/* 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![45, 15, -14, -2, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [5, 5, 2/3*w^2 + 1/3*w - 5], [5, 5, 2/3*w^2 - 5/3*w - 4], [9, 3, -w + 3], [9, 3, w + 2], [11, 11, 2/3*w^2 + 1/3*w - 4], [16, 2, 2], [19, 19, 1/3*w^3 - 10/3*w - 3], [19, 19, 1/3*w^3 - w^2 - 7/3*w + 6], [29, 29, -1/3*w^3 + 2/3*w^2 + 8/3*w - 2], [29, 29, 1/3*w^3 - 1/3*w^2 - 3*w + 1], [31, 31, -1/3*w^3 + 1/3*w^2 + 3*w + 1], [31, 31, -1/3*w^3 + 2/3*w^2 + 8/3*w - 4], [49, 7, 1/3*w^3 - 10/3*w - 1], [49, 7, -1/3*w^3 + w^2 + 7/3*w - 4], [59, 59, -1/3*w^3 + 5/3*w^2 + 5/3*w - 11], [59, 59, -2/3*w^3 + 5/3*w^2 + 5*w - 13], [61, 61, -1/3*w^3 + 5/3*w^2 + 2/3*w - 9], [61, 61, -1/3*w^3 + 1/3*w^2 + 4*w + 2], [61, 61, 2/3*w^2 - 5/3*w - 1], [61, 61, -1/3*w^2 + 4/3*w + 6], [71, 71, -4/3*w^2 + 7/3*w + 11], [79, 79, -1/3*w^2 + 7/3*w - 3], [79, 79, -2/3*w^3 + 1/3*w^2 + 16/3*w - 4], [101, 101, 2/3*w^2 - 5/3*w - 6], [101, 101, -2/3*w^2 - 1/3*w + 7], [109, 109, -4/3*w^2 + 1/3*w + 9], [109, 109, 1/3*w^3 - 1/3*w^2 - 3*w - 4], [121, 11, -1/3*w^2 + 1/3*w + 6], [131, 131, -2/3*w^3 + 2/3*w^2 + 5*w - 7], [131, 131, 2/3*w^3 - 4/3*w^2 - 13/3*w - 2], [151, 151, -1/3*w^3 - 2/3*w^2 + 5*w + 7], [151, 151, 1/3*w^3 - 10/3*w + 2], [169, 13, 5/3*w^2 + 1/3*w - 11], [169, 13, 5/3*w^2 - 11/3*w - 9], [181, 181, -1/3*w^3 + 4/3*w^2 + 3*w - 8], [181, 181, -1/3*w^3 - 1/3*w^2 + 14/3*w + 4], [191, 191, -2/3*w^3 + w^2 + 17/3*w - 9], [191, 191, 1/3*w^3 - 2*w^2 - 7/3*w + 13], [191, 191, 2/3*w^3 - 1/3*w^2 - 22/3*w - 7], [191, 191, 2/3*w^3 - w^2 - 17/3*w - 3], [199, 199, 1/3*w^3 + w^2 - 13/3*w - 9], [199, 199, 2/3*w^2 - 5/3*w - 7], [199, 199, 1/3*w^3 - w^2 - 4/3*w + 6], [199, 199, -1/3*w^3 + 2*w^2 + 4/3*w - 12], [269, 269, -1/3*w^3 - 4/3*w^2 + 11/3*w + 14], [269, 269, 1/3*w^3 - 7/3*w^2 + 16], [271, 271, 2/3*w^2 + 1/3*w - 9], [271, 271, 2/3*w^2 - 5/3*w - 8], [281, 281, 1/3*w^3 + w^2 - 13/3*w - 11], [281, 281, -2/3*w^3 + 20/3*w + 3], [281, 281, -w^2 + 3*w + 2], [281, 281, -1/3*w^3 + 2*w^2 + 4/3*w - 14], [289, 17, w^3 - 8/3*w^2 - 16/3*w + 