/* 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, 1, 6, -2, -6, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [5, 5, -w^4 + w^3 + 4*w^2 - 2*w - 1], [7, 7, w^5 - w^4 - 5*w^3 + 3*w^2 + 4*w - 3], [19, 19, w^5 - w^4 - 5*w^3 + 2*w^2 + 4*w], [23, 23, -w^2 + w + 2], [25, 5, w^5 + w^4 - 6*w^3 - 7*w^2 + 4*w + 2], [41, 41, 2*w^5 - w^4 - 10*w^3 + 6*w - 1], [47, 47, w^3 - w^2 - 4*w], [53, 53, w^5 - 5*w^3 - 3*w^2 + w + 3], [59, 59, 2*w^5 - 11*w^3 - 4*w^2 + 8*w], [61, 61, w^2 - 2*w - 2], [64, 2, -2], [67, 67, -w^4 + 6*w^2 + w - 3], [67, 67, -2*w^4 + w^3 + 10*w^2 - 6], [73, 73, w^5 - 5*w^3 - 2*w^2 + 2*w - 2], [73, 73, -w^5 - w^4 + 7*w^3 + 6*w^2 - 8*w - 2], [79, 79, w^5 - w^4 - 5*w^3 + 3*w^2 + 5*w - 1], [83, 83, 2*w^5 - 11*w^3 - 4*w^2 + 6*w], [89, 89, -2*w^5 + 11*w^3 + 4*w^2 - 7*w - 2], [97, 97, w^5 - w^4 - 5*w^3 + 3*w^2 + 2*w - 2], [97, 97, w^5 - w^4 - 6*w^3 + 3*w^2 + 7*w - 1], [97, 97, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 5*w + 4], [101, 101, w^5 - 6*w^3 - 3*w^2 + 6*w + 2], [103, 103, -w^5 - w^4 + 5*w^3 + 8*w^2 - 5], [107, 107, 2*w^4 - w^3 - 9*w^2 + w + 2], [109, 109, 2*w^5 - 10*w^3 - 5*w^2 + 4*w + 1], [113, 113, -w^4 + 6*w^2 + 2*w - 3], [125, 5, 2*w^5 - 10*w^3 - 6*w^2 + 5*w + 6], [127, 127, 2*w^5 - w^4 - 11*w^3 + 10*w], [131, 131, 2*w^5 - w^4 - 11*w^3 + 9*w + 1], [137, 137, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 9*w + 5], [139, 139, -w^5 - 2*w^4 + 7*w^3 + 11*w^2 - 5*w - 5], [157, 157, 2*w^5 - 10*w^3 - 5*w^2 + 5*w + 1], [167, 167, -2*w^5 + 10*w^3 + 5*w^2 - 3*w + 1], [173, 173, 4*w^5 + 2*w^4 - 22*w^3 - 19*w^2 + 10*w + 5], [179, 179, 2*w^5 - w^4 - 11*w^3 + 10*w + 1], [179, 179, 2*w^5 - 11*w^3 - 5*w^2 + 8*w + 1], [181, 181, -4*w^5 - w^4 + 22*w^3 + 14*w^2 - 12*w - 4], [191, 191, w^4 + w^3 - 6*w^2 - 4*w + 2], [191, 191, -3*w^5 + 18*w^3 + 5*w^2 - 15*w - 1], [193, 193, -w^4 + w^3 + 6*w^2 - 2*w - 7], [197, 197, -3*w^5 + 2*w^4 + 16*w^3 - 3*w^2 - 12*w + 1], [197, 197, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 7*w - 5], [197, 197, -w^5 + w^4 + 6*w^3 - 3*w^2 - 9*w + 1], [211, 211, -2*w^5 + w^4 + 10*w^3 - w^2 - 4*w + 4], [223, 223, 2*w^5 - 10*w^3 - 6*w^2 + 5*w + 4], [227, 227, w^5 - 7*w^3 - w^2 + 9*w], [227, 227, -2*w^5 + w^4 + 9*w^3 - 2*w + 1], [229, 229, w^5 + w^4 - 5*w^3 - 6*w^2 - 2*w - 2], [233, 233, -2*w^5 - w^4 + 12*w^3 + 8*w^2 - 8*w - 4], [269, 269, 3*w^5 - 16*w^3 - 6*w^2 + 8*w], [269, 269, -2*w^5 + 12*w^3 + 3*w^2 - 12*w - 1], [277, 277, 3*w^5 - w^4 - 17*w^3 - w^2 + 16*w], [289, 17, -2*w^5 - w^4 + 12*w^3 + 8*w^2 - 10*w - 3], [307, 307, w^4 - 5*w^2 - 2*w - 1], [307, 307, 3*w^5 - w^4 - 16*w^3 - 3*w^2 + 12*w + 3], [307, 307, -w^4 - w^3 + 5*w^2 + 6*w - 2], [311, 311, -w^4 - w^3 + 7*w^2 + 4*w - 4], [311, 311, 3*w^5 - w^4 - 17*w^3 - 2*w^2 + 16*w + 2], [313, 313, -w^5 + w^4 + 6*w^3 - 3*w^2 - 7*w], [317, 317, 2*w^5 + w^4 - 12*w^3 - 8*w^2 + 7*w], [331, 331, 2*w^5 - 2*w^4 - 10*w^3 + 5*w^2 + 8*w - 5], [337, 337, w^5 - 6*w^3 - 3*w^2 + 6*w], [347, 347, -2*w^5 + 2*w^4 + 10*w^3 - 6*w^2 - 7*w + 2], [347, 347, 3*w^5 + w^4 - 17*w^3 - 11*w^2 + 9*w + 2], [349, 349, -2*w^5 + 11*w^3 + 5*w^2 - 7*w - 6], [367, 367, -w^5 - 2*w^4 + 5*w^3 + 13*w^2 + 2*w - 5], [367, 367, 2*w^5 + 2*w^4 - 10*w^3 - 16*w^2 - w + 6], [373, 373, 3*w^5 - w^4 - 17*w^3 - w^2 + 15*w + 1], [373, 373, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 4], [379, 379, 2*w^5 + w^4 - 11*w^3 - 10*w^2 + 5*w + 7], [389, 389, w^2 - 5], [389, 389, w^5 - 7*w^3 + 8*w], [397, 397, -2*w^5 - w^4 + 12*w^3 + 9*w^2 - 7*w - 5], [397, 397, -3*w^5 - w^4 + 18*w^3 + 10*w^2 - 14*w - 2], [397, 397, -2*w^5 - 2*w^4 + 12*w^3 + 13*w^2 - 6*w - 4], [419, 419, 2*w^5 + 2*w^4 - 11*w^3 - 14*w^2 + 2*w + 5], [419, 419, -w^5 + 4*w^3 + 3*w^2 + w - 2], [433, 433, -w^5 + 5*w^3 + 4*w^2 - w - 3], [433, 433, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 6*w - 4], [439, 439, 4*w^5 + 2*w^4 - 22*w^3 - 18*w^2 + 9*w + 4], [439, 439, w^5 + w^4 - 5*w^3 - 9*w^2 + 6], [439, 439, -3*w^5 + 2*w^4 + 16*w^3 - 3*w^2 - 13*w + 2], [443, 443, -3*w^5 + w^4 + 15*w^3 + 3*w^2 - 8*w - 2], [449, 449, -w^5 + w^4 + 5*w^3 - 4*w^2 - 2*w + 3], [449, 449, 3*w^5 - 16*w^3 - 8*w^2 + 9*w + 4], [449, 449, w^4 + w^3 - 6*w^2 - 6*w + 5], [449, 449, -w^5 + w^4 + 6*w^3 - 5*w^2 - 7*w + 4], [457, 457, -2*w^5 + 2*w^4 + 10*w^3 - 5*w^2 - 6*w + 