/* 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![12, -9, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, w - 2], [3, 3, -w + 3], [3, 3, w - 1], [4, 2, w^2 + w - 7], [19, 19, w + 1], [23, 23, w^2 - 2*w - 1], [31, 31, -2*w^2 + 19], [31, 31, -w^2 + 5], [31, 31, -3*w + 5], [41, 41, w^2 + 2*w - 7], [43, 43, w^2 - 11], [47, 47, 3*w - 7], [53, 53, -3*w^2 - 6*w + 11], [59, 59, 2*w - 1], [67, 67, 2*w^2 + w - 19], [67, 67, 3*w^2 + 2*w - 25], [67, 67, w - 5], [73, 73, -4*w^2 - 3*w + 29], [73, 73, 2*w^2 - w - 11], [73, 73, w^2 + 2*w - 11], [79, 79, 2*w^2 + 2*w - 11], [83, 83, 2*w^2 - 13], [89, 89, -2*w + 7], [97, 97, -2*w^2 + 4*w - 1], [101, 101, -2*w - 5], [103, 103, 2*w^2 + w - 13], [107, 107, -2*w^2 - 2*w + 17], [109, 109, -4*w^2 - 2*w + 31], [113, 113, w^2 + 2*w - 1], [125, 5, -5], [127, 127, 2*w^2 + 3*w - 13], [137, 137, -2*w^2 + 7*w - 7], [139, 139, 2*w^2 + w - 7], [149, 149, -6*w^2 - 9*w + 31], [157, 157, w^2 - 4*w + 1], [163, 163, 2*w^2 + 5*w - 5], [173, 173, 5*w^2 + 4*w - 37], [179, 179, 2*w^2 + w - 11], [179, 179, 3*w^2 - 2*w - 17], [179, 179, -4*w^2 - w + 37], [191, 191, -6*w^2 - 9*w + 29], [193, 193, -2*w^2 - 4*w + 5], [223, 223, -2*w^2 + 2*w + 7], [227, 227, 6*w^2 + 4*w - 47], [233, 233, 2*w^2 + 4*w - 13], [233, 233, -4*w^2 - 4*w + 25], [233, 233, 4*w + 13], [239, 239, -4*w^2 - 8*w + 13], [239, 239, 2*w^2 + 2*w - 5], [239, 239, -w^2 + 8*w - 11], [241, 241, 3*w - 1], [251, 251, w^2 + 4*w - 19], [263, 263, -2*w^2 + 2*w + 25], [263, 263, -5*w^2 - 4*w + 35], [263, 263, 3*w^2 - 25], [269, 269, 6*w^2 + 3*w - 47], [269, 269, -w^2 - 1], [269, 269, 2*w^2 - 7], [277, 277, 2*w^2 - w - 5], [277, 277, -5*w^2 - 6*w + 31], [277, 277, 2*w^2 - 2*w - 11], [281, 281, 4*w^2 + 5*w - 25], [307, 307, 4*w^2 + 2*w - 37], [317, 317, -w - 7], [317, 317, 3*w^2 + 2*w - 19], [317, 317, -2*w^2 + 2*w + 5], [331, 331, 2*w^2 + 3*w - 19], [337, 337, w^2 - 4*w + 7], [343, 7, -7], [347, 347, 4*w^2 + 2*w - 29], [349, 349, 3*w^2 - 19], [361, 19, w^2 - 2*w - 7], [367, 367, 4*w^2 + 5*w - 19], [367, 367, -6*w + 11], [373, 373, -4*w^2 + w + 25], [379, 379, -5*w^2 - 10*w + 19], [383, 383, 4*w^2 + 6*w - 23], [401, 401, -3*w + 