4], [289, 17, -w^3 + 1/3*w^2 + 23/3*w - 3], [331, 331, -2/3*w^3 + 2*w^2 + 11/3*w - 7], [331, 331, 2/3*w^3 - 17/3*w - 2], [359, 359, -1/3*w^3 - 1/3*w^2 + 11/3*w - 2], [359, 359, 1/3*w^3 + 4/3*w^2 - 17/3*w - 12], [361, 19, 4/3*w^2 - 4/3*w - 11], [389, 389, 5/3*w^2 - 8/3*w - 13], [389, 389, 5/3*w^2 - 2/3*w - 14], [401, 401, -1/3*w^3 + 5/3*w^2 + 11/3*w - 12], [401, 401, 1/3*w^3 + 2/3*w^2 - 6*w - 7], [409, 409, 1/3*w^3 + 1/3*w^2 - 14/3*w - 3], [409, 409, -1/3*w^3 + 4/3*w^2 + 3*w - 7], [421, 421, -1/3*w^3 + 5/3*w^2 + 11/3*w - 13], [421, 421, 1/3*w^3 + 2/3*w^2 - 6*w - 8], [431, 431, 2/3*w^2 + 4/3*w - 9], [431, 431, 2/3*w^2 - 8/3*w - 7], [449, 449, 2/3*w^3 - 17/3*w + 2], [449, 449, 1/3*w^3 + 4/3*w^2 - 20/3*w - 11], [461, 461, 1/3*w^2 + 5/3*w - 8], [461, 461, 1/3*w^3 + 1/3*w^2 - 17/3*w - 3], [461, 461, -1/3*w^3 + 4/3*w^2 + 4*w - 8], [461, 461, 1/3*w^2 - 7/3*w - 6], [479, 479, -1/3*w^3 + 2/3*w^2 + 5/3*w - 9], [479, 479, 4/3*w^3 + w^2 - 40/3*w - 18], [479, 479, 4/3*w^3 - 5*w^2 - 22/3*w + 29], [479, 479, 1/3*w^3 - 1/3*w^2 - 2*w - 7], [491, 491, 2/3*w^3 - 5/3*w^2 - 5*w + 8], [491, 491, -1/3*w^3 + 3*w^2 + 10/3*w - 24], [491, 491, 1/3*w^3 + 2*w^2 - 25/3*w - 18], [491, 491, 2/3*w^3 - 1/3*w^2 - 19/3*w - 2], [499, 499, -1/3*w^3 + 10/3*w^2 - 4*w - 17], [499, 499, 1/3*w^3 - 7/3*w - 9], [499, 499, -1/3*w^3 + w^2 + 4/3*w - 11], [499, 499, -w^3 + 5/3*w^2 + 22/3*w - 14], [509, 509, w^3 - 14/3*w^2 - 10/3*w + 24], [509, 509, -2/3*w^2 + 5/3*w + 11], [509, 509, 2/3*w^2 + 1/3*w - 12], [509, 509, -2/3*w^3 + 3*w^2 + 8/3*w - 18], [521, 521, 1/3*w^3 - 13/3*w + 1], [521, 521, w^3 + 2/3*w^2 - 32/3*w - 12], [521, 521, 2/3*w^3 + 7/3*w^2 - 13*w - 23], [521, 521, -1/3*w^3 + w^2 + 10/3*w - 3], [541, 541, 2/3*w^3 - 17/3*w - 3], [541, 541, -1/3*w^3 + 4/3*w^2 + 2*w - 14], [541, 541, -1/3*w^3 + 4/3*w^2 + w - 1], [541, 541, -1/3*w^3 + 1/3*w^2 + 5*w - 4], [569, 569, 1/3*w^3 + 11/3*w^2 - 12*w - 26], [569, 569, -w^3 + 10/3*w^2 + 23/3*w - 21], [571, 571, w^3 - 7/3*w^2 - 23/3*w + 17], [571, 571, 2/3*w^3 - 3*w^2 - 11/3*w + 17], [599, 599, 1/3*w^3 + 2/3*w^2 - 4*w - 1], [599, 599, 1/3*w^3 - 5/3*w^2 - 5/3*w + 4], [601, 601, 1/3*w^3 - 7/3*w - 6], [601, 601, -2/3*w^3 + 2*w^2 + 11/3*w - 9], [601, 601, 2/3*w^3 - 17/3*w - 4], [601, 601, -1/3*w^3 + w^2 + 4/3*w - 8], [619, 619, -2/3*w^3 + 10/3*w^2 + 19/3*w - 26], [619, 619, 2/3*w^3 + 4/3*w^2 - 11*w - 17], [631, 631, 1/3*w^3 - 7/3*w^2 - w + 14], [631, 631, 1/3*w^3 + 4/3*w^2 - 14/3*w - 11], [659, 659, -2/3*w^3 + 2/3*w^2 + 6*w + 1], [659, 659, 2/3*w^3 - 4/3*w^2 - 16/3*w + 7], [661, 661, -2/3*w^3 + 13/3*w^2 + 16/3*w - 32], [661, 661, -2/3*w^3 - 7/3*w^2 + 12*w + 23], [691, 691, 1/3*w^3 - 11/3*w^2 - 11/3*w + 29], [691, 691, -2/3*w^3 + 10/3*w^2 + 10/3*w - 23], [701, 701, 2/3*w^3 - 4/3*w^2 - 13/3*w - 3], [701, 701, 2/3*w^3 - 4/3*w^2 - 16/3*w - 1], [701, 701, -2/3*w^3 + 2/3*w^2 + 6*w - 7], [701, 701, -2/3*w^3 + 2/3*w^2 + 5*w - 8], [709, 709, -4/3*w^3 + 31/3*w - 2], [709, 709, -w^3 + 1/3*w^2 + 32/3*w + 4], [719, 719, -2/3*w^3 + w^2 + 17/3*w - 2], [719, 719, 1/3*w^3 - 2*w^2 - 4/3*w + 9], [719, 719, 1/3*w^3 + w^2 - 13/3*w - 6], [719, 719, 2/3*w^3 - w^2 - 17/3*w + 4], [739, 739, 2/3*w^3 + 2/3*w^2 - 19/3*w - 11], [739, 739, -2/3*w^3 + 8/3*w^2 + 3*w - 16], [751, 751, -1/3*w^3 + w^2 + 4/3*w - 9], [751, 751, 1/3*w^3 - 7/3*w - 7], [761, 761, 1/3*w^3 - 16/3*w - 4], [761, 761, w^3 - 1/3*w^2 - 29/3*w - 12], [761, 761, -w^3 + 8/3*w^2 + 22/3*w - 21], [761, 761, -1/3*w^3 + w^2 + 13/3*w - 9], [769, 769, w^3 - 7/3*w^2 - 23/3*w + 8], [769, 769, 1/3*w^3 + 4/3*w^2 - 14/3*w - 14], [769, 769, -1/3*w^3 + 7/3*w^2 + w - 17], [769, 769, 1/3*w^3 - 3*w^2 + 11/3*w + 13], [811, 811, 1/3*w^3 + w^2 - 10/3*w - 12], [811, 811, -1/3*w^3 + 2*w^2 + 1/3*w - 14], [821, 821, -1/3*w^3 + 13/3*w - 3], [821, 821, -1/3*w^3 + w^2 + 10/3*w - 1], [839, 839, -w^3 - 5/3*w^2 + 32/3*w + 18], [839, 839, 1/3*w^3 + 3*w^2 - 7/3*w - 22], [839, 839, 1/3*w^3 - 4*w^2 + 14/3*w + 21], [839, 839, -w^3 + 14/3*w^2 + 13/3*w - 26], [841, 29, -5/3*w^2 + 5/3*w + 11], [859, 859, -w^3 + 5/3*w^2 + 25/3*w - 1], [859, 859, -w^3 + 4/3*w^2 + 26/3*w - 8], [911, 911, -1/3*w^3 + 1/3*w^2 + 5*w + 1], [911, 