/* 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, 5, -5, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [5, 5, w^3 - 4*w], [7, 7, w + 1], [7, 7, -w^3 + 4*w - 1], [13, 13, -w^2 + 3], [16, 2, 2], [17, 17, w^3 + w^2 - 4*w - 2], [17, 17, -w^3 + 5*w - 2], [19, 19, -w^2 - w + 4], [19, 19, -w^2 - w + 1], [29, 29, 2*w^3 - w^2 - 9*w + 5], [41, 41, -w^3 + w^2 + 5*w - 3], [43, 43, w^3 - w^2 - 4*w + 2], [43, 43, w^3 - 6*w], [47, 47, -w^3 - w^2 + 5*w], [49, 7, w^2 + 2*w - 2], [59, 59, 2*w^3 - 8*w + 3], [67, 67, w^2 - w - 4], [79, 79, w^3 - w^2 - 4*w + 1], [81, 3, -3], [97, 97, w^3 + w^2 - 5*w + 1], [107, 107, -2*w^3 + 3*w^2 + 10*w - 12], [109, 109, 2*w^3 - w^2 - 9*w + 3], [125, 5, -3*w^3 - w^2 + 12*w + 1], [127, 127, -2*w^3 + w^2 + 10*w - 8], [137, 137, -w^3 + w^2 + 3*w - 5], [139, 139, w^3 - w^2 - 5*w + 1], [149, 149, 2*w^2 - w - 6], [149, 149, 2*w^3 + w^2 - 9*w], [163, 163, w^3 - 5*w - 3], [163, 163, 2*w^3 - 7*w], [173, 173, -2*w^3 + 11*w - 3], [173, 173, -w^3 + w^2 + 4*w - 8], [179, 179, 2*w^3 - w^2 - 8*w + 5], [181, 181, 2*w^3 - w^2 - 8*w], [181, 181, 2*w^3 - 9*w - 3], [191, 191, 2*w^2 - 5], [191, 191, -2*w^2 - w + 10], [193, 193, w^3 + w^2 - 3*w - 4], [193, 193, -w^3 + 2*w^2 + 3*w - 3], [197, 197, -2*w^3 + 9*w - 4], [197, 197, w^2 - 7], [199, 199, 3*w^3 - 13*w + 1], [199, 199, w^3 + 2*w^2 - 5*w - 7], [227, 227, -3*w^3 + w^2 + 13*w - 4], [227, 227, -w^3 + 2*w^2 + 7*w - 6], [241, 241, -2*w^3 + 7*w - 1], [257, 257, -2*w^3 + w^2 + 7*w - 5], [257, 257, 2*w^3 - w^2 - 7*w + 2], [263, 263, -w^3 + 7*w - 3], [269, 269, -2*w^3 + 2*w^2 + 11*w - 8], [271, 271, w^2 + 3*w - 2], [271, 271, w^2 + 2*w - 7], [277, 277, -2*w^3 + 11*w - 2], [281, 281, 2*w^3 - 11*w + 1], [283, 283, 2*w^3 - 9*w + 5], [283, 283, -2*w^3 + w^2 + 11*w - 7], [289, 17, -w^3 + 2*w^2 + 4*w - 4], [293, 293, -2*w^3 - w^2 + 7*w + 3], [307, 307, -3*w^3 + 2*w^2 + 13*w - 9], [311, 311, -w^3 + w^2 + 3*w - 7], [313, 313, -w^3 + 2*w^2 + 3*w - 2], [313, 313, w^2 - w - 8], [317, 317, -2*w^3 - w^2 + 9*w - 1], [331, 331, w^3 + 2*w^2 - 5*w - 3], [337, 337, w^3 + 2*w^2 - 6*w - 4], [337, 337, w^3 - 2*w^2 - 6*w + 5], [347, 347, 2*w^3 + 