/* 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, -12, 12, 7, -7, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, -w^5 + 6*w^3 + w^2 - 8*w - 2], [11, 11, w + 1], [16, 2, -w^5 + 6*w^3 + w^2 - 7*w - 2], [19, 19, -w^3 + w^2 + 4*w - 3], [29, 29, 2*w^5 - 13*w^3 + 19*w - 2], [31, 31, -2*w^5 + 12*w^3 + w^2 - 16*w - 2], [41, 41, -w^5 + 7*w^3 + w^2 - 12*w - 2], [41, 41, 2*w^5 + w^4 - 13*w^3 - 5*w^2 + 19*w + 4], [59, 59, -w^5 + 7*w^3 + w^2 - 11*w], [61, 61, w^5 - 7*w^3 + 10*w], [61, 61, -2*w^5 + 12*w^3 + w^2 - 16*w - 3], [71, 71, -2*w^5 - w^4 + 12*w^3 + 5*w^2 - 16*w - 4], [71, 71, -3*w^5 - w^4 + 20*w^3 + 6*w^2 - 31*w - 5], [71, 71, -w^5 + 7*w^3 - w^2 - 12*w + 2], [79, 79, w^5 + w^4 - 7*w^3 - 5*w^2 + 10*w + 4], [79, 79, -w^5 - w^4 + 8*w^3 + 4*w^2 - 15*w + 1], [79, 79, 2*w^5 + w^4 - 13*w^3 - 5*w^2 + 20*w + 3], [89, 89, 2*w^5 + w^4 - 14*w^3 - 5*w^2 + 23*w + 1], [89, 89, -2*w^5 - w^4 + 12*w^3 + 5*w^2 - 15*w - 4], [89, 89, 2*w^5 + w^4 - 11*w^3 - 5*w^2 + 13*w + 3], [101, 101, w^5 + w^4 - 7*w^3 - 3*w^2 + 11*w - 2], [109, 109, w^5 + w^4 - 8*w^3 - 4*w^2 + 14*w], [121, 11, -w^5 - w^4 + 8*w^3 + 4*w^2 - 13*w - 2], [125, 5, -w^5 + 6*w^3 - 8*w - 2], [131, 131, -w^4 + 5*w^2 - w - 3], [131, 131, -w^5 - w^4 + 6*w^3 + 5*w^2 - 9*w - 3], [139, 139, 2*w^5 - w^4 - 11*w^3 + 4*w^2 + 12*w - 2], [149, 149, w^5 + w^4 - 7*w^3 - 5*w^2 + 9*w + 4], [151, 151, w^5 - 8*w^3 + 2*w^2 + 15*w - 6], [151, 151, -w^5 + 7*w^3 - 12*w + 3], [151, 151, w^5 - 5*w^3 + 4*w - 3], [151, 151, -w^4 + w^3 + 5*w^2 - 4*w - 5], [179, 179, -3*w^5 - w^4 + 20*w^3 + 4*w^2 - 29*w - 1], [179, 179, -3*w^5 - 2*w^4 + 20*w^3 + 10*w^2 - 31*w - 6], [179, 179, -w^4 + 3*w^2 - 2*w + 2], [179, 179, 3*w^5 + w^4 - 18*w^3 - 4*w^2 + 23*w + 1], [181, 181, -2*w^5 - w^4 + 13*w^3 + 4*w^2 - 20*w + 1], [181, 181, w^5 - 8*w^3 + 14*w - 3], [191, 191, w^5 + w^4 - 7*w^3 - 6*w^2 + 11*w + 5], [199, 199, -2*w^5 - 2*w^4 + 14*w^3 + 10*w^2 - 23*w - 5], [199, 199, w^5 + w^4 - 8*w^3 - 6*w^2 + 14*w + 4], [199, 199, -w^5 + 6*w^3 + 2*w^2 - 8*w - 2], [199, 199, w^5 - w^4 - 6*w^3 + 4*w^2 + 8*w - 2], [211, 211, -3*w^5 - 2*w^4 + 20*w^3 + 9*w^2 - 30*w - 5], [229, 229, 2*w^5 + w^4 - 14*w^3 - 4*w^2 + 22*w - 1], [229, 229, 4*w^5 + w^4 - 25*w^3 - 5*w^2 + 35*w + 1], [239, 239, 2*w^4 - w^3 - 8*w^2 + 5*w + 2], [241, 241, w^4 - 3*w^2 + w - 2], [251, 251, -3*w^5 - 2*w^4 + 21*w^3 + 9*w^2 - 33*w - 3], [251, 251, -3*w^5 - w^4 + 21*w^3 + 4*w^2 - 34*w + 1], [251, 251, -2*w^5 - 2*w^4 + 15*w^3 + 9*w^2 - 26*w - 4], [251, 251, 2*w^5 - 11*w^3 + w^2 + 12*w - 3], [269, 269, -2*w^5 + 11*w^3 - w^2 - 12*w + 1], [271, 271, -w^5 + 5*w^3 - w^2 - 5*w + 3], [281, 281, w^4 - w^3 - 5*w^2 + 5*w + 5], [289, 17, -2*w^5 - w^4 + 11*w^3 + 4*w^2 - 12*w - 2], [311, 311, -3*w^5 + 19*w^3 - w^2 - 26*w + 4], [311, 311, -w^4 + 3*w^2 - w + 3], [311, 311, 3*w^5 - 19*w^3 + w^2 + 27*w - 4], [311, 311, w^5 + 2*w^4 - 7*w^3 - 10*w^2 + 12*w + 8], [331, 331, -3*w^5 - w^4 + 19*w^3 + 4*w^2 - 27*w - 1], [349, 349, -2*w^5 - 2*w^4 + 14*w^3 + 8*w^2 - 22*w - 3], [349, 349, -2*w^5 - w^4 + 14*w^3 + 4*w^2 - 22*w + 2], [349, 349, 3*w^5 + w^4 - 19*w^3 - 6*w^2 + 25*w + 4], [359, 359, -4*w^5 - w^4 + 25*w^3 + 4*w^2 - 35*w + 2], [359, 359, 4*w^5 + w^4 - 25*w^3 - 6*w^2 + 34*w + 4], [361, 19, 2*w^5 - w^4 - 12*w^3 + 5*w^2 + 15*w - 6], [379, 379, 2*w^5 + 2*w^4 - 14*w^3 - 8*w^2 + 23*w + 1], [379, 379, -2*w^5 - 2*w^4 + 14*w^3 + 10*w^2 - 23*w - 7], [389, 389, w^5 - w^4 - 4*w^3 + 5*w^2 + w - 3], [401, 401, -w^5 + w^4 + 7*w^3 - 4*w^2 - 12*w + 1], [401, 401, -3*w^5 - 2*w^4 + 20*w^3 + 8*w^2 - 31*w - 2], [401, 401, 2*w^5 + w^4 - 11*w^3 - 6*w^2 + 12*w + 5], [401, 401, -4*w^5 - w^4 + 26*w^3 + 4*w^2 - 38*w + 1], [409, 409, -w^5 + w^4 + 7*w^3 - 4*w^2 - 11*w + 3], [409, 409, 3*w^5 - 17*w^3 - w^2 + 20*w + 2], [419, 419, -3*w^5 - w^4 + 21*w^3 + 4*w^2 - 33*w + 1], [419, 419, -2*w^5 + w^4 + 12*w^3 - 5*w^2 - 16*w + 4], [419, 419, -2*w^5 - 2*w^4 + 15*w^3 + 9*w^2 - 26*w - 3], [419, 419, w^2 + 2*w - 2], [421, 421, 3*w^5 + w^4 - 21*w^3 - 5*w^2 + 34*w], [421, 421, w^4 - 4*w^2 - 2*w + 2], [421, 421, -3*w^5 + 18*w^3 + w^2 - 24*w - 3], [421, 421, -2*w^5 + 14*w^3 + w^2 - 22*w + 1], [431, 431, 2*w^5 + w^4 - 13*w^3 - 3*w^2 + 18*w - 1], [431, 431, 3*w^5 + 2*w^4 - 20*w^3 - 8*w^2 + 30*w + 1], [431, 431, 2*w^5 + w^4 - 11*w^3 - 6*w^2 + 12*w + 7], [439, 439, -3*w^5 + 19*w^3 + 2*w^2 - 27*w - 3], [461, 461, -3*w^5 + 20*w^3 - w^2 - 30*w + 5], [461, 461, w^5 - 5*w^3 + 5*w - 3], [491, 491, -w^5 - w^4 + 9*w^3 + 4*w^2 - 18*w + 1], [491, 491, 2*w^5 - w^4 - 12*w^3 + 3*w^2 + 