/* 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![-34, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, w - 6], [3, 3, w + 1], [3, 3, w + 2], [5, 5, w + 2], [5, 5, w + 3], [11, 11, w + 1], [11, 11, w + 10], [17, 17, -3*w + 17], [29, 29, w + 11], [29, 29, w + 18], [37, 37, w + 16], [37, 37, w + 21], [47, 47, -w - 9], [47, 47, w - 9], [49, 7, -7], [61, 61, w + 20], [61, 61, w + 41], [89, 89, 2*w - 15], [89, 89, -2*w - 15], [103, 103, -14*w + 81], [103, 103, 4*w - 21], [107, 107, w + 26], [107, 107, w + 81], [109, 109, w + 19], [109, 109, w + 90], [127, 127, 2*w - 3], [127, 127, -2*w - 3], [131, 131, w + 54], [131, 131, w + 77], [137, 137, -3*w - 13], [137, 137, 3*w - 13], [139, 139, w + 27], [139, 139, w + 112], [151, 151, 8*w - 45], [151, 151, -10*w + 57], [163, 163, w + 69], [163, 163, w + 94], [169, 13, -13], [173, 173, w + 42], [173, 173, w + 131], [181, 181, w + 45], [181, 181, w + 136], [191, 191, -w - 15], [191, 191, w - 15], [197, 197, w + 25], [197, 197, w + 172], [211, 211, w + 33], [211, 211, w + 178], [223, 223, -3*w - 23], [223, 223, 3*w - 23], [227, 227, w + 48], [227, 227, w + 179], [239, 239, -5*w + 33], [239, 239, -23*w + 135], [257, 257, -3*w - 7], [257, 257, 3*w - 7], [263, 263, -24*w + 139], [263, 263, 6*w - 31], [269, 269, w + 29], [269, 269, w + 240], [271, 271, -9*w + 55], [271, 271, -15*w + 89], [277, 277, w + 119], [277, 277, w + 158], [281, 281, 3*w - 5], [281, 281, -3*w - 5], [283, 283, w + 113], [283, 283, w + 170], [317, 317, w + 44], [317, 317, w + 273], [347, 347, w + 46], [347, 347, w + 301], [353, 353, 9*w - 49], [353, 353, -21*w + 121], [359, 359, -25*w + 147], [359, 359, -7*w + 45], [361, 19, -19], [379, 379, w + 105], [379, 379, w + 274], [383, 383, 6*w - 29], [383, 383, -36*w + 209], [397, 397, w + 35], [397, 397, w + 362], [409, 409, -5*w - 21], [409, 409, 5*w - 21], [419, 419, w + 156], [419, 419, w + 263], [433, 433, -12*w + 73], [433, 433, -18*w + 107], [457, 457, 6*w + 41], [457, 457, -6*w + 41], [463, 463, -4*w - 9], [463, 463, 4*w - 9], [499, 499, w + 212], [499, 499, w + 287], [529, 23, -23], [541, 541, w + 121], [541, 541, w + 420], [547, 547, w + 157], [547, 547, w + 390], [569, 569, -28*w + 165], [569, 569, -10*w + 63], [571, 571, w + 264], [571, 571, w + 307], [577, 577, -47*w + 273], [577, 577, 7*w - 33], [593, 593, 2*w - 27], [593, 593, -2*w - 27], [599, 