/* 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![-49, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, 2], [7, 7, w - 7], [7, 7, w + 6], [9, 3, 3], [19, 19, w + 5], [19, 19, w - 6], [23, 23, w + 8], [23, 23, -w + 9], [25, 5, 5], [29, 29, -w - 4], [29, 29, w - 5], [37, 37, -w - 3], [37, 37, w - 4], [41, 41, -w - 9], [41, 41, w - 10], [43, 43, -w - 2], [43, 43, w - 3], [47, 47, -w - 1], [47, 47, w - 2], [53, 53, 2*w - 13], [53, 53, -2*w - 11], [59, 59, -4*w - 25], [59, 59, -4*w + 29], [61, 61, -w - 10], [61, 61, w - 11], [83, 83, -w - 11], [83, 83, w - 12], [97, 97, 2*w - 11], [97, 97, -2*w - 9], [101, 101, 3*w - 20], [101, 101, -3*w - 17], [107, 107, -w - 12], [107, 107, w - 13], [109, 109, -5*w - 31], [109, 109, -5*w + 36], [121, 11, -11], [127, 127, 2*w - 19], [127, 127, -2*w - 17], [137, 137, -3*w - 16], [137, 137, 3*w - 19], [157, 157, -3*w - 23], [157, 157, 3*w - 26], [163, 163, -4*w + 27], [163, 163, 4*w + 23], [169, 13, -13], [173, 173, -6*w - 37], [173, 173, -6*w + 43], [181, 181, 2*w - 5], [181, 181, -2*w - 3], [191, 191, -w - 15], [191, 191, w - 16], [193, 193, 2*w - 3], [193, 193, -2*w - 1], [197, 197, 2*w - 1], [223, 223, -w - 16], [223, 223, w - 17], [233, 233, -3*w - 13], [233, 233, 3*w - 16], [239, 239, -5*w + 34], [239, 239, -5*w - 29], [251, 251, 5*w - 41], [251, 251, 5*w + 36], [257, 257, -w - 17], [257, 257, w - 18], [289, 17, -17], [293, 293, -w - 18], [293, 293, w - 19], [311, 311, -3*w - 10], [311, 311, 3*w - 13], [313, 313, -3*w - 26], [313, 313, 3*w - 29], [331, 331, -w - 19], [331, 331, w - 20], [347, 347, 4*w - 23], [347, 347, -4*w - 19], [353, 353, 3*w - 11], [353, 353, -3*w - 8], [379, 379, 2*w - 25], [379, 379, -2*w - 23], [401, 401, 3*w - 8], [401, 401, -3*w - 5], [409, 409, 5*w - 43], [409, 409, 5*w + 38], [419, 419, -5*w - 26], [419, 419, 5*w - 31], [431, 431, 3*w - 5], [431, 431, -3*w - 2], [433, 433, -7*w + 48], [433, 433, -7*w - 41], [443, 443, 3*w - 2], [443, 443, 3*w - 1], [449, 449, -9*w - 55], [449, 449, -9*w + 64], [457, 457, -w - 22], [457, 457, w - 23], [479, 479, 2*w - 27], [479, 479, -2*w - 25], [487, 487, -3*w - 29], [487, 487, 3*w - 32], [491, 491, -5*w - 39], [491, 491, 5*w - 44], [499, 499, 4*w - 19], [499, 499, -4*w - 15], [503, 503, -w - 23], [503, 503, w - 24], [521, 521, 11*w - 86], [521, 521, -7*w + 47], [557, 557, 7*w + 51], [557, 557, 7*w - 58], [563, 563, -4*w - 13], [563, 563, 4*w - 17], [569, 569, -10*w - 