15*w], [499, 499, -3*w^5 - w^4 + 20*w^3 + 5*w^2 - 29*w - 3], [499, 499, -2*w^5 - w^4 + 12*w^3 + 4*w^2 - 16*w - 3], [509, 509, -w^5 + w^4 + 6*w^3 - 4*w^2 - 9*w], [509, 509, 2*w^5 - 14*w^3 + 24*w - 3], [509, 509, -2*w^5 - w^4 + 14*w^3 + 5*w^2 - 24*w], [509, 509, -2*w^5 - 2*w^4 + 14*w^3 + 9*w^2 - 22*w - 5], [569, 569, -2*w^5 + w^4 + 12*w^3 - 3*w^2 - 17*w + 3], [571, 571, -w^5 + w^4 + 5*w^3 - 4*w^2 - 5*w - 1], [571, 571, -w^5 - w^4 + 8*w^3 + 3*w^2 - 16*w + 2], [571, 571, -2*w^5 + 13*w^3 - 21*w + 2], [571, 571, 2*w^5 + w^4 - 12*w^3 - 7*w^2 + 15*w + 8], [571, 571, 2*w^5 + 2*w^4 - 13*w^3 - 10*w^2 + 19*w + 8], [571, 571, 2*w^5 - 12*w^3 + 15*w - 4], [601, 601, 3*w^5 + 2*w^4 - 21*w^3 - 9*w^2 + 33*w + 2], [601, 601, 3*w^5 + w^4 - 19*w^3 - 3*w^2 + 25*w - 4], [601, 601, -2*w^5 - 2*w^4 + 14*w^3 + 10*w^2 - 22*w - 5], [601, 601, -3*w^5 - w^4 + 18*w^3 + 6*w^2 - 23*w - 6], [619, 619, w^5 + w^4 - 7*w^3 - 5*w^2 + 13*w + 4], [631, 631, 2*w^5 + w^4 - 15*w^3 - 4*w^2 + 26*w - 1], [631, 631, -2*w^5 + 12*w^3 + w^2 - 14*w - 4], [641, 641, -w^5 + w^4 + 6*w^3 - 6*w^2 - 9*w + 7], [641, 641, -2*w^5 - w^4 + 13*w^3 + 7*w^2 - 19*w - 8], [659, 659, 2*w^5 - w^4 - 12*w^3 + 4*w^2 + 16*w - 4], [659, 659, 2*w^5 - 13*w^3 + 20*w - 4], [661, 661, -2*w^5 - w^4 + 13*w^3 + 3*w^2 - 19*w], [661, 661, -w^5 - w^4 + 6*w^3 + 7*w^2 - 8*w - 8], [661, 661, -3*w^5 - 2*w^4 + 21*w^3 + 10*w^2 - 34*w - 5], [691, 691, 3*w^5 + w^4 - 20*w^3 - 3*w^2 + 31*w - 3], [691, 691, 5*w^5 + 2*w^4 - 34*w^3 - 9*w^2 + 54*w + 3], [691, 691, -2*w^5 - w^4 + 13*w^3 + 5*w^2 - 22*w - 1], [691, 691, 3*w^5 + w^4 - 19*w^3 - 5*w^2 + 28*w], [691, 691, w^5 + w^4 - 8*w^3 - 3*w^2 + 14*w - 4], [691, 691, -w^5 - 2*w^4 + 8*w^3 + 9*w^2 - 15*w - 5], [701, 701, -3*w^5 - w^4 + 19*w^3 + 7*w^2 - 26*w - 11], [709, 709, -w^5 + 7*w^3 - w^2 - 13*w + 4], [709, 709, -w^5 + 4*w^3 - 2*w - 4], [709, 709, 2*w^5 + 2*w^4 - 15*w^3 - 8*w^2 + 27*w + 1], [709, 709, -w^5 - w^4 + 6*w^3 + 6*w^2 - 7*w - 4], [719, 719, -w^4 + w^3 + 4*w^2 - 3*w + 2], [719, 719, -3*w^5 - w^4 + 20*w^3 + 6*w^2 - 32*w - 2], [729, 3, -3], [739, 739, w^5 + 2*w^4 - 7*w^3 - 9*w^2 + 11*w + 7], [739, 739, -3*w^5 - 2*w^4 + 21*w^3 + 10*w^2 - 34*w - 6], [739, 739, w^5 + w^4 - 6*w^3 - 3*w^2 + 9*w - 3], [739, 739, w^4 + w^3 - 4*w^2 - 2*w + 