599, -6*w - 25], [599, 599, 6*w - 25], [619, 619, w + 167], [619, 619, w + 452], [631, 631, -15*w + 91], [631, 631, -21*w + 125], [643, 643, w + 57], [643, 643, w + 586], [647, 647, -11*w + 69], [647, 647, -29*w + 171], [653, 653, w + 120], [653, 653, w + 533], [677, 677, w + 64], [677, 677, w + 613], [683, 683, w + 248], [683, 683, w + 435], [691, 691, w + 322], [691, 691, w + 369], [709, 709, w + 265], [709, 709, w + 444], [727, 727, -9*w + 59], [727, 727, -39*w + 229], [761, 761, -27*w + 155], [761, 761, 15*w - 83], [769, 769, 5*w - 9], [769, 769, -5*w - 9], [787, 787, w + 63], [787, 787, w + 724], [811, 811, w + 70], [811, 811, w + 741], [821, 821, w + 149], [821, 821, w + 672], [827, 827, w + 193], [827, 827, w + 634], [853, 853, w + 253], [853, 853, w + 600], [863, 863, -6*w - 19], [863, 863, 6*w - 19], [877, 877, w + 339], [877, 877, w + 538], [907, 907, w + 74], [907, 907, w + 833], [919, 919, -3*w - 35], [919, 919, 3*w - 35], [937, 937, 7*w - 27], [937, 937, -7*w - 27], [941, 941, w + 144], [941, 941, w + 797], [947, 947, w + 292], [947, 947, w + 655], [953, 953, 2*w - 33], [953, 953, -2*w - 33], [961, 31, -31], [967, 967, 9*w - 61], [967, 967, -9*w - 61], [977, 977, -4*w - 39], [977, 977, 4*w - 39], [997, 997, w + 55], [997, 997, w + 942]]; primes := [ideal : I in primesArray]; heckePol := x^5 - x^4 - 5*x^3 + 3*x^2 + 4*x - 1; K := NumberField(heckePol); heckeEigenvaluesArray := [e, e^4 - 5*e^2 - e + 3, -e^3 + e^2 + 3*e - 1, -e^4 + e^3 + 5*e^2 - 3*e - 2, -1, -2*e^4 + 2*e^3 + 9*e^2 - 3*e - 5, e^4 - 2*e^3 - 4*e^2 + 7*e + 3, e^4 - e^3 - 4*e^2 + e, 2*e^4 - 2*e^3 - 9*e^2 + 5*e + 7, e^4 - 3*e^3 - 3*e^2 + 9*e + 3, -e^4 + 2*e^3 + 3*e^2 - 5*e + 4, -e^3 - 3*e^2 + 5*e + 11, e^4 - 3*e^3 - 2*e^2 + 9*e, 2*e^4 + e^3 - 13*e^2 - 3*e + 11, -e^4 - e^3 + 6*e^2 + e - 3, 4*e^4 - e^3 - 20*e^2 - 5*e + 16, -3*e^4 + e^3 + 17*e^2 - 2*e - 10, -e^4 + 2*e^3 + 3*e^2 - 6*e + 4, 2*e^4 - 4*e^3 - 10*e^2 + 7*e + 14, -e^4 - 3*e^3 + 6*e^2 + 13*e - 9, -4*e^4 + 2*e^3 + 23*e^2 + e - 18, -6*e^4 + 4*e^3 + 27*e^2 - 9*e - 10, -4*e^4 + 8*e^3 + 16*e^2 - 24*e - 13, 2*e^4 - e^3 - 13*e^2 - 2*e + 20, -e^4 + 5*e^3 + e^2 - 19*e + 1, 4*e^4 - 2*e^3 - 16*e^2 + 3*e - 3, -2*e^4 + 3*e^3 + 13*e^2 - 9*e - 7, -4*e^4 + 2*e^3 + 18*e^2 - 5*e + 6, -3*e^3 + e^2 + 20*e - 1, -4*e^4 + 7*e^3 + 10*e^2 - 16*e + 7, -3*e^4 + 4*e^3 + 15*e^2 - 11*e - 14, e^4 - e^3 - 7*e^2 + 13*e + 10, 3*e^4 + 3*e^3 - 19*e^2 - 14*e + 18, -6*e^4 + 4*e^3 + 28*e^2 - 14*e - 10, 2*e^4 - 4*e^3 - 9*e^2 + 13*e + 5, 7*e^4 - 7*e^3 - 35*e^2 + 14*e + 21, -2*e^4 + 2*e^3 + 9*e^2 - 8*e - 10, -3*e^4 + 2*e^3 + 12*e^2 + 3*e - 5, -2*e^4 + 9*e^2 - 4*e + 4, -e^4 + 4*e^3 - 3*e^2 - 4*e + 23, 4*e^4 - 8*e^3 - 12*e^2 + 25*e, -3*e^4 - 2*e^3 + 15*e^2 + 15*e - 10, e^4 - 5*e^3 - 3*e^2 + 11*e + 6, 2*e^4 - 8*e^2 + 5*e - 4, -2*e^4 + 18*e^2 - 18, 2*e^4 + 3*e^3 - 11*e^2 - 10*e + 8, 4*e^4 - 2*e^3 - 19*e^2 + 4*e + 9, 2*e^4 - 5*e^3 - 12*e^2 + 18*e + 11, 2*e^4 - 4*e^3 - 4*e^2 + 16*e - 11, -4*e^4 + 7*e^3 + 16*e^2 - 16*e - 10, -5*e^4 + 4*e^3 + 23*e^2 - 17*e - 14, -6*e^4 + 6*e^3 + 23*e^2 - 17*e - 2, 7*e^4 - 4*e^3 - 29*e^2 + 6*e + 11, 3*e^4 + 3*e^3 - 20*e^2 - 21*e + 16, e^4 + e^3 - 2*e^2 - 2*e - 17, -2*e^4 + 2*e^3 + 3*e^2 + e + 15, 7*e^4 - 9*e^3 - 30*e^2 + 21*e + 9, -2*e^4 + 8*e^3 + 3*e^2 - 24*e, -e^4 - e^3 + 6*e^2 + e - 13, -3*e^4 + 3*e^3 + 5*e^2 - 3*e + 21, -4*e^4 + 5*e^3 + 13*e^2 - 10*e + 10, -3*e^4 - 2*e^3 + 19*e^2 + 13*e - 18, -e^4 - 3*e^3 + 7*e^2 + 26*e - 7, 3*e^4 + 3*e^3 - 18*e^2 - 21*e + 20, 11*e^4 - 11*e^3 - 50*e^2 + 18*e + 27, -e^4 + 2*e^3 + 6*e^2 - 2*e - 4, 6*e^4 - 3*e^3 - 20*e^2 + e - 1, -2*e^4 + 8*e^3 + 3*e^2 - 29*e, 4*e^4 + 2*e^3 - 18*e^2 - 14*e + 15, 8*e^4 - 6*e^3 - 41*e^2 + 32, 3*e^4 + 3*e^3 - 13*e^2 - 16*e + 9, 2*e^4 + 7*e^3 - 10*e^2 - 23*e - 9, -7*e^4 + 14*e^3 + 21*e^2 - 39*e - 3, -7*e^4 + 5*e^3 + 27*e^2 - 2*e - 16, -4*e^4 + 3*e^3 + 16*e^2 - 9*e + 5, -5*e^4 + 10*e^3 + 13*e^2 - 36*e + 5, -7*e^4 + 8*e^3 + 30*e^2 - 27*e - 22, 5*e^4 - 5*e^3 - 19*e^2 + 15*e + 4, 8*e^4 - 5*e^3 - 31*e^2 + 7*e, 8*e^4 - 3*e^3 - 43*e^2 + 13*e + 21, 4*e^4 - 2*e^3 - 22*e^2 + 2*e - 7, -15*e^4 + 11*e^3 + 69*e^2 - 16*e - 28, -5*e^4 - e^3 + 30*e^2 + 11*e - 18, -e^4 + 8*e^3 - 10*e^2 - 25*e + 23, 9*e^4 - 6*e^3 - 40*e^2 + 16*e + 24, 6*e^4 + 3*e^3 - 26*e^2 - 20*e + 3, 9*e^4 - 3*e^3 - 49*e^2 + 2*e + 31, -9*e^4 + 14*e^3 + 28*e^2 - 38*e - 3, 4*e^4 + e^3 - 31*e^2 - e + 40, 2*e^4 + 5*e^3 - 6*e^2 - 22*e - 18, -5*e^4 + 7*e^3 + 27*e^2 - 19*e - 6, 4*e^4 + e^3 - 26*e^2 - 6*e + 35, -6*e^4 + 8*e^3 + 30*e^2 - 23*e - 18, -4*e^4 + 8*e^3 + 14*e^2 - 10*e - 8, -2*e^4 + 4*e^3 + e^2 - 5*e + 15, 11*e^4 - 3*e^3 - 50*e^2 + 14, e^4 - 4*e^3 + 7*e^2 + 6*e - 19, -3*e^4 + 11*e^2 + 12*e - 5, -e^4 + e^3 + 3*e^2 - 10*e + 18, 3*e^4 - 15*e^2 - 3*e + 18, e^4 + 8*e^3 - 12*e^2 - 32*e + 22, -16*e^4 + 