/* 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![8, -7, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -w + 2], [4, 2, -w^2 - w + 5], [5, 5, -w + 3], [7, 7, -w^2 - 2*w + 3], [11, 11, -w^2 + 5], [13, 13, w + 1], [23, 23, w^2 - 3], [25, 5, w^2 + 2*w - 1], [27, 3, -3], [29, 29, -3*w + 7], [43, 43, -3*w^2 - 2*w + 17], [49, 7, -2*w^2 + w + 11], [67, 67, -2*w^2 - 4*w + 3], [71, 71, -2*w^2 + w + 9], [73, 73, w^2 + 2*w - 7], [73, 73, -2*w^2 - 2*w + 11], [73, 73, w - 5], [89, 89, 2*w^2 + w - 9], [89, 89, -w^2 - 2*w + 9], [89, 89, -2*w - 1], [101, 101, 4*w^2 + 4*w - 19], [103, 103, 2*w + 3], [107, 107, w + 5], [109, 109, 3*w^2 + 2*w - 15], [113, 113, 2*w - 7], [121, 11, 2*w^2 + w - 7], [127, 127, 2*w^2 - 9], [127, 127, -4*w + 11], [127, 127, -2*w^2 + 2*w + 9], [131, 131, 3*w^2 + 4*w - 13], [137, 137, -w^2 + 2*w - 3], [149, 149, 2*w^2 - 6*w + 1], [151, 151, 3*w - 1], [157, 157, -3*w^2 - 6*w + 7], [157, 157, -2*w^2 + 3*w + 3], [157, 157, 2*w^2 + 2*w - 15], [163, 163, 3*w^2 + 2*w - 19], [163, 163, 4*w^2 + 5*w - 17], [163, 163, -5*w^2 - 6*w + 19], [167, 167, w^2 - 11], [167, 167, -3*w - 7], [167, 167, -w^2 - 2*w + 13], [169, 13, w^2 - 2*w - 5], [181, 181, -2*w - 7], [181, 181, 2*w^2 - 7], [181, 181, 2*w^2 - 2*w - 5], [193, 193, 2*w^2 + 2*w - 3], [197, 197, -4*w^2 - 7*w + 9], [197, 197, 2*w^2 - 2*w - 19], [197, 197, 2*w^2 + 3*w - 11], [199, 199, -6*w^2 + 37], [227, 227, 2*w^2 - 3*w - 7], [233, 233, w^2 + 4*w - 15], [251, 251, -4*w^2 + 23], [257, 257, w^2 - 2*w - 7], [269, 269, -4*w^2 - w + 23], [271, 271, 2*w^2 - w - 19], [277, 277, -3*w^2 - 2*w + 9], [281, 281, w^2 - 2*w - 9], [283, 283, 6*w^2 + 5*w - 31], [293, 293, 2*w^2 - 2*w - 21], [311, 311, -w^2 - 4*w - 1], [317, 317, 3*w^2 + 2*w - 11], [317, 317, 2*w^2 + 3*w - 13], [317, 317, 2*w^2 - 2*w - 11], [331, 331, 4*w^2 - w - 21], [331, 331, 3*w + 5], [331, 331, 4*w^2 + 2*w - 21], [337, 337, -2*w^2 + 4*w + 5], [349, 349, w^2 - 4*w - 1], [353, 353, -w^2 - 4*w - 5], [367, 367, 6*w^2 + 7*w - 23], [379, 379, 2*w - 9], [389, 389, 2*w^2 + 3*w - 15], [389, 389, w^2 + 4*w - 13], [389, 389, 2*w^2 + 4*w - 1], [397, 397, 4*w - 1], [409, 409, 2*w^2 + w - 19], [409, 409, -w^2 + 6*w - 11], [409, 409, 2*w^2 + 4*w - 21], [419, 419, 4*w^2 + 4*w - 21], [421, 421, -w^2 - 3], [421, 421, -4*w^2 - 6*w + 17], [421, 421, 4*w^2 + 3*w - 31], [439, 439, 5*w^2 - 29], [443, 443, w^2 - 13], [449, 449, 4*w^2 - w - 27], [457, 457, 2*w^2 - 19], [467, 467, 4*w + 9], [467, 467, 3*w^2 - 2*w - 17], [467, 467, 2*w^2 - 2*w - 13], [479, 479, -w^2 + 4*w + 19], [487, 487, 4*w^2 + 5*w - 19], [491, 491, 3*w - 11], [503, 503, 4*w^2 + 3*w - 17], [523, 523, -2*w^2 - 5*w - 1], [529, 23, -4*w^2 - w + 21], [547, 547, 5*w^2 + 4*w - 23], [557, 557, 4*w^2 - 2*w - 33], [563, 563, 2*w^2 + 2*w - 1], [563, 563, 3*w^2 - 8*w - 1], [563, 563, 3*w^2 + 4*w - 5], [571, 571, 5*w^2 + 10*w - 11], [577, 577, -4*w + 13], [587, 587, 3*w^2 + 4*w - 17], [593, 593, 3*w^2 - 11], [593, 593, 4*w^2 + 2*w - 19], [593, 593, w - 9], [601, 601, -4*w^2 - 10*w + 3], [607, 607, 3*w^2 + 4*w - 27], [607, 607, -4*w^2 - 2*w + 33], [607, 607, 2*w^2 - 7*w + 9], [617, 617, -2*w^2 + 9*w - 5], [619, 619, -4*w - 1], [631, 631, -8*w^2 - 10*w + 29], [641, 641, w^2 - 4*w - 3], [653, 653, 3*w^2 - 2*w - 9], [659, 659, 2*w^2 + 4*w - 19], [661, 661, 6*w^2 + 6*w - 25], [673, 673, 6*w^2 + 7*w - 27], [683, 683, 5*w^2 + 2*w - 27], [701, 701, 7*w - 13], [709, 709, 6*w^2 + 8*w - 25], [719, 719, -2*w^2 + 4*w + 7], [719, 719, -7*w^2 - 10*w + 27], [719, 719, 4*w^2 - w - 19], [727, 727, 3*w^2 + 8*w + 1], [733, 733, 3*w^2 - 2*w - 7], [733, 733, 3*w^2 - 2*w - 29], [739, 739, -w - 9], [751, 751, -w^2 - 4*w + 19], [757, 757, -4*w - 7], [761, 761, -5*w^2 - 4*w + 29], [787, 787, -2*w^2 - 6*w + 3], [809, 809, -w^2 + 4*w - 9], [809, 809, -4*w^2 + 3*w + 15], [809, 809, 2*w^2 - 9*w + 13], [821, 821, -4*w^2 - 9*w + 7], [821, 821, 5*w - 1], [821, 821, 8*w^2 + 11*w - 31], [827, 827, 8*w^2 + 8*w - 35], [829, 829, 2*w^2 - 5*w - 27], [839, 839, -w^2 - 6*w - 7], [841, 29, 3*w^2 + 4*w - 31], [863, 863, 4*w^2 - 19], [883, 883, -5*w^2 + 27], [887, 887, 5*w^2 + 4*w - 21], [907, 907, -6*w^2 + 2*w + 31], [907, 907, 2*w^2 - 5*w - 29], [907, 907, 2*w^2 + 5*w - 15], [911, 911, w^2 - 15], [919, 919, 8*w - 17], [941, 941, -4*w^2 + 5*w + 15], [947, 947, 4*w^2 + 2*w - 13], [947, 947, 3*w^2 - 6*w - 7], [947, 947, -5*w^2 - 4*w + 15]]; primes := [ideal : I in primesArray]; heckePol := x^4 - x^3 - 7*x^2 + 5*x + 4; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -1, 1, -e^2 + e + 4, e^3 - e^2 - 6*e + 2, e + 1, e^3 - 7*e - 2, 2*e, -e^2 + 2*e + 1, -e^3 + 5*e + 2, -e^3 + e^2 + 5*e - 3, -2*e^3 + 12*e - 2, e^2 - 1, e^3 + e^2 - 6*e + 4, -3*e^2 + e + 8, -e^3 + 7*e, -2*e^2 + 2*e + 6, -e^3 - e^2 + 4*e + 4, 2*e^3 - 4*e^2 - 10*e + 12, -e^2 - 2*e + 15, e^3 + 2*e^2 - 6*e - 5, -2*e, -3*e^3 + 2*e^2 + 17*e, e^3 - 2*e^2 - 9*e + 8, -2*e^2 + 12, 2*e^3 - 12*e - 6, -e^3 - 3*e^2 + 10*e + 12, -2*e^3 + 15*e - 3, -3*e^3 + 15*e + 4, -3*e^3 + e^2 + 19*e - 3, e^3 + 2*e^2 - 10*e - 7, -e^3 + e^2 + 7*e - 3, 6*e - 6, -e^3 - 3*e^2 + 6*e + 14, -e^3 + 3*e^2 + 9*e - 19, -e^3 + 4*e^2 + 5*e - 6, -3*e^3 + 3*e^2 + 20*e - 12, -3*e^3 + 17*e + 4, 2*e^3 - 2*e^2 - 18*e + 12, e^3 + 4*e^2 - 6*e - 17, 4*e^3 - 3*e^2 - 26*e + 5, 3*e^3 - e^2 - 15*e + 11, 3*e^3 - 3*e^2 - 14*e + 18, -2*e^3 - e^2 + 11*e + 2, -3*e^3 + 19*e - 4, 5*e^3 - 4*e^2 - 26*e + 11, 4*e^2 - 6*e - 14, 2*e^3 - e^2 - 11*e - 14, -e^2 + 4*e + 15, -3*e^2 + 5*e + 10, -e^3 - 2*e^2 + 5*e + 10, 2*e^3 - 20*e + 4, 3*e^3 - e^2 - 20*e + 2, -2*e^3 + 2*e^2 + 16*e, 3*e^3 + 3*e^2 - 11*e - 27, 2*e^3 + 2*e^2 - 16*e + 6, 4*e^3 - 2*e^2 - 25*e + 7, 3*e^3 + 2*e^2 - 15*e - 16, e^3 + 4*e^2 - e - 18, -4*e^3 + 2*e^2 + 20*e - 6, 4*e^3 - 16*e - 6, -e^3 - 4*e^2 - 3*e + 24, 3*e^3 + e^2 - 27*e - 1, -e^3 + 5*e^2 + 13*e - 19, 2*e^2 - 6*e - 2, -4*e^3 - 3*e^2 + 21*e + 14, -2*e^3 + 3*e^2 + 22*e - 17, 2*e^3 + 3*e^2 - 11*e - 16, -4*e^3 + 3*e^2 + 19*e - 18, 2*e^3 - e^2 - 13*e - 16, -e^3 + e^2 - e - 5, 3*e^3 + e^2 - 19*e + 5, -5*e^3 + e^2 + 37*e - 15, -e^3 + e^2 + 3*e - 11, -4*e^3 - 2*e^2 + 22*e + 10, 3*e^3 - 3*e^2 - 32*e + 22, -4*e^3 + 2*e^2 + 18*e + 4, e^3 + e^2 - 12*e + 10, 4*e^3 - e^2 - 29*e - 2, -6*e^3 + 4*e^2 + 28*e - 8, -e^3 + 3*e^2 - e + 3, -3*e^3 + 2*e^2 + 13*e + 6, -e^3 + 8*e^2 + 3*e - 22, 3*e^2 - 35, -3*e^3 - 4*e^2 + 15*e + 12, e^3 + 2*e^2 - 5*e - 16, e^3 + e^2 - 11*e - 3, 5*e^3 + e^2 - 32*e + 10, e^2 - 5*e + 4, e^3 - 4*e^2 - 14*e + 19, 5*e^3 + 2*e^2 - 31*e + 8, e^3 + 4*e^2 - 4*e - 21, -8*e^3 + 3*e^2 + 38*e - 7, -4*e^2 - 2*e + 14, 6*e^3 - 9*e^2 - 39*e + 20, e^3 + e^2 - 7*e - 11, -2*e^3 + 2*e^2 + 18*e + 12, -6*e^3 + 8*e^2 + 44*e - 14, 3*e^3 - 3*e^2 - 12*e + 20, e^3 + 4*e^2 - 10*e - 9, 2*e^3 + 4*e^2 - 29*e - 19, 5*e^3 - e^2 - 41*e + 11, 2*e^2 - 6, 2*e^3 + 2*e^2 - 26*e - 16, -e^3 + e^2 + 21*e - 7, -3*e^3 + 4*e^2 + 15*e - 16, e^3 - 2*e^2 - 13*e - 16, 2*e^3 + 2*e^2 - 12*e + 18, -5*e^3 + 3*e^2 + 46*e - 8, -6*e^3 + 4*e^2 + 28*e - 30, e^3 - 2*e^2 + e + 36, 5*e^3 - 4*e^2 - 35*e + 24, 2*e^3 - 4*e^2 - 14*e + 12, 4*e^3 - e^2 - 19*e + 18, -4*e^3 + 2*e^2 + 22*e - 4, -e^3 + 9*e^2 + 7*e - 23, -6*e^3 + e^2 + 24*e + 5, -2*e^3 + e^2 + 3*e - 4, -2*e^3 + 4*e^2 + 4*e - 32, 2*e^2 - 2*e - 18, -6*e^3 + 4*e^2 + 42*e - 16, 5*e^3 - 5*e^2 - 43*e + 21, 10*e^3 - e^2 - 52*e - 5, 2*e^3 - 4*e^2 + 6*e + 24, -e^3 + 6*e^2 + 11*e - 20, -2*e^3 + 4*e^2 + 22*e - 8, -6*e^3 + 9*e^2 + 30*e - 41, -e^3 + 3*e^2 + 7*e + 23, e^3 - 3*e + 6, 4*e^3 - 8*e^2 - 12*e + 40, e^3 - 10*e^2 - 2*e + 31, -4*e^2 - 16, -4*e^3 + e^2 + 34*e + 9, 3*e^3 + 6*e^2 - 9*e - 40, e^3 + 6*e^2 - e - 48, -e^3 + 7*e^2 + e + 1, -4*e^3 + 4*e^2 + 28*e - 32, 3*e^3 - 11*e^2 - 10*e + 30, e^3 - 5*e^2 - 15*e + 25, 4*e^3 - 5*e^2 - 16*e - 3, 2*e^3 + 6*e^2 - 20*e - 10, -3*e^3 + 5*e^2 + 11*e - 17, 2*e^3 + 7*e^2 - 26*e - 39, -5*e^3 - 2*e^2 + 37*e - 16, 3*e^3 + 4*e^2 - 27*e - 22, -4*e^3 + 18*e - 12, -e^3 - 4*e^2 + e + 8, -3*e^3 - 2*e^2 + 17*e + 16, 4*e^3 - 28*e + 4, -6*e^3 - 2*e^2 + 50*e - 2, -7*e^3 - e^2 + 55*e - 5, e^3 - 3*e^2 - 23*e + 29, -2*e^3 - 6*e^2 + 16*e + 42, e^3 - 14*e^2 + 2*e + 61, -8*e^3 + 6*e^2 + 48*e - 4, 2*e^3 + e^2 - 18*e + 9]; 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;