61], [569, 569, -10*w + 71], [587, 587, 2*w - 29], [587, 587, -2*w - 27], [601, 601, -w - 25], [601, 601, w - 26], [607, 607, -7*w - 39], [607, 607, 7*w - 46], [613, 613, 3*w - 34], [613, 613, -3*w - 31], [617, 617, -6*w - 31], [617, 617, 6*w - 37], [619, 619, 4*w - 15], [619, 619, -4*w - 11], [631, 631, 5*w - 27], [631, 631, -5*w - 22], [653, 653, -w - 26], [653, 653, w - 27], [661, 661, 5*w - 46], [661, 661, -5*w - 41], [683, 683, -9*w + 62], [683, 683, -9*w - 53], [691, 691, 7*w - 45], [691, 691, -7*w - 38], [727, 727, -6*w - 47], [727, 727, 6*w - 53], [733, 733, 4*w - 41], [733, 733, -4*w - 37], [739, 739, -4*w - 5], [739, 739, 4*w - 9], [751, 751, 8*w - 53], [751, 751, -8*w - 45], [769, 769, 5*w - 24], [769, 769, -5*w - 19], [773, 773, 7*w - 44], [773, 773, -7*w - 37], [787, 787, 4*w - 3], [787, 787, 4*w - 1], [797, 797, -9*w - 52], [797, 797, -9*w + 61], [811, 811, -5*w - 18], [811, 811, 5*w - 23], [821, 821, -w - 29], [821, 821, w - 30], [827, 827, 2*w - 33], [827, 827, -2*w - 31], [829, 829, -10*w + 69], [829, 829, -10*w - 59], [839, 839, 5*w - 48], [839, 839, -5*w - 43], [853, 853, -7*w - 36], [853, 853, 7*w - 43], [881, 881, -w - 30], [881, 881, w - 31], [961, 31, -31], [991, 991, -5*w - 13], [991, 991, 5*w - 18]]; primes := [ideal : I in primesArray]; heckePol := x^4 - x^3 - 18*x^2 + 41*x - 19; K := NumberField(heckePol); heckeEigenvaluesArray := [0, -1/7*e^3 - 3/7*e^2 + 6/7*e + 11/7, e, 0, 1/7*e^3 - 4/7*e^2 - 27/7*e + 59/7, -4/7*e^3 - 5/7*e^2 + 66/7*e - 40/7, 2/7*e^3 - 1/7*e^2 - 26/7*e + 62/7, -5/7*e^3 - 8/7*e^2 + 65/7*e - 15/7, 1/7*e^3 + 3/7*e^2 - 13/7*e - 39/7, -1/7*e^3 - 3/7*e^2 + 6/7*e + 25/7, e + 2, 4/7*e^3 + 5/7*e^2 - 59/7*e + 47/7, -2/7*e^3 + 1/7*e^2 + 33/7*e - 41/7, -8/7*e^3 - 3/7*e^2 + 139/7*e - 115/7, 8/7*e^3 + 3/7*e^2 - 139/7*e + 171/7, -e^3 - e^2 + 15*e - 16, 5/7*e^3 + 1/7*e^2 - 79/7*e + 64/7, -9/7*e^3 - 6/7*e^2 + 152/7*e - 146/7, e^3 - 18*e + 20, 5/7*e^3 + 1/7*e^2 - 79/7*e + 127/7, -e^3 - e^2 + 15*e - 7, -3/7*e^3 - 2/7*e^2 + 53/7*e - 44/7, 2/7*e^3 - 1/7*e^2 - 40/7*e + 55/7, 10/7*e^3 + 9/7*e^2 - 158/7*e + 156/7, -e^3 + 17*e - 17, -5/7*e^3 - 1/7*e^2 + 79/7*e - 36/7, e^3 + e^2 - 15*e + 20, -4/7*e^3 - 5/7*e^2 + 73/7*e - 54/7, -e^2 - 3*e + 8, 3/7*e^3 + 2/7*e^2 - 39/7*e + 23/7, -4/7*e^3 - 5/7*e^2 + 52/7*e - 54/7, 10/7*e^3 + 9/7*e^2 - 165/7*e + 205/7, -6/7*e^3 + 3/7*e^2 + 113/7*e - 81/7, e^3 - 20*e + 14, -e^3 + 20*e - 30, 6/7*e^3 + 18/7*e^2 - 78/7*e - 52/7, 