1], [739, 739, -3*w^5 + 18*w^3 + w^2 - 23*w + 1], [739, 739, 3*w^5 - 18*w^3 - 2*w^2 + 24*w + 5], [751, 751, -w^5 + w^4 + 5*w^3 - 5*w^2 - 5*w + 2], [761, 761, -w^5 - 2*w^4 + 8*w^3 + 8*w^2 - 15*w - 4], [761, 761, -w^5 + 6*w^3 - 6*w + 2], [769, 769, w^3 - 2*w^2 - 3*w + 3], [811, 811, -3*w^5 + 19*w^3 + w^2 - 25*w - 1], [829, 829, -2*w^5 + w^4 + 13*w^3 - 4*w^2 - 19*w + 2], [829, 829, 3*w^5 + w^4 - 20*w^3 - 4*w^2 + 29*w - 1], [839, 839, -2*w^5 - w^4 + 15*w^3 + 5*w^2 - 25*w - 1], [839, 839, -w^5 + 6*w^3 + 2*w^2 - 6*w - 6], [841, 29, 4*w^5 + 2*w^4 - 27*w^3 - 10*w^2 + 41*w + 5], [859, 859, -2*w^5 + 13*w^3 - 17*w + 2], [881, 881, -2*w^5 - 2*w^4 + 14*w^3 + 8*w^2 - 23*w - 3], [881, 881, w^5 - 7*w^3 + w^2 + 11*w], [911, 911, 4*w^5 - 25*w^3 + 35*w - 3], [911, 911, 6*w^5 + 2*w^4 - 39*w^3 - 11*w^2 + 58*w + 5], [919, 919, w^4 + w^3 - 6*w^2 - 3*w + 6], [919, 919, -3*w^5 + 17*w^3 - 20*w + 1], [929, 929, -w^3 + 2*w^2 + 3*w - 8], [941, 941, 2*w^5 + 2*w^4 - 15*w^3 - 8*w^2 + 27*w], [941, 941, 2*w^5 + w^4 - 15*w^3 - 3*w^2 + 25*w - 3], [961, 31, -3*w^5 - w^4 + 18*w^3 + 6*w^2 - 21*w - 6], [971, 971, -w^5 + 4*w^3 + 2*w^2 - 5], [991, 991, -2*w^5 - 3*w^4 + 14*w^3 + 13*w^2 - 22*w - 6], [1009, 1009, -3*w^5 + 17*w^3 - 20*w], [1009, 1009, 4*w^5 - 25*w^3 + 35*w - 1], [1019, 1019, -3*w^5 + 17*w^3 + 2*w^2 - 20*w - 4], [1019, 1019, -w^5 + 5*w^3 - w^2 - 3*w + 1], [1019, 1019, -w^3 + 2*w^2 + 5*w - 5], [1021, 1021, 2*w^5 + w^4 - 13*w^3 - 4*w^2 + 18*w + 3], [1031, 1031, 2*w^5 - 11*w^3 - 2*w^2 + 12*w + 6], [1031, 1031, 2*w^5 - 14*w^3 + w^2 + 23*w - 2], [1039, 1039, w^4 - 2*w^3 - 5*w^2 + 7*w + 5], [1039, 1039, 2*w^5 - w^4 - 11*w^3 + 5*w^2 + 13*w - 5], [1039, 1039, -w^5 - w^4 + 7*w^3 + 6*w^2 - 13*w - 6], [1039, 1039, 4*w^5 + w^4 - 23*w^3 - 5*w^2 + 28*w + 3], [1049, 1049, -w^5 + w^4 + 6*w^3 - 6*w^2 - 9*w + 6], [1049, 1049, -3*w^5 + 19*w^3 + w^2 - 25*w + 2], [1051, 1051, w^5 + w^4 - 7*w^3 - 2*w^2 + 12*w - 5], [1069, 1069, w^5 + 2*w^4 - 7*w^3 - 9*w^2 + 12*w + 6], [1091, 1091, -4*w^5 + w^4 + 24*w^3 - 4*w^2 - 30*w + 2], [1109, 1109, w^5 + w^4 - 5*w^3 - 5*w^2 + 6*w + 3], [1129, 1129, -3*w^5 + 19*w^3 + 2*w^2 - 26*w - 3], [1129, 1129, -2*w^5 - w^4 + 14*w^3 + 2*w^2 - 22*w + 2], [1129, 1129, 2*w^5 - 11*w^3 - 2*w^2 + 11*w + 3], [1129, 