11], [401, 401, -3*w - 1], [401, 401, -3*w - 7], [409, 409, -w^2 - 4*w + 13], [419, 419, 2*w^2 - w - 17], [421, 421, -3*w^2 + 31], [431, 431, w^2 + 4*w - 1], [449, 449, -w^2 + 4*w + 1], [457, 457, -w^2 + 2*w + 13], [461, 461, 4*w^2 - 4*w - 19], [463, 463, 2*w^2 + 4*w - 23], [467, 467, 2*w^2 - 5*w + 5], [479, 479, 3*w + 11], [491, 491, 8*w^2 + 9*w - 49], [499, 499, -6*w^2 - 6*w + 41], [499, 499, 2*w^2 + 2*w - 23], [499, 499, 2*w^2 + 5*w - 29], [523, 523, -4*w^2 + w + 41], [529, 23, 6*w^2 + 7*w - 35], [547, 547, -2*w^2 + 6*w + 1], [557, 557, 3*w^2 + 2*w - 13], [563, 563, -w^2 + 4*w + 23], [569, 569, 10*w^2 + 8*w - 73], [587, 587, 2*w^2 - 2*w - 13], [593, 593, -6*w^2 - 8*w + 31], [601, 601, 4*w^2 + 5*w - 41], [601, 601, 10*w^2 + 16*w - 49], [601, 601, w^2 - 2*w + 5], [613, 613, -8*w^2 - 7*w + 55], [617, 617, -8*w^2 - 12*w + 37], [617, 617, 2*w^2 + 3*w - 25], [617, 617, 3*w^2 - 17], [641, 641, 4*w^2 - 35], [643, 643, 4*w^2 + 4*w - 37], [659, 659, -3*w^2 - 4*w + 23], [661, 661, -w^2 - 4*w - 5], [673, 673, 4*w^2 + 4*w - 31], [683, 683, 2*w^2 + 4*w - 19], [683, 683, 2*w^2 + w - 23], [683, 683, 3*w^2 + 6*w - 19], [701, 701, -6*w^2 - 12*w + 23], [701, 701, -2*w^2 + 7*w - 1], [701, 701, 3*w^2 + 2*w - 31], [709, 709, 5*w^2 + 4*w - 41], [719, 719, -6*w^2 - 2*w + 55], [719, 719, w^2 + 2*w - 17], [719, 719, 3*w^2 + 4*w - 29], [727, 727, -9*w^2 - 12*w + 49], [733, 733, 13*w^2 + 20*w - 65], [739, 739, 11*w^2 + 6*w - 91], [743, 743, -2*w^2 - 8*w + 35], [751, 751, 3*w^2 - 4*w - 13], [769, 769, 2*w^2 - 1], [797, 797, 2*w^2 - 8*w + 11], [811, 811, 4*w^2 - w - 29], [821, 821, 2*w^2 + 5*w - 17], [827, 827, -3*w^2 - 6*w + 7], [827, 827, -11*w^2 - 8*w + 83], [827, 827, -5*w^2 - 6*w + 25], [829, 829, -4*w^2 + 41], [829, 829, 4*w^2 + 8*w - 23], [829, 829, 8*w - 19], [839, 839, 2*w^2 + 6*w - 7], [853, 853, 10*w^2 + 6*w - 77], [859, 859, -5*w^2 - 2*w + 47], [863, 863, 12*w^2 + 19*w - 59], [877, 877, 8*w^2 + 6*w - 61], [907, 907, -4*w - 1], [907, 907, 9*w^2 + 4*w - 71], [907, 907, 2*w^2 + 7*w - 17], [919, 919, -10*w^2 - 5*w + 79], [929, 929, 2*w^2 + 7*w - 11], [937, 937, -4*w^2 + 6*w + 11], [941, 941, 3*w^2 - 11], [947, 947, 6*w - 