 2*w^2 + 1], [113, 113, 2*w^5 - w^4 - 10*w^3 + 2*w^2 + 8*w - 3], [121, 11, 2*w^5 - w^4 - 10*w^3 + 2*w^2 + 8*w - 1], [125, 5, -w^3 + 4*w - 1], [125, 5, -2*w^5 + w^4 + 10*w^3 - 3*w^2 - 9*w + 3], [131, 131, w^5 + 2*w^4 - 7*w^3 - 10*w^2 + 9*w + 5], [137, 137, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 7], [137, 137, w^4 - 3*w^2 - 2*w + 1], [139, 139, -w^5 + w^4 + 5*w^3 - 4*w^2 - 5*w + 1], [139, 139, w^5 - w^4 - 5*w^3 + 4*w^2 + 3*w - 3], [149, 149, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 - w + 7], [151, 151, w^4 - 6*w^2 - w + 5], [163, 163, -w^5 + w^4 + 4*w^3 - 3*w^2 - w + 3], [167, 167, -w^5 + w^4 + 5*w^3 - 2*w^2 - 6*w + 1], [169, 13, -2*w^5 + 12*w^3 + w^2 - 13*w + 1], [179, 179, -w^5 + 6*w^3 + 2*w^2 - 9*w - 3], [179, 179, -w^3 + 5*w - 1], [191, 191, w^4 - 3*w^2 - 1], [193, 193, -w^5 - 3*w^4 + 6*w^3 + 17*w^2 - 5*w - 15], [197, 197, 2*w^5 - 11*w^3 - 2*w^2 + 10*w + 1], [197, 197, w^5 - 5*w^3 - 2*w^2 + 3*w - 1], [227, 227, w^5 - w^4 - 5*w^3 + 5*w^2 + 4*w - 5], [229, 229, -2*w^4 + w^3 + 10*w^2 - w - 9], [241, 241, 3*w^5 - 2*w^4 - 16*w^3 + 6*w^2 + 17*w - 3], [263, 263, -2*w^5 + 11*w^3 + w^2 - 11*w - 1], [269, 269, -w^5 + w^4 + 5*w^3 - 4*w^2 - 7*w + 3], [277, 277, 2*w^5 - w^4 - 10*w^3 + w^2 + 7*w + 1], [277, 277, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 11*w - 5], [283, 283, -w^4 + w^3 + 4*w^2 - 3], [289, 17, -w^5 - w^4 + 7*w^3 + 4*w^2 - 10*w - 1], [293, 293, -3*w^5 + 16*w^3 + 3*w^2 - 15*w - 1], [293, 293, -2*w^5 - w^4 + 10*w^3 + 7*w^2 - 6*w - 5], [307, 307, -2*w^5 + w^4 + 9*w^3 - w^2 - 4*w - 1], [311, 311, w^4 - w^3 - 5*w^2 + 4*w + 1], [311, 311, -3*w^5 + 17*w^3 + 4*w^2 - 18*w - 7], [311, 311, -w^5 - w^4 + 5*w^3 + 6*w^2 - 2*w - 5], [311, 311, 2*w^4 - w^3 - 10*w^2 + w + 5], [313, 313, w^5 - 5*w^3 - 3*w^2 + 3*w + 3], [313, 313, -w^5 - w^4 + 5*w^3 + 7*w^2 - 3*w - 3], [317, 317, -w^5 + 2*w^4 + 3*w^3 - 8*w^2 + w + 5], [337, 337, 3*w^5 + w^4 - 16*w^3 - 10*w^2 + 14*w + 11], [337, 337, -w^5 - 2*w^4 + 6*w^3 + 10*w^2 - 6*w - 7], [337, 337, 2*w^5 + w^4 - 11*w^3 - 7*w^2 + 10*w + 3], [337, 337, w^4 - w^3 - 6*w^2 + 4*w + 7], [347, 347, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 7*w + 3], [347, 347, -w^5 - w^4 + 5*w^3 + 7*w^2 - 2*w - 7], [347, 347, -w^5 - w^4 + 6*w^3 + 7*w^2 - 5*w - 7], [349, 349, 3*w^5 - w^4 - 17*w^3 + 3*w^2 + 19*w - 3], [349, 349, w^4 + w^3 - 6*w^2 - 4*w + 7], [359, 359, -2*w^5 + w^4 + 11*w^3 - 3*w^2 - 13*w + 1], [373, 373, -2*w^5 + 10*w^3 + 2*w^2 - 9*w - 1], [379, 379, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 - w + 