911, 1/3*w^3 - 2/3*w^2 - 14/3*w + 6], [919, 919, 2/3*w^3 - 2/3*w^2 - 5*w - 7], [919, 919, -2/3*w^3 + 7/3*w^2 + 10/3*w - 7], [919, 919, 2/3*w^3 + 1/3*w^2 - 6*w - 2], [919, 919, -2/3*w^3 + 4/3*w^2 + 13/3*w - 12], [941, 941, 5/3*w^2 + 1/3*w - 9], [941, 941, 5/3*w^2 - 11/3*w - 7], [961, 31, 5/3*w^2 - 5/3*w - 13], [971, 971, -1/3*w^3 + w^2 + 13/3*w - 8], [971, 971, 2/3*w^3 - 2/3*w^2 - 5*w + 2], [971, 971, -2/3*w^3 + 4/3*w^2 + 13/3*w - 3], [971, 971, 1/3*w^3 - 16/3*w - 3], [991, 991, 10/3*w^2 - 22/3*w - 19], [991, 991, 10/3*w^2 + 2/3*w - 23], [1009, 1009, -2*w^2 + 3*w + 16], [1009, 1009, 2*w^2 - w - 17], [1019, 1019, 1/3*w^3 + w^2 - 19/3*w - 14], [1019, 1019, -1/3*w^3 + 2*w^2 + 10/3*w - 19], [1021, 1021, 2/3*w^3 - 3*w^2 - 14/3*w + 19], [1021, 1021, 11/3*w^2 + 4/3*w - 27], [1021, 1021, 11/3*w^2 - 26/3*w - 22], [1021, 1021, -2/3*w^3 - w^2 + 26/3*w + 12], [1031, 1031, -2/3*w^3 + 5/3*w^2 + 4*w + 1], [1031, 1031, -2/3*w^3 + 1/3*w^2 + 16/3*w - 6], [1049, 1049, -1/3*w^3 - 2/3*w^2 + 4*w - 1], [1049, 1049, 1/3*w^3 + 5/3*w^2 - 6*w - 14], [1051, 1051, 1/3*w^3 - 16/3*w - 2], [1051, 1051, -1/3*w^3 + w^2 + 13/3*w - 7], [1061, 1061, 11/3*w^2 + 1/3*w - 27], [1061, 1061, 11/3*w^2 - 23/3*w - 23], [1069, 1069, w^2 + 2*w - 11], [1069, 1069, w^2 - 4*w - 8], [1109, 1109, w^3 - 5/3*w^2 - 19/3*w - 1], [1109, 1109, -w^3 - w^2 + 12*w + 16], [1129, 1129, -w^3 + 5/3*w^2 + 25/3*w - 11], [1129, 1129, -w^3 + 4/3*w^2 + 26/3*w + 2], [1151, 1151, 1/3*w^3 + w^2 - 16/3*w - 7], [1151, 1151, -1/3*w^3 + 2*w^2 + 7/3*w - 11], [1171, 1171, w^3 - 8/3*w^2 - 16/3*w + 1], [1171, 1171, -w^3 + 1/3*w^2 + 23/3*w - 6], [1201, 1201, -1/3*w^3 + 2/3*w^2 + 14/3*w - 4], [1201, 1201, -1/3*w^3 - 4/3*w^2 + 8/3*w + 13], [1201, 1201, 1/3*w^3 - 7/3*w^2 + w + 14], [1201, 1201, 1/3*w^3 - 1/3*w^2 - 5*w + 1], [1231, 1231, -1/3*w^3 + 5/3*w^2 + 8/3*w - 8], [1231, 1231, 1/3*w^3 + 2/3*w^2 - 5*w - 4], [1249, 1249, -4/3*w^3 + 7/3*w^2 + 9*w + 4], [1249, 1249, 4/3*w^2 - 13/3*w - 9], [1249, 1249, 4/3*w^2 + 5/3*w - 12], [1249, 1249, -4/3*w^3 + 5/3*w^2 + 29/3*w - 14], [1259, 1259, 