4], [457, 457, w^5 - 3*w^4 - 4*w^3 + 12*w^2 + 5*w - 4], [461, 461, -2*w^4 + w^3 + 10*w^2 + w - 5], [463, 463, -w^5 + w^4 + 6*w^3 - 4*w^2 - 5*w + 4], [487, 487, -3*w^4 + 2*w^3 + 14*w^2 - w - 7], [487, 487, 3*w^5 + w^4 - 18*w^3 - 10*w^2 + 14*w + 3], [491, 491, -3*w^5 + w^4 + 16*w^3 + w^2 - 12*w + 2], [491, 491, 3*w^5 - w^4 - 16*w^3 - 2*w^2 + 12*w + 3], [499, 499, 4*w^5 - 22*w^3 - 9*w^2 + 15*w + 4], [499, 499, w^4 + w^3 - 7*w^2 - 6*w + 6], [523, 523, w^4 - 2*w^3 - 4*w^2 + 5*w + 2], [529, 23, -w^5 + 7*w^3 + w^2 - 10*w - 1], [541, 541, w^5 + 2*w^4 - 8*w^3 - 10*w^2 + 9*w + 3], [557, 557, w^5 - w^4 - 5*w^3 + 3*w^2 + 5*w - 5], [557, 557, 4*w^5 - 22*w^3 - 8*w^2 + 14*w + 1], [563, 563, 2*w^5 - w^4 - 12*w^3 + 2*w^2 + 13*w - 1], [563, 563, -w^5 - 2*w^4 + 6*w^3 + 13*w^2 - 2*w - 6], [563, 563, -w^5 + w^4 + 4*w^3 - 2*w^2 + w + 3], [569, 569, -3*w^5 + w^4 + 16*w^3 + w^2 - 9*w + 1], [599, 599, -w^4 - w^3 + 7*w^2 + 6*w - 5], [607, 607, -w^5 + 6*w^3 + 3*w^2 - 7*w - 1], [613, 613, -w^5 - 2*w^4 + 7*w^3 + 12*w^2 - 4*w - 7], [619, 619, 4*w^5 - w^4 - 22*w^3 - 3*w^2 + 16*w], [653, 653, 2*w^5 - 2*w^4 - 10*w^3 + 4*w^2 + 7*w], [653, 653, -2*w^5 + w^4 + 12*w^3 - w^2 - 14*w + 1], [659, 659, -2*w^5 - w^4 + 11*w^3 + 11*w^2 - 5*w - 7], [661, 661, 2*w^5 - 11*w^3 - 3*w^2 + 7*w], [661, 661, 2*w^5 - w^4 - 11*w^3 + 11*w + 2], [673, 673, -w^4 - w^3 + 5*w^2 + 7*w - 1], [677, 677, 3*w^5 - 16*w^3 - 7*w^2 + 10*w + 3], [677, 677, w^5 + 3*w^4 - 6*w^3 - 18*w^2 + 12], [691, 691, w^5 - w^4 - 4*w^3 + 2*w^2 - 4], [709, 709, -w^4 + w^3 + 3*w^2 - 2*w + 2], [709, 709, -2*w^5 - w^4 + 11*w^3 + 9*w^2 - 6*w - 1], [719, 719, 2*w^5 + 2*w^4 - 12*w^3 - 14*w^2 + 8*w + 5], [729, 3, -3], [739, 739, 4*w^5 - w^4 - 21*w^3 - 5*w^2 + 12*w + 3], [743, 743, -2*w^5 - w^4 + 10*w^3 + 11*w^2 - w - 6], [751, 751, -3*w^5 + w^4 + 17*w^3 + 2*w^2 - 15*w - 2], [751, 751, -2*w^5 + 3*w^4 + 10*w^3 - 11*w^2 - 10*w + 6], [757, 757, w^5 - 3*w^4 - 5*w^3 + 13*w^2 + 7*w - 6], [761, 761, -w^5 + w^4 + 6*w^3 - 3*w^2 - 6*w - 1], [787, 787, -3*w^5 - w^4 + 18*w^3 + 10*w^2 - 15*w - 2], [797, 797, 2*w^5 - 2*w^4 - 11*w^3 + 6*w^2 + 12*w - 1], [797, 797, w^2 - 2*w - 5], [809, 