9], [389, 389, -2*w^5 + w^4 + 11*w^3 - 2*w^2 - 13*w - 1], [397, 397, w^5 - 3*w^4 - 5*w^3 + 14*w^2 + 7*w - 11], [419, 419, w^5 - 6*w^3 + w^2 + 7*w - 1], [419, 419, -w^5 + w^4 + 3*w^3 - 2*w^2 + w - 1], [433, 433, -w^5 + w^4 + 4*w^3 - 4*w^2 - w + 5], [433, 433, w^5 + w^4 - 7*w^3 - 4*w^2 + 9*w - 1], [439, 439, w^4 - 2*w^3 - 4*w^2 + 7*w + 3], [443, 443, -w^3 + w^2 + 3*w - 5], [443, 443, w^5 - w^4 - 5*w^3 + 2*w^2 + 5*w + 3], [449, 449, 2*w^5 - 2*w^4 - 11*w^3 + 8*w^2 + 12*w - 5], [457, 457, -2*w^5 - 2*w^4 + 11*w^3 + 13*w^2 - 10*w - 11], [461, 461, w^5 + 2*w^4 - 6*w^3 - 10*w^2 + 5*w + 7], [479, 479, -2*w^5 + 10*w^3 + w^2 - 6*w + 1], [479, 479, w^5 - w^4 - 6*w^3 + 3*w^2 + 9*w - 1], [491, 491, w^5 - w^4 - 4*w^3 + w^2 + w + 3], [499, 499, -w^3 + 5*w - 3], [499, 499, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 6*w - 1], [509, 509, -2*w^5 + 2*w^4 + 10*w^3 - 9*w^2 - 9*w + 11], [521, 521, -w^5 + 4*w^3 + 2*w^2 - w - 3], [521, 521, 2*w^5 + 2*w^4 - 11*w^3 - 13*w^2 + 9*w + 13], [521, 521, -w^4 + 2*w^3 + 4*w^2 - 5*w - 3], [523, 523, -w^5 + w^4 + 6*w^3 - 5*w^2 - 8*w + 3], [523, 523, w^5 + w^4 - 5*w^3 - 8*w^2 + 3*w + 7], [541, 541, w^5 + 3*w^4 - 7*w^3 - 16*w^2 + 8*w + 13], [547, 547, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 7*w - 3], [557, 557, 3*w^5 - w^4 - 15*w^3 + w^2 + 11*w - 1], [557, 557, 2*w^5 - w^4 - 11*w^3 + 2*w^2 + 11*w + 1], [563, 563, w^5 - 3*w^4 - 4*w^3 + 13*w^2 + 3*w - 9], [569, 569, -w^5 - w^4 + 5*w^3 + 6*w^2 - 4*w - 1], [571, 571, -3*w^5 + 2*w^4 + 15*w^3 - 7*w^2 - 10*w + 3], [577, 577, 2*w^5 - 11*w^3 - 2*w^2 + 9*w + 3], [587, 587, w^4 - 6*w^2 - 2*w + 7], [599, 599, -2*w^5 + 2*w^4 + 10*w^3 - 7*w^2 - 10*w + 3], [599, 599, w^5 - 2*w^4 - 4*w^3 + 9*w^2 + 3*w - 7], [601, 601, 2*w^2 - 5], [607, 607, -w^5 - w^4 + 4*w^3 + 7*w^2 - 5], [613, 613, -w^5 + 5*w^3 + w^2 - 3*w + 3], [617, 617, -2*w^4 + 11*w^2 + 2*w - 9], [619, 619, -2*w^5 + 10*w^3 + w^2 - 7*w + 1], [619, 619, -2*w^5 + 4*w^4 + 8*w^3 - 15*w^2 - 5*w + 7], [643, 643, -2*w^5 - w^4 + 12*w^3 + 6*w^2 - 12*w - 3], [647, 647, 3*w^5 - w^4 - 18*w^3 + w^2 + 23*w + 3], [647, 647, 2*w^5 + w^4 - 10*w^3 - 9*w^2 + 9*w + 9], [653, 653, -w^4 + 7*w^2 + w - 7], [673, 673, -2*w^4 + w^3 + 8*w^2 - w - 5], [673, 673, -2*w^5 + 11*w^3 + w^2 - 9*w + 1], [677, 677, 4*w^5 - w^4 - 20*w^3 + w^2 + 15*w - 3], [683, 683, 3*w^5 - 2*w^4 - 14*w^3 + 6*w^2 + 11*w - 5], [683, 683, 2*w^5 - w^4 - 11*w^3 + 3*w^2 + 10*w - 3], [683, 683, -w^5 + 5*w^3 + w^2 - 4*w + 3], [691, 691, w^5 + w^4 - 6*w^3 - 6*w^2 + 8*w + 3], [709, 709, -2*w^5 + 2*w^4 + 11*w^3 - 7*w^2 - 13*w + 3], [719, 719, 2*w^5 - 2*w^4 - 10*w^3 + 8*w^2 + 9*w - 9], [719, 719, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 12*w + 3], [727, 727, w^5 - w^4 - 6*w^3 + 4*w^2 + 7*w - 1], [729, 3, -3], [739, 739, 2*w^4 - w^3 - 9*w^2 + 5], [739, 739, -w^5 + w^4 + 4*w^3 - 3*w^2 - 3*w - 1], [751, 751, -2*w^4 + w^3 + 9*w^2 - 3*w - 3], [761, 761, -w^5 - 2*w^4 + 7*w^3 + 12*w^2 - 9*w - 13], [773, 773, -w^5 + 5*w^3 + 2*w^2 - 2*w - 3], [787, 787, 2*w^5 - 12*w^3 - 2*w^2 + 13*w + 3], [787, 787, -2*w^5 + w^4 + 10*w^3 - 3*w^2 - 10*w + 1], [797, 797, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 7*w + 3], [811, 811, -2*w^5 + 12*w^3 + w^2 - 15*w - 1], [821, 821, 4*w^5 + w^4 - 22*w^3 - 8*w^2 + 21*w + 3], [821, 821, w^5 - 2*w^4 - 3*w^3 + 8*w^2 - 3*w - 3], [821, 821, -w^5 + 7*w^3 - 9*w - 1], [827, 827, w^5 - 7*w^3 - w^2 + 10*w + 1], [829, 829, -2*w^4 + 10*w^2 + w - 7], [839, 839, -w^5 + 2*w^4 + 6*w^3 - 10*w^2 - 8*w + 9], [853, 853, -w^5 + w^4 + 3*w^3 - 3*w^2 + 2*w + 1], [859, 859, -w^5 + 6*w^3 - 9*w + 1], [859, 859, -2*w^5 + 3*w^4 + 8*w^3 - 10*w^2 - 6*w + 5], [859, 859, w^5 - 2*w^4 - 4*w^3 + 8*w^2 + 2*w - 7], [859, 859, 2*w^5 - 2*w^4 - 10*w^3 + 6*w^2 + 10*w - 3], [863, 863, 3*w^5 - 4*w^4 - 14*w^3 + 14*w^2 + 13*w - 7], [877, 877, w^3 + w^2 - 4*w - 1], [881, 881, -w^5 + w^4 + 4*w^3 - 4*w^2 - 2*w + 5], [881, 881, -2*w^5 + 10*w^3 + 4*w^2 - 7*w - 5], [907, 907, -w^4 + w^3 + 4*w^2 - w + 1], [937, 937, w^5 - 7*w^3 + w^2 + 11*w - 3], [941, 941, -2*w^5 + w^4 + 11*w^3 - w^2 - 12*w - 1], [947, 947, -w^5 + 5*w^3 + 3*w^2 - 5*w - 7], [953, 953, 2*w^5 - w^4 - 11*w^3 + 4*w^2 + 10*w - 7], [971, 971, w^5 - w^4 - 5*w^3 + 3*w^2 + 7*w + 1], [983, 983, -2*w^5 + 9*w^3 + 3*w^2 - 5*w - 3], [997, 997, 3*w^5 - 18*w^3 - 2*w^2 + 22*w + 1]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [1, -2, 1, 0, 8, 1, -4, 0, 10, 6, 4, -8, 0, 4, -4, 2, 10, -6, -6, 6, -4, 4, -12, 18, -6, 6, 14, -4, -6, 2, -20, 12, 22, 8, -12, -8, 10, -4, 20, 16, 2, -18, -2, 28, 14, -22, -24, -2, 14, -2, -4, -14, 14, 6, -20, -16, -24, 0, -32, -6, 26, -2, -22, -22, -22, 2, 12, -12, 20, 14, 30, 8, -10, 12, -10, -34, -12, -12, 26, -14, 24, 12, 4, 2, 18, -2, 16, 32, -36, 4, 20, 30, 2, 18, -6, 28, -4, 14, -28, -2, 30, -12, -6, 44, 2, 28, 8, -24, -22, 32, 38, -22, -20, 44, 20, -40, 0, -10, 2, 26, 6, -12, -36, -36, -28, 38, 0, -48, -8, 26, 12, 28, 40, 18, 6, -4, 52, 30, -20, -2, 22, -26, -12, -2, 40, 46, -20, -52, -4, -36, 0, 30, -46, -46, -12, -54, -18, -52, 42, 12, -24, 38]; 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;