2*w^2 - 9*w - 3], [347, 347, -2*w^3 + w^2 + 7*w - 4], [349, 349, w^2 - 2*w - 4], [353, 353, -w^3 + 2*w^2 + 2*w - 6], [359, 359, -w^3 + w^2 + 7*w - 6], [359, 359, -w^2 - w - 2], [361, 19, 2*w^3 - w^2 - 11*w], [367, 367, -2*w^3 + 2*w^2 + 8*w - 5], [373, 373, w^3 + w^2 - 2*w - 4], [379, 379, 2*w^3 - w^2 - 11*w + 4], [389, 389, 3*w^2 + w - 11], [397, 397, 3*w^3 - 13*w + 5], [401, 401, w - 5], [409, 409, 3*w^3 - 12*w + 4], [409, 409, 3*w^3 + 2*w^2 - 14*w - 4], [421, 421, 2*w^3 - w^2 - 10*w + 1], [421, 421, w^3 + 2*w^2 - 5*w - 4], [439, 439, -2*w^3 + 10*w - 5], [443, 443, -2*w^3 + 12*w - 5], [443, 443, -4*w^3 + w^2 + 18*w - 9], [443, 443, -2*w^3 + w^2 + 7*w - 9], [443, 443, 3*w^3 + 2*w^2 - 14*w - 5], [449, 449, w^3 + 2*w^2 - 2*w - 6], [449, 449, -2*w^3 + w^2 + 11*w - 5], [461, 461, 2*w^3 + 2*w^2 - 9*w - 4], [467, 467, -w^3 + w^2 + 7*w - 5], [487, 487, 2*w^2 + 4*w - 7], [491, 491, -w^3 + w^2 + 7*w - 4], [503, 503, 2*w^3 + w^2 - 9*w + 2], [521, 521, w^2 - 3*w - 3], [523, 523, -w^3 + 2*w^2 + 5*w - 4], [523, 523, -w^3 + 8*w - 5], [563, 563, w^3 + 3*w^2 - w - 8], [563, 563, w^3 - 2*w^2 - 8*w + 4], [563, 563, -w^3 + w^2 + 2*w - 5], [563, 563, -3*w^3 - w^2 + 10*w - 2], [577, 577, 3*w^3 - 16*w], [587, 587, w^3 + 2*w^2 - 6], [587, 587, -w^3 + 4*w - 6], [601, 601, -w - 5], [607, 607, w^3 - 2*w^2 - 4*w + 1], [613, 613, 4*w^3 - 17*w], [617, 617, -3*w^3 + w^2 + 14*w - 2], [617, 617, -4*w^3 + 17*w - 4], [619, 619, -3*w^3 + 12*w - 2], [643, 643, 3*w^3 - w^2 - 12*w + 3], [643, 643, -2*w^3 - w^2 + 11*w - 2], [647, 647, 2*w^3 + w^2 - 9*w + 3], [653, 653, w^3 + w^2 - 7*w - 4], [661, 661, -w^3 + 4*w^2 + 6*w - 15], [661, 661, 3*w^3 - 2*w^2 - 12*w + 10], [677, 677, -3*w^2 - 2*w + 12], [691, 691, -2*w^3 + 3*w^2 + 11*w - 10], [691, 691, -3*w^3 + 16*w - 3], [701, 701, 3*w^3 - 13*w + 6], [719, 719, -3*w^2 + w + 10], [727, 727, w^3 - 3*w^2 - 4*w + 6], [743, 743, -w^3 + 2*w^2 + 5*w - 3], [761, 761, -w^3 + 3*w^2 + 3*w - 12], [761, 761, 3*w^3 - 2*w^2 - 14*w + 7], [769, 769, -3*w^3 + 3*w^2 + 16*w - 13], [769, 769, w^3 + 2*w^2 - 7*w - 6], [773, 773, w^3 - 8*w + 4], [773, 773, 4*w^2 + w - 16], [787, 787, w^2 + 4*w - 