1129, -4*w^5 - 2*w^4 + 25*w^3 + 11*w^2 - 35*w - 11], [1151, 1151, 2*w^5 + w^4 - 14*w^3 - 5*w^2 + 21*w + 4], [1181, 1181, w^5 - 7*w^3 + w^2 + 12*w], [1201, 1201, w^5 + 2*w^4 - 6*w^3 - 9*w^2 + 8*w + 5], [1229, 1229, w^5 + 2*w^4 - 8*w^3 - 10*w^2 + 14*w + 6], [1229, 1229, 2*w^5 + w^4 - 13*w^3 - 5*w^2 + 17*w + 4], [1231, 1231, -2*w^5 - 2*w^4 + 15*w^3 + 9*w^2 - 25*w - 3], [1249, 1249, -2*w^5 - 2*w^4 + 14*w^3 + 10*w^2 - 21*w - 7], [1249, 1249, -w^5 - w^4 + 7*w^3 + 7*w^2 - 11*w - 8], [1259, 1259, w^3 + w^2 - 3*w - 5], [1259, 1259, -w^5 - w^4 + 9*w^3 + 3*w^2 - 19*w + 2], [1259, 1259, -w^4 + w^3 + 3*w^2 - 3*w + 4], [1279, 1279, -5*w^5 - 2*w^4 + 35*w^3 + 9*w^2 - 57*w - 2], [1279, 1279, 3*w^5 + 2*w^4 - 20*w^3 - 9*w^2 + 31*w + 5], [1289, 1289, -w^5 + w^4 + 5*w^3 - 5*w^2 - 6*w + 4], [1291, 1291, w^5 + w^4 - 7*w^3 - 3*w^2 + 10*w - 4], [1291, 1291, -w^5 - w^4 + 9*w^3 + 3*w^2 - 18*w + 4], [1291, 1291, 2*w^5 - 13*w^3 + 17*w], [1291, 1291, -5*w^5 - 3*w^4 + 32*w^3 + 15*w^2 - 46*w - 8], [1321, 1321, w^4 - w^3 - 4*w^2 + 2*w - 1], [1321, 1321, 2*w^5 - 2*w^4 - 12*w^3 + 8*w^2 + 14*w - 3], [1321, 1321, 3*w^5 - 19*w^3 - 2*w^2 + 26*w + 4], [1321, 1321, -2*w^5 - w^4 + 12*w^3 + 3*w^2 - 15*w + 1], [1331, 11, -3*w^5 + 18*w^3 - 24*w + 1], [1361, 1361, -4*w^5 - w^4 + 25*w^3 + 6*w^2 - 35*w - 6], [1381, 1381, -2*w^5 - 3*w^4 + 14*w^3 + 14*w^2 - 24*w - 6], [1381, 1381, -3*w^5 - w^4 + 19*w^3 + 4*w^2 - 28*w + 3], [1381, 1381, 2*w^5 + w^4 - 12*w^3 - 3*w^2 + 15*w - 3], [1399, 1399, 4*w^5 + 2*w^4 - 27*w^3 - 10*w^2 + 44*w + 4], [1409, 1409, 2*w^5 - 14*w^3 + 21*w - 1], [1429, 1429, -3*w^5 - w^4 + 20*w^3 + 6*w^2 - 29*w - 5], [1429, 1429, -3*w^5 + 18*w^3 - w^2 - 23*w + 3], [1429, 1429, w^4 - 2*w^2 - 2*w - 5], [1439, 1439, 3*w^5 + 3*w^4 - 19*w^3 - 14*w^2 + 27*w + 5], [1451, 1451, w^5 + 2*w^4 - 10*w^3 - 6*w^2 + 22*w - 4], [1459, 1459, -w^5 + w^4 + 7*w^3 - 5*w^2 - 10*w + 4], [1471, 1471, -w^5 - 2*w^4 + 9*w^3 + 9*w^2 - 17*w - 6], [1481, 1481, 2*w^5 + w^4 - 14*w^3 - 5*w^2 + 21*w + 5], [1489, 1489, 3*w^5 - w^4 - 17*w^3 + 3*w^2 + 22*w - 2], [1499, 1499, -2*w^4 + w^3 + 8*w^2 - 3*w - 4], [1511, 1511, 3*w^5 - 18*w^3 - 2*w^2 + 23*w + 4], [1511, 1511, w^5 + 2*w^4 - 8*w^3 - 10*w^2 + 14*w + 8], [1511, 1511, w^5 + 2*w^4 - 10*w^3 - 7*w^2 + 22*w - 3], [1511, 