13*e^3 + 71*e^2 - 39*e - 34, -4*e^4 - e^3 + 27*e^2 + 9*e - 26, 10*e^4 - 3*e^3 - 56*e^2 - 3*e + 34, -10*e^4 + 18*e^3 + 47*e^2 - 53*e - 24, 7*e^4 - 16*e^3 - 27*e^2 + 49*e + 7, -13*e^4 + 10*e^3 + 48*e^2 - 21*e - 6, 4*e^4 - 8*e^3 - 7*e^2 + 19*e - 15, 8*e^4 - 5*e^3 - 38*e^2 + 19*e + 30, 4*e^4 + 4*e^3 - 19*e^2 - 28*e + 16, 12*e^4 - 15*e^3 - 56*e^2 + 28*e + 39, 5*e^4 - 11*e^3 - 27*e^2 + 45*e + 9, -4*e^3 + 9*e^2 + 10*e - 21, -8*e^4 + 3*e^3 + 39*e^2 + 13*e - 10, 3*e^4 - 6*e^3 - 7*e^2 + 2*e - 12, -2*e^4 + 6*e^3 - 5*e^2 - 25*e + 30, 16*e^4 - 11*e^3 - 83*e^2 + 29*e + 36, 12*e^4 - 17*e^3 - 42*e^2 + 50*e + 12, 6*e^4 - 14*e^3 - 17*e^2 + 41*e + 10, -2*e^4 + 7*e + 27, -11*e^4 + 9*e^3 + 50*e^2 - 24*e - 16, -7*e^4 + 7*e^3 + 33*e^2 - 12*e + 13, 3*e^4 - 3*e^3 + 3*e^2 - 2*e - 36, -5*e^4 + e^3 + 32*e^2 + 2*e - 50, -3*e^4 + 2*e^3 + 25*e^2 - 3*e - 35, 9*e^4 - 5*e^3 - 33*e^2 + 10, 4*e^4 - 3*e^3 - 12*e^2 + 9*e - 15, -4*e^4 + 6*e^3 + 20*e^2 - 31*e - 20, 13*e^4 - 9*e^3 - 57*e^2 + 13*e + 41, e^4 + e^3 - 8*e^2 - 5*e - 34, -e^4 - 5*e^3 - e^2 + 28*e + 16, 15*e^4 - 14*e^3 - 70*e^2 + 43*e + 33, 10*e^4 - 13*e^3 - 48*e^2 + 36*e + 44, 15*e^4 - 69*e^2 - 25*e + 50, -e^4 - e^2 + 6*e - 10, 4*e^4 - 8*e^3 - 7*e^2 + 27*e - 18, -e^4 - 2*e^3 + 3*e^2 + 8*e + 12, -e^4 + 7*e^3 + 11*e^2 - 30*e - 32, -8*e^4 + 19*e^3 + 35*e^2 - 56*e - 17, -3*e^4 - 7*e^3 + 30*e^2 + 28*e - 21, 7*e^4 + e^3 - 39*e^2 - 22*e + 27, -9*e^4 + e^3 + 37*e^2 + 12*e - 14, 7*e^4 - 11*e^3 - 25*e^2 + 39*e + 3, 14*e^4 - 6*e^3 - 75*e^2 - e + 35, 4*e^4 + e^3 - 21*e^2 - 17*e + 26, -5*e^4 + 4*e^3 + 13*e^2 - 13*e + 18, -3*e^4 - 3*e^3 + 15*e^2 + 19*e - 25, 7*e^4 + 2*e^3 - 32*e^2 - 19*e + 11, -e^4 + 9*e^2 + 3*e - 42, -4*e^4 - e^3 + 2*e^2 + 26*e + 28, 6*e^4 - 12*e^3 - 27*e^2 + 26*e + 26, -4*e^4 + 8*e^3 + 17*e^2 - 25*e - 38, -2*e^4 - 4*e^3 + 9*e^2 + 16*e - 17, -13*e^4 + 19*e^3 + 53*e^2 - 61*e - 29, 2*e^4 + 2*e^3 - 10*e^2 - 20*e + 18, -6*e^4 - 2*e^3 + 38*e^2 + 14*e - 19, 3*e^4 - 8*e^3 + 6*e^2 + 23*e - 31, -10*e^3 + 18*e^2 + 26*e - 24, -2*e^4 - 3*e^3 + 21*e^2 + 4*e - 31, 6*e^4 - 11*e^3 - 19*e^2 + 35*e + 12, 10*e^4 - 8*e^3 - 48*e^2 + 27*e + 4, -4*e^4 + 12*e^3 + 14*e^2 - 32*e - 11, -17*e^4 + 20*e^3 + 68*e^2 - 57*e - 40, -14*e^4 + 18*e^3 + 51*e^2 - 39*e - 3, e^4 - 17*e^3 + 4*e^2 + 48*e - 7, -7*e^4 + e^3 + 32*e^2 + 25*e - 32, -8*e^4 + 5*e^3 + 40*e^2 - 3*e - 19]; 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;