5], [967, 967, -2*w^2 + 5*w + 5], [971, 971, 5*w^2 - 2*w - 29], [991, 991, 4*w^2 + 2*w - 25], [997, 997, 3*w^2 - 2*w - 11]]; primes := [ideal : I in primesArray]; heckePol := x^4 - x^3 - 6*x^2 + 4*x + 4; K := NumberField(heckePol); heckeEigenvaluesArray := [0, e, 1/2*e^3 - 1/2*e^2 - 2*e + 2, -1/2*e^3 - 1/2*e^2 + 2*e + 3, -1/2*e^3 + 1/2*e^2 + 5*e, -3/2*e^3 - 1/2*e^2 + 9*e, -1/2*e^3 - 3/2*e^2 + 4*e + 8, 2*e^3 - e^2 - 9*e + 2, 2*e^3 - 2*e^2 - 10*e + 6, 1/2*e^3 - 3/2*e^2 + 12, 3*e^2 - 2*e - 6, 2*e^3 + e^2 - 10*e - 4, 1/2*e^3 + 1/2*e^2 - 5*e - 4, e^3 - e^2 - 3*e, -e^3 - e^2 + 3*e + 4, 3*e^3 - e^2 - 16*e + 2, -2*e^3 - 2*e^2 + 8*e + 12, 3/2*e^3 - 7/2*e^2 - 5*e + 12, e^3 + 3*e^2 - 4*e - 8, -e^3 + 4*e^2 + 5*e - 10, 2*e^2 + 3*e - 4, e^3 - 5*e - 2, 2*e^3 - e^2 - 9*e - 2, -1/2*e^3 - 1/2*e^2 - 2*e + 6, -7/2*e^3 + 5/2*e^2 + 17*e - 10, -5/2*e^3 + 9/2*e^2 + 9*e - 10, -3/2*e^3 + 3/2*e^2 + 5*e - 10, -e^3 + 3*e^2 + 2*e - 8, -3*e^2 + e + 2, -4*e^3 + e^2 + 20*e - 6, -1/2*e^3 - 3/2*e^2 - e + 6, -7/2*e^3 + 1/2*e^2 + 14*e - 2, e^3 + 6*e^2 - 7*e - 16, -e^3 + 5*e - 2, e^3 - 4*e^2 - 9*e + 18, -2*e^3 + 5*e^2 + 6*e - 12, 5*e^3 - 6*e^2 - 20*e + 14, 7/2*e^3 - 11/2*e^2 - 14*e + 18, 2*e^2 + 2*e + 4, 2*e^3 + 6*e^2 - 14*e - 20, -1/2*e^3 + 1/2*e^2 - 3*e - 6, -2*e^3 + 2*e^2 + 12*e, -3/2*e^3 + 3/2*e^2 + 2*e + 2, -2*e^3 + 5*e^2 + 18*e - 18, 2*e^3 - 18*e - 6, -1/2*e^3 - 5/2*e^2 - 3*e + 18, e^3 - 12*e + 12, -5/2*e^3 + 17/2*e^2 + 10*e - 16, 6*e^3 - 4*e^2 - 32*e + 10, 4*e^3 - 5*e^2 - 13*e + 8, 2*e^3 - 6*e - 10, 7*e^2 - 2*e - 16, -3*e^3 + 2*e^2 + 25*e - 6, 2*e^2 - 5*e - 8, -3*e^3 + 5*e^2 + 10*e - 16, e^3 + 2*e^2 - 9*e - 18, -4*e^3 + 4*e^2 + 16*e - 10, 1/2*e^3 + 1/2*e^2 - 8*e - 10, -e^3 + 10*e^2 + 2*e - 32, 1/2*e^3 - 3/2*e^2 - 4*e + 16, -7/2*e^3 + 5/2*e^2 + 24*e + 2, -5*e^3 + e^2 + 30*e - 14, -e^3 - e^2 + 18*e, -3*e^3 + 3*e^2 + 14*e - 18, e^3 - e^2 - 10*e - 6, -4*e^2 + 5*e, 9/2*e^3 - 3/2*e^2 - 20*e + 20, -3*e^3 - 4*e^2 + 17*e + 14, 4*e^2 - 5*e - 12, e^3 + 7*e^2 - 36, e^3 + 4*e^2 - 7*e - 22, 2*e^2 - 2*e - 6, -5*e^2 + 4*e + 10, -e^3 - 6*e^2 + 7*e + 4, 4*e^3 - 9*e^2 - 24*e + 18, -7/2*e^3 + 7/2*e^2 + 15*e - 12, -3/2*e^3 - 9/2*e^2 + 10*e + 22, e^3 + 5*e^2 - 6*e - 10, -2*e^3 - e^2 + 10*e, 2*e^3 - 16*e + 18, 2*e^3 + 7*e^2 - 16*e - 14, 6*e^2 - 8*e - 18, e^3 - 5*e^2 - 4*e + 6, -5*e^3 + 5*e^2 + 29*e - 16, 3*e^3 - e^2 - 17*e - 12, 3/2*e^3 + 7/2*e^2 - 7*e - 18, -e^3 - 3*e^2 + 16, -e^3 - 7*e^2 + 8*e + 14, 2*e^3 - 8*e^2 - 8*e + 8, -6*e^3 - 2*e^2 + 29*e + 12, -e^2 - 9*e + 6, -1/2*e^3 - 7/2*e^2 + 5*e + 6, -3*e^3 - 6*e^2 + 16*e + 32, 5*e^3 - 35*e + 18, -5/2*e^3 + 5/2*e^2 + 5*e + 2, 3*e^3 - 3*e^2 - 10*e + 26, -7*e^3 + e^2 + 26*e + 20, -2*e^3 + 8*e^2 + 10*e - 34, -3*e^3 - e^2 + 13*e + 16, 3/2*e^3 + 1/2*e^2 - 7*e - 22, 4*e^3 - 2*e^2 - 18*e + 6, 9/2*e^3 - 7/2*e^2 - 26*e + 16, 5*e^3 - 6*e^2 - 38*e + 26, -7/2*e^3 + 1/2*e^2 + 17*e - 18, -10*e^3 + 7*e^2 + 52*e - 12, -3*e^3 - 2*e^2 + 21*e + 14, 3*e^3 - 4*e^2 - 8*e + 42, 1/2*e^3 - 3/2*e^2 + e - 12, 4*e^3 + 2*e^2 - 24*e - 8, 9*e^3 + e^2 - 50*e - 2, -e^3 - 3*e^2 + 10*e + 12, -3*e^3 + 2*e^2 + 4*e - 18, -9/2*e^3 + 7/2*e^2 + 27*e - 24, 19/2*e^3 - 13/2*e^2 - 48*e + 26, -6*e^3 + 26*e - 20, -e^3 - 10*e^2 - 2*e + 44, 9*e^2 - 11*e - 20, -6*e^3 + 6*e^2 + 36*e - 30, 15/2*e^3 - 13/2*e^2 - 52*e + 18, -11*e^3 + 10*e^2 + 47*e - 22, 5/2*e^3 - 11/2*e^2 - 5*e + 22, 3*e^3 - 5*e^2 - 11*e + 24, e^3 + 16*e^2 - 13*e - 46, 2*e^3 + 6*e^2 - 14*e - 20, -e^3 + 3*e^2 + 10*e - 40, 19/2*e^3 - 19/2*e^2 - 41*e + 18, -3*e^2 - 7*e + 16, -5*e^3 - 10*e^2 + 20*e + 48, 7*e^3 - 3*e^2 - 34*e + 8, -3*e^3 - 10*e^2 + 23*e + 26, 2*e^3 - 4*e^2 - 4*e + 40, -9*e^3 - 3*e^2 + 34*e + 28, -8*e^3 + e^2 + 40*e - 26, -3*e^3 - e^2 + 10*e - 4, -6*e^3 - 4*e^2 + 38*e - 8, -1/2*e^3 + 9/2*e^2 + 15*e - 6, 4*e^3 + 2*e^2 - 25*e, 5/2*e^3 + 17/2*e^2 - 24*e - 32, 6*e^3 - 7*e^2 - 25*e + 22, e^2 - 19*e - 2, e^3 + e^2 - 9*e - 24, -e^3 + 5*e^2 - 19*e - 24, -10*e^3 + e^2 + 46*e - 16, -3*e^3 + 2*e^2 + 4*e + 14, 3*e^3 + 7*e^2 - 10*e - 24, 11/2*e^3 - 15/2*e^2 - 32*e + 24, 2*e^3 + 2*e^2 - 22*e + 8, -e^3 - 5*e^2 + 13*e + 32, e^3 + 12*e^2 - 12*e - 30, -3*e^3 + 5*e^2 + 20*e - 30, -3*e^3 - 10*e^2 + 21*e + 22, -7*e^3 + 10*e^2 + 14*e - 42, 6*e^3 + 4*e^2 - 24*e - 28, -5*e^3 + e^2 + 26*e + 16, -1/2*e^3 + 9/2*e^2 - 3*e - 42, 10*e^3 - 2*e^2 - 36*e - 2]; 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;