2/3*w^3 - w^2 - 20/3*w + 11], [1259, 1259, -2/3*w^3 + w^2 + 20/3*w + 4], [1279, 1279, -2/3*w^3 - 1/3*w^2 + 5*w + 8], [1279, 1279, 1/3*w^3 - w^2 - 7/3*w + 12], [1279, 1279, -2*w^2 + 3*w + 13], [1279, 1279, -2/3*w^3 + 7/3*w^2 + 7/3*w - 12], [1289, 1289, w^3 + 1/3*w^2 - 28/3*w - 8], [1289, 1289, -2/3*w^3 + 3*w^2 + 8/3*w - 19], [1289, 1289, 2/3*w^3 + w^2 - 20/3*w - 14], [1289, 1289, 5/3*w^3 - 11/3*w^2 - 11*w - 1], [1291, 1291, -w^3 + 2/3*w^2 + 25/3*w - 7], [1291, 1291, w^3 - 7/3*w^2 - 20/3*w + 1], [1301, 1301, w^3 - 10/3*w^2 - 14/3*w + 16], [1301, 1301, -w^3 + 14/3*w^2 + 25/3*w - 34], [1319, 1319, -5/3*w^3 + 6*w^2 + 17/3*w - 14], [1319, 1319, 1/3*w^3 - 2/3*w^2 - 8/3*w - 4], [1319, 1319, -1/3*w^3 + 1/3*w^2 + 3*w - 7], [1319, 1319, -5/3*w^3 - w^2 + 38/3*w + 4], [1361, 1361, w^3 - 4*w^2 - 6*w + 23], [1361, 1361, w^3 + w^2 - 11*w - 14], [1369, 37, -w^3 + 7*w - 4], [1369, 37, -1/3*w^3 - 2*w^2 + 10/3*w + 16], [1381, 1381, -2/3*w^3 + 20/3*w + 1], [1381, 1381, 2/3*w^3 - 2*w^2 - 14/3*w + 7], [1399, 1399, -w^3 + 4/3*w^2 + 20/3*w + 4], [1399, 1399, -w^3 + 5/3*w^2 + 19/3*w - 11], [1429, 1429, -2/3*w^3 + 2/3*w^2 + 4*w - 7], [1429, 1429, w^3 + 1/3*w^2 - 19/3*w + 1], [1451, 1451, w^3 - 4/3*w^2 - 23/3*w + 9], [1451, 1451, -1/3*w^3 + 2*w^2 - 2/3*w - 12], [1451, 1451, -1/3*w^3 - w^2 + 7/3*w + 11], [1451, 1451, 1/3*w^3 - 8/3*w^2 + 10/3*w + 12], [1459, 1459, -2/3*w^3 + 8/3*w^2 + 5*w - 16], [1459, 1459, -1/3*w^3 + 8/3*w^2 + 2/3*w - 16], [1459, 1459, -1/3*w^3 - 5/3*w^2 + 5*w + 13], [1459, 1459, 2/3*w^3 + 2/3*w^2 - 25/3*w - 9], [1471, 1471, -2/3*w^3 + 3*w^2 + 5/3*w - 16], [1471, 1471, -13/3*w^2 - 5/3*w + 33], [1471, 1471, -13/3*w^2 + 31/3*w + 27], [1471, 1471, 2/3*w^3 + w^2 - 17/3*w - 12], [1489, 1489, 1/3*w^3 - 7/3*w^2 - 4*w + 17], [1489, 1489, 1/3*w^3 + 4/3*w^2 - 23/3*w - 11], [1531, 1531, -w^3 + 3*w^2 + 7*w - 17], [1531, 1531, w^3 - 10*w - 8], [1549, 1549, w^3 + 2/3*w^2 - 35/3*w - 13], [1549, 1549, -2/3*w^3 + 5*w^2 + 17/3*w - 36], [1549, 1549, -2/3*w^3 - 3*w^2 + 41/3*w + 26], [1549, 1549, -w^3 + 11/3*w^2 + 22/3*w - 23], [1571, 