1511, 5*w^5 + w^4 - 29*w^3 - 5*w^2 + 34*w + 4], [1531, 1531, 2*w^4 - 3*w^3 - 8*w^2 + 8*w + 2], [1549, 1549, -4*w^5 + 27*w^3 - 41*w + 6], [1549, 1549, w^5 + w^4 - 7*w^3 - 4*w^2 + 13*w + 5], [1559, 1559, 2*w^5 - 11*w^3 + w^2 + 12*w - 4], [1559, 1559, -w^5 + 6*w^3 + w^2 - 10*w - 3], [1571, 1571, w^3 + 2*w^2 - 4*w - 6], [1579, 1579, 2*w^5 + 2*w^4 - 15*w^3 - 11*w^2 + 26*w + 8], [1601, 1601, -w^5 + 8*w^3 - 2*w^2 - 16*w + 3], [1601, 1601, 3*w^5 + 2*w^4 - 19*w^3 - 7*w^2 + 27*w - 2], [1609, 1609, -5*w^5 - 2*w^4 + 32*w^3 + 10*w^2 - 47*w - 2], [1619, 1619, 4*w^5 + 3*w^4 - 25*w^3 - 14*w^2 + 35*w + 8], [1619, 1619, -2*w^5 + 13*w^3 - 17*w + 1], [1621, 1621, -5*w^5 - 3*w^4 + 34*w^3 + 15*w^2 - 53*w - 7], [1621, 1621, 4*w^5 + 2*w^4 - 26*w^3 - 7*w^2 + 38*w - 5], [1669, 1669, w^5 + w^4 - 9*w^3 - 4*w^2 + 17*w - 1], [1681, 41, 4*w^5 - 24*w^3 - 2*w^2 + 33*w + 5], [1681, 41, 2*w^4 - 2*w^3 - 9*w^2 + 6*w + 4], [1699, 1699, -3*w^5 - 2*w^4 + 18*w^3 + 10*w^2 - 24*w - 3], [1699, 1699, -2*w^5 + w^4 + 13*w^3 - 3*w^2 - 19*w + 4], [1699, 1699, 4*w^5 - w^4 - 23*w^3 + 3*w^2 + 28*w - 1], [1699, 1699, 2*w^4 - w^3 - 8*w^2 + 3*w + 2], [1709, 1709, -4*w^5 - 2*w^4 + 26*w^3 + 11*w^2 - 38*w - 7], [1709, 1709, 3*w^5 - 17*w^3 + w^2 + 20*w - 4], [1721, 1721, 2*w^5 + 2*w^4 - 14*w^3 - 9*w^2 + 22*w - 1], [1721, 1721, -2*w^5 + w^4 + 12*w^3 - 5*w^2 - 17*w + 5], [1721, 1721, w^5 - w^4 - 7*w^3 + 7*w^2 + 10*w - 12], [1721, 1721, -2*w^5 - w^4 + 16*w^3 + 4*w^2 - 30*w + 4], [1741, 1741, -2*w^5 + w^4 + 12*w^3 - 3*w^2 - 15*w + 1], [1741, 1741, 3*w^5 + 2*w^4 - 19*w^3 - 10*w^2 + 26*w + 8], [1759, 1759, 3*w^5 + w^4 - 19*w^3 - 6*w^2 + 25*w + 5], [1789, 1789, w^4 + w^3 - 4*w^2 - 5*w + 2], [1789, 1789, w^5 - 3*w^3 - 3*w + 3], [1789, 1789, -3*w^5 - w^4 + 19*w^3 + 7*w^2 - 26*w - 8], [1789, 1789, 2*w^5 - 13*w^3 + w^2 + 20*w - 1], [1801, 1801, 6*w^5 + 2*w^4 - 39*w^3 - 9*w^2 + 56*w + 3], [1811, 1811, 2*w^5 + w^4 - 15*w^3 - 4*w^2 + 26*w - 3], [1811, 1811, -4*w^5 - w^4 + 26*w^3 + 5*w^2 - 37*w - 5], [1811, 1811, -4*w^5 + w^4 + 23*w^3 - 3*w^2 - 27*w], [1831, 1831, 5*w^5 + w^4 - 30*w^3 - 4*w^2 + 39*w], [1831, 1831, 2*w^5 - 12*w^3 - w^2 + 16*w + 6], [1831, 1831, 2*w^5 + 2*w^4 - 14*w^3 - 12*w^2 + 22*w + 13], [1831, 1831, w^5 - 7*w^3 + 11*w + 3], [1871, 1871, 