2], [797, 797, -2*w^3 + 3*w^2 + 10*w - 10], [797, 797, -w^3 - w^2 + 4*w - 4], [797, 797, w^3 + 2*w^2 - 3*w - 9], [797, 797, -3*w^3 + 3*w^2 + 11*w - 10], [821, 821, w^2 + w - 9], [827, 827, w^3 - 8*w + 1], [827, 827, -2*w^3 + w^2 + 9*w - 11], [827, 827, 3*w^3 - 14*w + 5], [827, 827, 2*w^3 + w^2 - 7*w - 5], [829, 829, -2*w^3 - 2*w^2 + 9*w - 1], [839, 839, -3*w^3 + 15*w - 5], [853, 853, -4*w^3 + 15*w - 8], [853, 853, -w^3 + 2*w^2 + 6*w - 3], [859, 859, -w^3 + w^2 + w - 4], [859, 859, w^3 + w^2 - w - 5], [881, 881, -w^3 + w^2 + 2*w - 6], [881, 881, w^3 + w^2 - 2*w - 6], [883, 883, w^3 + 3*w^2 - 3*w - 6], [887, 887, w^3 + 3*w^2 - 3*w - 8], [919, 919, 2*w^3 - w^2 - 13*w], [929, 929, -w^3 - 3*w^2 + 2*w + 10], [929, 929, w^3 + w^2 - 5*w - 8], [941, 941, w^3 + 2*w^2 - 3*w - 10], [953, 953, -w^3 + 5*w - 7], [953, 953, -2*w^3 + 3*w^2 + 9*w - 9], [953, 953, -2*w^3 - w^2 + 10*w - 3], [953, 953, -3*w^3 + 11*w - 1], [961, 31, -w^3 + 3*w^2 + 2*w - 10], [961, 31, -3*w^2 + 2*w + 7], [967, 967, w^3 + 3*w^2 - 3*w - 7], [967, 967, 4*w^3 + w^2 - 20*w + 1], [983, 983, 2*w^3 - w^2 - 12*w + 8], [983, 983, 2*w^3 + w^2 - 6*w - 4], [991, 991, 2*w^3 - 12*w + 3]]; primes := [ideal : I in primesArray]; heckePol := x^5 + 7*x^4 + 5*x^3 - 52*x^2 - 101*x - 25; K := NumberField(heckePol); heckeEigenvaluesArray := [e, 2/5*e^4 + 7/5*e^3 - 17/5*e^2 - 52/5*e + 1, e^4 + 4*e^3 - 7*e^2 - 32*e - 10, -6/5*e^4 - 26/5*e^3 + 36/5*e^2 + 201/5*e + 15, 2/5*e^4 + 7/5*e^3 - 12/5*e^2 - 52/5*e - 10, -1/5*e^4 - 6/5*e^3 + 6/5*e^2 + 51/5*e, 3/5*e^4 + 13/5*e^3 - 18/5*e^2 - 103/5*e - 7, 1/5*e^4 + 6/5*e^3 - 1/5*e^2 - 51/5*e - 9, 1, -1/5*e^4 - 1/5*e^3 + 11/5*e^2 + 6/5*e - 6, -11/5*e^4 - 41/5*e^3 + 81/5*e^2 + 311/5*e + 8, e^3 + 2*e^2 - 8*e - 8, 7/5*e^4 + 27/5*e^3 - 52/5*e^2 - 222/5*e - 11, 6/5*e^4 + 26/5*e^3 - 46/5*e^2 - 206/5*e, 6/5*e^4 + 26/5*e^3 - 36/5*e^2 - 191/5*e - 17, -14/5*e^4 - 59/5*e^3 + 89/5*e^2 + 454/5*e + 29, 2*e^4 + 8*e^3 - 13*e^2 - 65*e - 29, 2*e^4 + 8*e^3 - 15*e^2 - 64*e - 8, -7/5*e^4 - 32/5*e^3 + 42/5*e^2 + 252/5*e + 14, -12/5*e^4 - 47/5*e^3 + 82/5*e^2 + 372/5*e + 19, 2*e^4 + 7*e^3 - 13*e^2 - 53*e - 33, -14/5*e^4 - 49/5*e^3 + 104/5*e^2 + 374/5*e + 17, -17/5*e^4 - 62/5*e^3 + 127/5*e^2 + 487/5*e + 22, -19/5*e^4 - 79/5*e^3 + 119/5*e^2 + 624/5*e + 52, -6/5*e^4 - 21/5*e^3 + 56/5*e^2 + 166/5*e - 14, 2*e^3 + 2*e^2 - 20*e - 15, -8/5*e^4 - 38/5*e^3 + 48/5*e^2 + 298/5*e + 14, -11/5*e^4 - 46/5*e^3 + 71/5*e^2 + 381/5*e + 35, -3*e^4 - 12*e^3 + 20*e^2 + 95*e + 30, -4*e^4 - 17*e^3 + 28*e^2 + 136*e + 31, -6/5*e^4 - 31/5*e^3 + 16/5*e^2 + 236/5*e + 38, -6/5*e^4 - 31/5*e^3 + 26/5*e^2 + 236/5*e + 16, 11/5*e^4 + 51/5*e^3 - 66/5*e^2 - 411/5*e - 36, 3*e^2 + 6*e - 19, 3/5*e^4 + 8/5*e^3 - 38/5*e^2 - 78/5*e + 10, 12/5*e^4 + 47/5*e^3 - 92/5*e^2 - 377/5*e - 2, 2*e^4 + 8*e^3 - 14*e^2 - 57*e - 5, -4/5*e^4 - 19/5*e^3 + 29/5*e^2 + 159/5*e + 2, -4/5*e^4 - 14/5*e^3 + 24/5*e^2 + 109/5*e + 6, -1/5*e^4 + 4/5*e^3 + 21/5*e^2 - 44/5*e - 14, -e^4 - 2*e^3 + 12*e^2 + 16*e - 19, -3/5*e^4 - 13/5*e^3 + 28/5*e^2 + 103/5*e - 12, 2/5*e^4 + 2/5*e^3 - 37/5*e^2 - 17/5*e + 14, 3*e^4 + 12*e^3 - 17*e^2 - 91*e - 48, -3*e^4 - 13*e^3 + 16*e^2 + 98*e + 55, 11/5*e^4 + 41/5*e^3 - 71/5*e^2 - 316/5*e - 25, -2/5*e^4 - 17/5*e^3 - 8/5*e^2 + 132/5*e + 37, 6/5*e^4 + 21/5*e^3 - 51/5*e^2 - 141/5*e + 17, 18/5*e^4 + 73/5*e^3 - 113/5*e^2 - 563/5*e - 50, 4/5*e^4 + 4/5*e^3 - 44/5*e^2 + 21/5*e + 18, 18/5*e^4 + 78/5*e^3 - 118/5*e^2 - 648/5*e - 47, 7/5*e^4 + 17/5*e^3 - 67/5*e^2 - 127/5*e + 17, -6*e^4 - 26*e^3 + 37*e^2 + 207*e + 83, -e^4 - 2*e^3 + 13*e^2 + 16*e - 21, -e^4 - 3*e^3 + 8*e^2 + 23*e + 2, 2*e^4 + 8*e^3 - 12*e^2 - 63*e - 21, 3/5*e^4 + 8/5*e^3 - 23/5*e^2 - 48/5*e - 14, -17/5*e^4 - 57/5*e^3 + 132/5*e^2 + 407/5*e + 1, -28/5*e^4 - 108/5*e^3 + 203/5*e^2 + 838/5*e + 33, -2*e^4 - 7*e^3 + 17*e^2 + 57*e - 2, -12/5*e^4 - 52/5*e^3 + 82/5*e^2 + 442/5*e + 25, -7/5*e^4 - 32/5*e^3 + 57/5*e^2 + 262/5*e - 15, 12/5*e^4 + 62/5*e^3 - 62/5*e^2 - 487/5*e - 42, 13/5*e^4 + 48/5*e^3 - 113/5*e^2 - 388/5*e + 5, -13/5*e^4 - 58/5*e^3 + 68/5*e^2 + 443/5*e + 43, 3/5*e^4 + 18/5*e^3 - 13/5*e^2 - 178/5*e - 18, e^4 + 6*e^3 - 5*e^2 - 49*e - 7, 8/5*e^4 + 43/5*e^3 - 48/5*e^2 - 368/5*e - 20, 5*e^4 + 20*e^3 - 36*e^2 - 160*e - 34, 18/5*e^4 + 63/5*e^3 - 143/5*e^2 - 503/5*e - 17, 3*e^4 + 9*e^3 - 25*e^2 - 