3/7*e^3 - 12/7*e^2 - 88/7*e + 177/7, -11/7*e^3 - 12/7*e^2 + 192/7*e - 131/7, -8/7*e^3 + 4/7*e^2 + 153/7*e - 206/7, 13/7*e^3 + 11/7*e^2 - 218/7*e + 179/7, 2*e^3 - 37*e + 36, -17/7*e^3 - 9/7*e^2 + 298/7*e - 331/7, 1/7*e^3 + 10/7*e^2 + 22/7*e - 165/7, 3/7*e^3 + 2/7*e^2 - 74/7*e - 33/7, 4/7*e^3 + 12/7*e^2 - 52/7*e + 26/7, 9/7*e^3 - 1/7*e^2 - 194/7*e + 244/7, -8/7*e^3 + 4/7*e^2 + 181/7*e - 185/7, 3/7*e^3 + 23/7*e^2 - 11/7*e - 145/7, 13/7*e^3 + 25/7*e^2 - 197/7*e + 53/7, -12/7*e^3 - 1/7*e^2 + 219/7*e - 225/7, 2*e^3 + e^2 - 35*e + 37, -25/7*e^3 - 12/7*e^2 + 395/7*e - 432/7, 4*e^3 + 3*e^2 - 62*e + 53, 5/7*e^3 + 15/7*e^2 - 65/7*e + 57/7, 2/7*e^3 - 22/7*e^2 - 68/7*e + 251/7, -20/7*e^3 - 32/7*e^2 + 302/7*e - 123/7, 2*e^2 + 3*e - 8, 11/7*e^3 + 19/7*e^2 - 164/7*e + 131/7, -4/7*e^3 + 9/7*e^2 + 87/7*e - 103/7, 12/7*e^3 + 15/7*e^2 - 191/7*e + 183/7, 5/7*e^3 - 6/7*e^2 - 72/7*e + 190/7, -15/7*e^3 - 24/7*e^2 + 202/7*e - 52/7, 6/7*e^3 - 10/7*e^2 - 148/7*e + 319/7, -12/7*e^3 - 8/7*e^2 + 226/7*e - 99/7, -5/7*e^3 - 15/7*e^2 + 65/7*e - 15/7, 19/7*e^3 + 22/7*e^2 - 268/7*e + 148/7, -13/7*e^3 - 4/7*e^2 + 190/7*e - 270/7, 2/7*e^3 - 8/7*e^2 - 54/7*e + 125/7, -8/7*e^3 - 10/7*e^2 + 132/7*e - 73/7, 5/7*e^3 - 6/7*e^2 - 135/7*e + 197/7, -6/7*e^3 + 3/7*e^2 + 148/7*e - 144/7, -8/7*e^3 + 11/7*e^2 + 167/7*e - 213/7, 18/7*e^3 + 19/7*e^2 - 297/7*e + 271/7, 2/7*e^3 + 13/7*e^2 - 5/7*e - 50/7, 6/7*e^3 + 11/7*e^2 - 99/7*e + 60/7, -8/7*e^3 - 3/7*e^2 + 125/7*e - 73/7, 10/7*e^3 + 9/7*e^2 - 151/7*e + 191/7, -10/7*e^3 + 5/7*e^2 + 172/7*e - 219/7, 19/7*e^3 + 22/7*e^2 - 289/7*e + 232/7, -13/7*e^3 - 4/7*e^2 + 162/7*e - 312/7, 23/7*e^3 + 34/7*e^2 - 292/7*e + 62/7, 13/7*e^3 - 3/7*e^2 - 274/7*e + 347/7, -2*e^3 + 41*e - 40, 26/7*e^3 + 15/7*e^2 - 429/7*e + 379/7, -24/7*e^3 - 9/7*e^2 + 403/7*e - 457/7, 6/7*e^3 + 25/7*e^2 - 92/7*e - 164/7, 15/7*e^3 + 38/7*e^2 - 181/7*e - 109/7, -17/7*e^3 - 16/7*e^2 + 298/7*e - 324/7, e^3 - 2*e^2 - 24*e + 26, -3*e^2 - 6*e + 26, -15/7*e^3 - 24/7*e^2 + 237/7*e - 115/7, 18/7*e^3 + 33/7*e^2 - 248/7*e + 82/7, -1/7*e^3 + 18/7*e^2 + 27/7*e - 171/7, -2*e^3 - 4*e^2 + 31*e + 2, -5/7*e^3 - 29/7*e^2 + 30/7*e + 223/7, 18/7*e^3 + 26/7*e^2 - 262/7*e + 264/7, -6/7*e^3 + 10/7*e^2 + 106/7*e - 88/7, -4/7*e^3 - 5/7*e^2 + 87/7*e - 47/7, -2/7*e^3 - 13/7*e^2 - 9/7*e + 85/7, 3/7*e^3 - 12/7*e^2 - 18/7*e + 254/7, -3*e^3 - 6*e^2 + 36*e + 8, 17/7*e^3 + 30/7*e^2 - 242/7*e + 149/7, -1/7*e^3 + 18/7*e^2 + 34/7*e - 115/7, -4*e^3 - 