1571, -w^3 + 3*w^2 + 7*w - 23], [1571, 1571, -1/3*w^3 - 2*w^2 + 4/3*w + 17], [1571, 1571, -1/3*w^3 + 1/3*w^2 + 6*w - 13], [1571, 1571, -2/3*w^3 + 4*w^2 + 8/3*w - 23], [1601, 1601, -w^3 + 4/3*w^2 + 26/3*w - 11], [1601, 1601, -w^3 + 8/3*w^2 + 25/3*w - 24], [1601, 1601, -1/3*w^3 - 7/3*w^2 + 23/3*w + 18], [1601, 1601, -w^3 + 5/3*w^2 + 25/3*w + 2], [1609, 1609, -5/3*w^3 + 2/3*w^2 + 12*w - 8], [1609, 1609, 2/3*w^3 + 4/3*w^2 - 7*w - 17], [1619, 1619, w^3 - 4/3*w^2 - 23/3*w - 3], [1619, 1619, -w^3 + 5/3*w^2 + 22/3*w - 11], [1621, 1621, 7/3*w^2 - 13/3*w - 11], [1621, 1621, -2/3*w^3 + 8/3*w^2 + 6*w - 17], [1669, 1669, 5/3*w^2 + 1/3*w - 16], [1669, 1669, 5/3*w^2 - 11/3*w - 14], [1681, 41, 2*w^2 - 2*w - 17], [1681, 41, 2*w^2 - 2*w - 13], [1699, 1699, 1/3*w^3 - 2*w^2 + 5/3*w + 11], [1699, 1699, -1/3*w^3 - w^2 + 4/3*w + 11], [1721, 1721, 1/3*w^2 + 8/3*w - 11], [1721, 1721, 1/3*w^2 - 10/3*w - 8], [1789, 1789, -2/3*w^2 - 7/3*w - 4], [1789, 1789, -w^3 + 2*w^2 + 8*w - 13], [1789, 1789, 4/3*w^2 - 13/3*w - 8], [1789, 1789, 2/3*w^2 - 11/3*w + 7], [1811, 1811, 2/3*w^2 - 11/3*w - 9], [1811, 1811, 2/3*w^2 + 7/3*w - 12], [1831, 1831, 4/3*w^2 + 5/3*w - 9], [1831, 1831, -w^3 + 5/3*w^2 + 25/3*w - 2], [1861, 1861, 14/3*w^2 + 1/3*w - 33], [1861, 1861, 14/3*w^2 - 29/3*w - 28], [1871, 1871, -w^3 + 2/3*w^2 + 28/3*w - 3], [1871, 1871, 1/3*w^3 + w^2 - 22/3*w - 7], [1871, 1871, -1/3*w^3 + 2*w^2 + 13/3*w - 13], [1871, 1871, -w^3 + 7/3*w^2 + 23/3*w - 6], [1879, 1879, -w^3 + 8/3*w^2 + 22/3*w - 13], [1879, 1879, -w^3 + 1/3*w^2 + 29/3*w + 4], [1889, 1889, -5/3*w^3 + 2*w^2 + 38/3*w - 16], [1889, 1889, 5/3*w^3 - 3*w^2 - 35/3*w - 3], [1901, 1901, w^2 - 3*w - 11], [1901, 1901, w^2 + w - 13], [1931, 1931, 2/3*w^3 + 2*w^2 - 38/3*w - 21], [1931, 1931, w^3 - 2/3*w^2 - 25/3*w - 9], [1931, 1931, -w^3 + 7/3*w^2 + 20/3*w - 17], [1931, 1931, -2/3*w^3 + 4*w^2 + 20/3*w - 31], [1949, 1949, 2/3*w^3 + 5/3*w^2 - 28/3*w - 21], [1949, 1949, -1/3*w^3 + 3*w^2 + 1/3*w - 17], [1951, 1951, -1/3*w^3 + 8/3*w^2 + 2/3*w - 19], [1951, 1951, 1/3*w^3 + 5/3*w^2 - 5*w - 16], [1999, 