3*w^5 + w^4 - 21*w^3 - 3*w^2 + 32*w - 1], [1871, 1871, 3*w^5 + w^4 - 17*w^3 - 7*w^2 + 22*w + 8], [1871, 1871, w - 4], [1879, 1879, -3*w^5 - 2*w^4 + 21*w^3 + 9*w^2 - 32*w - 6], [1901, 1901, 3*w^5 + 2*w^4 - 20*w^3 - 7*w^2 + 31*w - 5], [1931, 1931, -4*w^5 - 3*w^4 + 27*w^3 + 15*w^2 - 41*w - 10], [1931, 1931, -3*w^5 - w^4 + 19*w^3 + 7*w^2 - 26*w - 7], [1949, 1949, -2*w^5 - 3*w^4 + 13*w^3 + 13*w^2 - 19*w - 8], [1949, 1949, -5*w^5 - 2*w^4 + 33*w^3 + 8*w^2 - 50*w - 2], [1949, 1949, -4*w^5 - 2*w^4 + 28*w^3 + 11*w^2 - 45*w - 6], [1951, 1951, 5*w^5 + w^4 - 31*w^3 - 3*w^2 + 42*w - 5], [1951, 1951, 3*w^5 + 2*w^4 - 21*w^3 - 11*w^2 + 35*w + 8], [1979, 1979, 5*w^5 + w^4 - 31*w^3 - 3*w^2 + 43*w - 6], [1979, 1979, -5*w^5 - w^4 + 31*w^3 + 7*w^2 - 42*w - 9], [1999, 1999, -3*w^5 - w^4 + 21*w^3 + 4*w^2 - 33*w + 2], [1999, 1999, 3*w^5 + 2*w^4 - 17*w^3 - 10*w^2 + 20*w + 7], [1999, 1999, w^5 + w^4 - 9*w^3 - 5*w^2 + 19*w + 6]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [1, -1, -3, -4, 10, 8, 6, -10, 4, -14, 10, 8, 0, 8, 8, 8, -16, 6, -6, 14, -6, 10, 14, -2, 12, 20, -12, 2, -16, 16, 16, -16, -20, -4, 12, -4, -10, -26, -8, 24, 0, 8, -16, 4, 6, 2, 24, -26, -20, -20, 12, 12, 26, 16, -6, -2, -16, 16, 32, -32, -12, -2, -2, -34, -8, 0, -26, 4, 20, 6, 38, 14, -30, 30, 6, 26, -28, 36, -4, 12, 10, -30, -26, 6, -24, 8, 0, 40, -30, -14, -20, -20, -20, 44, 14, 30, -2, 30, 42, 12, 20, 44, -20, -4, -28, -38, 10, 42, 38, -36, 32, 16, 38, 2, -36, -44, -38, 22, 10, -12, -44, -28, 20, 44, -36, -50, 6, -10, -14, -22, -16, 0, 26, -12, 4, 36, -28, 4, -4, 0, 22, -22, 54, 28, -34, 30, -48, 48, -18, 20, 2, -14, 40, -8, 0, 48, -18, 30, -14, 50, 12, 40, -18, 14, 20, -52, -12, 30, 48, -48, 40, -48, 16, -24, -10, 30, 4, -22, 28, 34, 46, 26, -38, 22, -16, 10, 6, 50, -18, -8, 18, -46, 28, 44, -4, 0, 8, 10, 52, 44, -28, -60, -42, 58, -66, -38, -4, 2, -26, -46, 10, -8, -46, 34, 50, 6, 48, -36, 52, 32, -18, -62, 44, -72, 0, -40, 24, -52, -54, -2, 24, 8, 20, 4, -42, 54, -70, -60, 12, -22, -42, 2, 6, 70, -28, 68, 4, 44, -6, -34, -6, -38, 58, -70, 58, -78, 56, 14, -66, 58, 2, -22, 60, -36, -12, 32, 40, -16, 8, 0, 8, -32, 24, -18, -44, -28, -14, -66, 2, -64, -56, -20, -36, -24, 72, 40]; 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;