809, 2*w^5 - w^4 - 12*w^3 + 3*w^2 + 12*w - 3], [827, 827, 4*w^5 - 22*w^3 - 8*w^2 + 15*w + 1], [839, 839, 3*w^5 - w^4 - 16*w^3 - 3*w^2 + 10*w + 2], [839, 839, -w^5 + w^4 + 6*w^3 - 4*w^2 - 8*w + 1], [841, 29, -2*w^5 - w^4 + 11*w^3 + 9*w^2 - 7*w - 3], [853, 853, w^5 - 7*w^3 - w^2 + 7*w + 1], [857, 857, -2*w^5 + w^4 + 10*w^3 - w^2 - 6*w + 4], [863, 863, 3*w^5 - w^4 - 17*w^3 - 2*w^2 + 16*w], [877, 877, 2*w^5 - 10*w^3 - 5*w^2 + 3*w + 4], [887, 887, 2*w^5 + w^4 - 13*w^3 - 8*w^2 + 13*w + 5], [887, 887, 3*w^5 - 2*w^4 - 15*w^3 + 3*w^2 + 8*w - 4], [907, 907, 2*w^5 - w^4 - 13*w^3 + 2*w^2 + 17*w], [907, 907, -2*w^5 + 2*w^4 + 10*w^3 - 4*w^2 - 9*w + 1], [919, 919, -3*w^5 + 17*w^3 + 5*w^2 - 11*w], [929, 929, -3*w^5 + 17*w^3 + 6*w^2 - 11*w], [941, 941, 2*w^5 - w^4 - 11*w^3 - w^2 + 9*w + 2], [947, 947, -w^4 + 6*w^2 + 4*w - 4], [947, 947, -2*w^5 + w^4 + 11*w^3 - 11*w - 3], [947, 947, -4*w^5 + w^4 + 22*w^3 + 3*w^2 - 16*w - 1], [953, 953, -w^5 - 2*w^4 + 6*w^3 + 13*w^2 - 3*w - 8], [953, 953, 2*w^5 + 2*w^4 - 13*w^3 - 14*w^2 + 10*w + 6], [961, 31, 2*w^5 + w^4 - 12*w^3 - 10*w^2 + 9*w + 4], [967, 967, 2*w^5 - w^4 - 11*w^3 + w^2 + 11*w + 1], [967, 967, 2*w^5 - 2*w^4 - 11*w^3 + 6*w^2 + 13*w - 3], [983, 983, -4*w^5 + w^4 + 22*w^3 + 4*w^2 - 18*w - 2], [991, 991, 3*w^5 - 16*w^3 - 7*w^2 + 7*w + 2], [991, 991, -2*w^5 - w^4 + 12*w^3 + 8*w^2 - 11*w - 2]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [0, -3, -7, 4, 5, 8, -9, 6, 8, -6, 5, -12, 2, -6, 11, -6, 0, -8, -8, 16, -9, 6, 2, -15, -4, -10, -6, -8, -15, -2, 11, -18, -6, 9, -24, 4, 13, -7, 9, -20, 5, 3, -4, -20, -1, 3, 14, 9, -3, 21, 0, -22, -22, 16, -4, -29, -21, -3, -1, 4, -14, -29, -16, -27, -33, -37, 9, -25, 10, -24, 22, -15, 10, -20, 19, -14, 30, 2, -21, 26, -28, 34, 17, -32, 13, 24, 14, -18, -22, -18, -7, -8, -2, -30, 22, -36, -4, -16, 26, -10, 36, -37, -35, -29, -18, -4, -37, 30, 44, 14, 0, 30, -35, -31, -35, -9, -6, 18, 2, 20, 32, 24, 10, 10, -5, 4, 43, 17, 22, -12, 34, -3, 0, -54, 0, -32, -34, 42, -30, -11, 31, 32, 38, 8, 22, 45, 38, -59, -8, 21, 18, -20, 15, 46, -32, -16, -48, 47, 42]; 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;