4*e^2 + 59*e - 40, 3*e^3 + e^2 - 46*e + 59, 11/7*e^3 - 2/7*e^2 - 206/7*e + 334/7, -15/7*e^3 - 10/7*e^2 + 258/7*e - 150/7, 11/7*e^3 + 26/7*e^2 - 150/7*e + 96/7, 5/7*e^3 + 22/7*e^2 - 58/7*e + 8/7, 8/7*e^3 + 10/7*e^2 - 97/7*e + 73/7, -e^3 - e^2 + 12*e - 10, -33/7*e^3 - 1/7*e^2 + 590/7*e - 624/7, 6*e^3 + 4*e^2 - 101*e + 101, 8/7*e^3 + 24/7*e^2 - 69/7*e - 235/7, 3/7*e^3 + 9/7*e^2 - 74/7*e - 180/7, -29/7*e^3 - 31/7*e^2 + 426/7*e - 283/7, 20/7*e^3 + 4/7*e^2 - 309/7*e + 410/7, -4/7*e^3 - 26/7*e^2 + 17/7*e + 86/7, -13/7*e^3 - 25/7*e^2 + 204/7*e - 123/7, 9/7*e^3 + 27/7*e^2 - 117/7*e + 62/7, 9/7*e^3 + 27/7*e^2 - 117/7*e + 62/7, 18/7*e^3 + 5/7*e^2 - 311/7*e + 278/7, -20/7*e^3 - 11/7*e^2 + 337/7*e - 382/7, 9/7*e^3 + 6/7*e^2 - 173/7*e + 132/7, -4/7*e^3 + 9/7*e^2 + 108/7*e - 187/7, 32/7*e^3 + 26/7*e^2 - 507/7*e + 572/7, -25/7*e^3 - 5/7*e^2 + 416/7*e - 341/7, -8/7*e^3 - 24/7*e^2 + 76/7*e + 221/7, -4/7*e^3 - 12/7*e^2 + 80/7*e + 177/7, -3*e^3 + 48*e - 78, 33/7*e^3 + 36/7*e^2 - 492/7*e + 246/7, 15/7*e^3 - 25/7*e^2 - 321/7*e + 514/7, -37/7*e^3 - 41/7*e^2 + 607/7*e - 454/7, -20/7*e^3 - 32/7*e^2 + 274/7*e - 39/7, 6/7*e^3 - 10/7*e^2 - 92/7*e + 291/7, 1/7*e^3 - 11/7*e^2 - 97/7*e + 31/7, -1/7*e^3 + 11/7*e^2 + 97/7*e - 255/7, -18/7*e^3 + 2/7*e^2 + 297/7*e - 327/7, 29/7*e^3 + 31/7*e^2 - 440/7*e + 388/7, 15/7*e^3 + 17/7*e^2 - 293/7*e + 115/7, 1/7*e^3 + 31/7*e^2 + 85/7*e - 347/7, 8/7*e^3 + 10/7*e^2 - 174/7*e + 150/7, 4/7*e^3 + 26/7*e^2 + 18/7*e - 114/7, 15/7*e^3 + 24/7*e^2 - 216/7*e + 108/7, -3/7*e^3 + 12/7*e^2 + 60/7*e - 156/7, -9/7*e^3 - 6/7*e^2 + 194/7*e - 160/7, 1/7*e^3 - 18/7*e^2 - 90/7*e + 192/7, -43/7*e^3 - 38/7*e^2 + 643/7*e - 458/7, 36/7*e^3 + 17/7*e^2 - 552/7*e + 675/7, 15/7*e^3 + 3/7*e^2 - 223/7*e + 178/7, -23/7*e^3 - 27/7*e^2 + 327/7*e - 328/7, 18/7*e^3 + 26/7*e^2 - 311/7*e + 40/7, 1/7*e^3 + 31/7*e^2 + 64/7*e - 389/7, -5/7*e^3 - 29/7*e^2 + 79/7*e + 258/7, -3*e^3 - 7*e^2 + 37*e + 18, 12/7*e^3 + 15/7*e^2 - 142/7*e + 225/7, -11/7*e^3 - 12/7*e^2 + 129/7*e + 16/7, -2*e^2 - 2*e + 30, -12/7*e^3 - 22/7*e^2 + 170/7*e + 34/7, -40/7*e^3 - 36/7*e^2 + 681/7*e - 582/7, 3*e^3 - 3*e^2 - 62*e + 85, -2/7*e^3 - 27/7*e^2 + 26/7*e + 120/7, -23/7*e^3 - 48/7*e^2 + 299/7*e - 111/7, 7*e^3 + 7*e^2 - 111*e + 86, -29/7*e^3 + 11/7*e^2 + 517/7*e - 696/7, 10/7*e^3 + 30/7*e^2 - 130/7*e - 180/7, 22/7*e^3 - 11/7*e^2 - 412/7*e + 640/7, -37/7*e^3 - 34/7*e^2 + 607/7*e - 405/7]; 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;