66*e, -32/5*e^4 - 132/5*e^3 + 212/5*e^2 + 1027/5*e + 58, 11/5*e^4 + 36/5*e^3 - 91/5*e^2 - 246/5*e + 2, 13/5*e^4 + 58/5*e^3 - 93/5*e^2 - 483/5*e - 17, -2*e^4 - 10*e^3 + 14*e^2 + 81*e + 8, -6/5*e^4 - 11/5*e^3 + 61/5*e^2 + 66/5*e - 15, 19/5*e^4 + 84/5*e^3 - 104/5*e^2 - 654/5*e - 62, 2/5*e^4 + 7/5*e^3 - 7/5*e^2 - 42/5*e - 3, -4/5*e^4 - 19/5*e^3 + 9/5*e^2 + 104/5*e + 33, -21/5*e^4 - 81/5*e^3 + 126/5*e^2 + 586/5*e + 65, -12/5*e^4 - 62/5*e^3 + 57/5*e^2 + 472/5*e + 48, 11/5*e^4 + 36/5*e^3 - 91/5*e^2 - 276/5*e - 24, 2*e^4 + 11*e^3 - 8*e^2 - 95*e - 55, 16/5*e^4 + 66/5*e^3 - 96/5*e^2 - 506/5*e - 42, -4*e^4 - 16*e^3 + 26*e^2 + 127*e + 33, 3/5*e^4 - 2/5*e^3 - 33/5*e^2 + 47/5*e + 9, -3/5*e^4 - 8/5*e^3 + 38/5*e^2 + 38/5*e - 25, -19/5*e^4 - 84/5*e^3 + 114/5*e^2 + 674/5*e + 55, 3*e^4 + 10*e^3 - 23*e^2 - 72*e - 12, 12/5*e^4 + 22/5*e^3 - 122/5*e^2 - 127/5*e + 26, 2*e^4 + 10*e^3 - 10*e^2 - 75*e - 14, 3*e^2 + 7*e - 18, -26/5*e^4 - 106/5*e^3 + 161/5*e^2 + 806/5*e + 71, -7/5*e^4 - 22/5*e^3 + 47/5*e^2 + 187/5*e + 37, 31/5*e^4 + 136/5*e^3 - 191/5*e^2 - 1111/5*e - 94, 1/5*e^4 - 19/5*e^3 - 46/5*e^2 + 189/5*e + 41, -29/5*e^4 - 129/5*e^3 + 189/5*e^2 + 1014/5*e + 56, e^4 + 4*e^3 - 10*e^2 - 39*e + 13, -3/5*e^4 - 28/5*e^3 - 17/5*e^2 + 228/5*e + 33, -16/5*e^4 - 56/5*e^3 + 116/5*e^2 + 411/5*e, 13/5*e^4 + 43/5*e^3 - 83/5*e^2 - 293/5*e - 40, -28/5*e^4 - 118/5*e^3 + 188/5*e^2 + 928/5*e + 46, -24/5*e^4 - 89/5*e^3 + 179/5*e^2 + 704/5*e + 10, 5*e^4 + 23*e^3 - 29*e^2 - 189*e - 84, 21/5*e^4 + 81/5*e^3 - 171/5*e^2 - 626/5*e + 4, 1/5*e^4 - 4/5*e^3 - 16/5*e^2 + 74/5*e + 4, 13/5*e^4 + 38/5*e^3 - 113/5*e^2 - 278/5*e + 9, 3*e^4 + 10*e^3 - 23*e^2 - 69*e + 2, 29/5*e^4 + 124/5*e^3 - 154/5*e^2 - 954/5*e - 99, -4/5*e^4 - 39/5*e^3 - 21/5*e^2 + 299/5*e + 39, -4/5*e^4 + 1/5*e^3 + 39/5*e^2 - 81/5*e - 18, 6*e^4 + 28*e^3 - 35*e^2 - 216*e - 78, 18/5*e^4 + 68/5*e^3 - 128/5*e^2 - 493/5*e + 1, 4*e^4 + 18*e^3 - 20*e^2 - 143*e - 81, 24/5*e^4 + 114/5*e^3 - 134/5*e^2 - 879/5*e - 80, 2*e^3 + 7*e^2 - 6*e - 38, -4/5*e^4 - 9/5*e^3 + 24/5*e^2 + 59/5*e + 24, -4/5*e^4 - 19/5*e^3 + 9/5*e^2 + 174/5*e + 18, 28/5*e^4 + 118/5*e^3 - 148/5*e^2 - 903/5*e - 95, -7/5*e^4 - 22/5*e^3 + 57/5*e^2 + 142/5*e - 14, 13/5*e^4 + 48/5*e^3 - 123/5*e^2 - 408/5*e + 12, 44/5*e^4 + 169/5*e^3 - 314/5*e^2 - 1334/5*e - 64, -21/5*e^4 - 86/5*e^3 + 111/5*e^2 + 656/5*e + 73, 18/5*e^4 + 63/5*e^3 - 168/5*e^2 - 523/5*e + 19, 27/5*e^4 + 107/5*e^3 - 162/5*e^2 - 812/5*e - 86, -13/5*e^4 - 48/5*e^3 + 83/5*e^2 + 338/5*e + 52, -37/5*e^4 - 167/5*e^3 + 227/5*e^2 + 1312/5*e + 106, -41/5*e^4 - 166/5*e^3 + 271/5*e^2 + 1291/5*e + 54, -8*e^4 - 32*e^3 + 58*e^2 + 254*e + 51, -8*e^4 - 33*e^3 + 54*e^2 + 253*e + 65, 19/5*e^4 + 74/5*e^3 - 114/5*e^2 - 574/5*e - 53, 56/5*e^4 + 241/5*e^3 - 341/5*e^2 - 1881/5*e - 149, 4*e^4 + 20*e^3 - 21*e^2 - 153*e - 53, -36/5*e^4 - 141/5*e^3 + 271/5*e^2 + 1166/5*e + 36, 2*e^4 + 10*e^3 - 6*e^2 - 68*e - 52, e^4 + 5*e^3 - 4*e^2 - 46*e - 31, -51/5*e^4 - 211/5*e^3 + 331/5*e^2 + 1661/5*e + 117, 34/5*e^4 + 124/5*e^3 - 239/5*e^2 - 914/5*e - 24, -3*e^4 - 11*e^3 + 23*e^2 + 98*e + 15, 1/5*e^4 + 16/5*e^3 + 19/5*e^2 - 151/5*e - 53, 4*e^3 + 11*e^2 - 30*e - 42, 44/5*e^4 + 179/5*e^3 - 319/5*e^2 - 1404/5*e - 50, -4*e^4 - 12*e^3 + 34*e^2 + 79*e - 27, -2*e^4 - 7*e^3 + 16*e^2 + 54*e - 26, 6/5*e^4 + 36/5*e^3 - 21/5*e^2 - 261/5*e - 32, -2*e^4 - 9*e^3 + 9*e^2 + 62*e + 46, 7/5*e^4 + 37/5*e^3 - 17/5*e^2 - 262/5*e - 45, 29/5*e^4 + 129/5*e^3 - 169/5*e^2 - 959/5*e - 76, 26/5*e^4 + 111/5*e^3 - 166/5*e^2 - 841/5*e - 37, -23/5*e^4 - 88/5*e^3 + 153/5*e^2 + 653/5*e + 44, -24/5*e^4 - 109/5*e^3 + 159/5*e^2 + 874/5*e + 41, -33/5*e^4 - 143/5*e^3 + 208/5*e^2 + 1193/5*e + 90, 9/5*e^4 + 44/5*e^3 - 69/5*e^2 - 399/5*e + 11, 12/5*e^4 + 42/5*e^3 - 52/5*e^2 - 292/5*e - 70, 32/5*e^4 + 142/5*e^3 - 197/5*e^2 - 1097/5*e - 79, 2*e^4 + 5*e^3 - 18*e^2 - 37*e - 30, 4*e^4 + 15*e^3 - 30*e^2 - 121*e - 6, -2*e^4 - 8*e^3 + 17*e^2 + 66*e + 14, -11/5*e^4 - 41/5*e^3 + 66/5*e^2 + 376/5*e + 49, -39/5*e^4 - 144/5*e^3 + 279/5*e^2 + 1079/5*e + 47, -13/5*e^4 - 58/5*e^3 + 98/5*e^2 + 463/5*e - 4, 59/5*e^4 + 234/5*e^3 - 409/5*e^2 - 1814/5*e - 68, -4*e^4 - 17*e^3 + 28*e^2 + 123*e + 11, 2*e^4 + 5*e^3 - 23*e^2 - 44*e + 36, 21/5*e^4 + 111/5*e^3 - 91/5*e^2 - 836/5*e - 89]; 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;