1999, -5/3*w^3 + 3*w^2 + 38/3*w - 23], [1999, 1999, -1/3*w^3 - 5/3*w^2 + 4*w + 19], [1999, 1999, 1/3*w^3 - 8/3*w^2 + 1/3*w + 21], [1999, 1999, 5/3*w^3 - 2*w^2 - 41/3*w - 9]]; primes := [ideal : I in primesArray]; heckePol := x^8 - 28*x^6 + 168*x^4 - 344*x^2 + 196; K := NumberField(heckePol); heckeEigenvaluesArray := [0, -5/56*e^7 + 9/4*e^5 - 35/4*e^3 + 47/7*e, -5/56*e^7 + 9/4*e^5 - 35/4*e^3 + 54/7*e, -2, -1/8*e^6 + 3*e^4 - 35/4*e^2 + 5/2, -3/56*e^7 + 11/8*e^5 - 6*e^3 + 243/28*e, 3/14*e^7 - 21/4*e^5 + 35/2*e^3 - 115/14*e, -1/8*e^6 + 3*e^4 - 35/4*e^2 + 5/2, -1/8*e^7 + 3*e^5 - 35/4*e^3 + 3/2*e, 3/8*e^6 - 9*e^4 + 105/4*e^2 - 15/2, 1/8*e^6 - 13/4*e^4 + 57/4*e^2 - 13, 1/4*e^4 - 11/2*e^2 + 21/2, 5/28*e^7 - 9/2*e^5 + 35/2*e^3 - 94/7*e, -3/8*e^6 + 19/2*e^4 - 149/4*e^2 + 57/2, 11/56*e^7 - 5*e^5 + 83/4*e^3 - 351/14*e, -e^6 + 25*e^4 - 94*e^2 + 76, -5/56*e^7 + 9/4*e^5 - 35/4*e^3 + 54/7*e, 3/7*e^7 - 21/2*e^5 + 35*e^3 - 115/7*e, 3/56*e^7 - 3/2*e^5 + 35/4*e^3 - 167/14*e, -29/56*e^7 + 13*e^5 - 197/4*e^3 + 513/14*e, 3/7*e^7 - 43/4*e^5 + 81/2*e^3 - 405/14*e, -1/8*e^6 + 5/2*e^4 + 9/4*e^2 - 37/2, -5/28*e^7 + 9/2*e^5 - 35/2*e^3 + 94/7*e, 3/14*e^7 - 11/2*e^5 + 24*e^3 - 243/7*e, 5/8*e^7 - 63/4*e^5 + 245/4*e^3 - 54*e, 1/7*e^7 - 7/2*e^5 + 11*e^3 + 27/7*e, -4, e^6 - 25*e^4 + 94*e^2 - 82, 5/56*e^7 - 9/4*e^5 + 35/4*e^3 - 54/7*e, 41/56*e^7 - 37/2*e^5 + 293/4*e^3 - 999/14*e, 5/8*e^6 - 61/4*e^4 + 197/4*e^2 - 23, 5/8*e^6 - 16*e^4 + 263/4*e^2 - 109/2, 1/2*e^6 - 47/4*e^4 + 59/2*e^2 + 1/2, -15/28*e^7 + 51/4*e^5 - 35*e^3 + 11/14*e, 2/7*e^7 - 15/2*e^5 + 35*e^3 - 261/7*e, -11/28*e^7 + 39/4*e^5 - 35*e^3 + 303/14*e, 15/56*e^7 - 15/2*e^5 + 175/4*e^3 - 835/14*e, -1/14*e^7 + 3/2*e^5 - 73/7*e, -5/28*e^7 + 21/4*e^5 - 35*e^3 + 741/14*e, -2/7*e^7 + 15/2*e^5 - 35*e^3 + 261/7*e, 11/56*e^7 - 5*e^5 + 83/4*e^3 - 351/14*e, -e^6 + 25*e^4 - 94*e^2 + 60, 41/56*e^7 - 37/2*e^5 + 293/4*e^3 - 999/14*e, e^6 - 25*e^4 + 94*e^2 - 88, 1/2*e^6 - 25/2*e^4 + 47*e^2 - 53, 71/56*e^7 - 32*e^5 + 503/4*e^3 - 1647/14*e, -3/2*e^6 + 75/2*e^4 - 141*e^2 + 113, e^6 - 25*e^4 + 94*e^2 - 94, 23/56*e^7 - 21/2*e^5 + 179/4*e^3 - 837/14*e, -9/4*e^6 + 111/2*e^4 - 381/2*e^2 + 108, 9/8*e^6 - 111/4*e^4 + 381/4*e^2 - 54, -53/56*e^7 + 95/4*e^5 - 359/4*e^3 + 459/7*e, 3/2*e^6 - 75/2*e^4 + 138*e^2 - 93, -1/14*e^7 + 9/4*e^5 - 35/2*e^3 + 407/14*e, 19/56*e^7 - 15/2*e^5 + 35/4*e^3 + 417/14*e, -47/56*e^7 + 81/4*e^5 - 245/4*e^3 + 110/7*e, -99/56*e^7 + 89/2*e^5 - 687/4*e^3 + 2025/14*e, 3*e^6 - 75*e^4 + 282*e^2 - 232, -12/7*e^7 + 173/4*e^5 - 339/2*e^3 + 2187/14*e, -3/2*e^6 + 75/2*e^4 - 141*e^2 + 121, 15/8*e^7 - 189/4*e^5 + 735/4*e^3 - 162*e, -5/8*e^6 + 65/4*e^4 - 285/4*e^2 + 65, 1/8*e^6 - 3*e^4 + 35/4*e^2 - 5/2, 7/8*e^6 - 43/2*e^4 + 289/4*e^2 - 77/2, 25/28*e^7 - 45/2*e^5 + 175/2*e^3 - 470/7*e, 1/8*e^6 - 5/2*e^4 - 9/4*e^2 + 37/2, 3/4*e^6 - 37/2*e^4 + 127/2*e^2 - 36, 8, -34, -13/8*e^6 + 81/2*e^4 - 587/4*e^2 + 191/2, 2/7*e^7 - 27/4*e^5 + 35/2*e^3 + 31/14*e, -39/56*e^7 + 71/4*e^5 - 297/4*e^3 + 648/7*e, 7/4*e^6 - 177/4*e^4 + 172*e^2 - 259/2, -7/4*e^6 + 175/4*e^4 - 161*e^2 + 217/2, 3/8*e^7 - 19/2*e^5 + 153/4*e^3 - 81/2*e, 3*e^6 - 75*e^4 + 282*e^2 - 214, 47/56*e^7 - 21*e^5 + 315/4*e^3 - 773/14*e, 3/4*e^6 - 83/4*e^4 + 113*e^2 - 261/2, -75/56*e^7 + 135/4*e^5 - 525/4*e^3 + 810/7*e, 1/56*e^7 - 3/4*e^5 + 35/4*e^3 - 120/7*e, -2*e^6 + 50*e^4 - 188*e^2 + 134, -e^6 + 25*e^4 - 94*e^2 + 54, -3/14*e^7 + 21/4*e^5 - 35/2*e^3 + 115/14*e, 3/2*e^6 - 75/2*e^4 + 141*e^2 - 133, e^6 - 25*e^4 + 94*e^2 - 86, -7/8*e^7 + 22*e^5 - 337/4*e^3 + 135/2*e, -1/8*e^7 + 13/4*e^5 - 61/4*e^3 + 27*e, 25/56*e^7 - 11*e^5 + 145/4*e^3 + 27/14*e, 5/2*e^6 - 125/2*e^4 + 235*e^2 - 183, 57/56*e^7 - 103/4*e^5 + 411/4*e^3 - 729/7*e, -e^6 + 25*e^4 - 94*e^2 + 98, -11/14*e^7 + 75/4*e^5 - 105/2*e^3 + 53/14*e, 11/8*e^6 - 135/4*e^4 + 451/4*e^2 - 59, 13/8*e^6 - 79/2*e^4 + 499/4*e^2 - 107/2, 17/14*e^7 - 30*e^5 + 105*e^3 - 418/7*e, 27/56*e^7 - 45/4*e^5 + 105/4*e^3 + 78/7*e, -4*e^6 + 100*e^4 - 376*e^2 + 304, 3*e^6 - 75*e^4 + 282*e^2 - 222, 31/56*e^7 - 57/4*e^5 + 245/4*e^3 - 402/7*e, 33/56*e^7 - 27/2*e^5 + 105/4*e^3 + 375/14*e, -5/4*e^6 + 63/2*e^4 - 241/2*e^2 + 88, -1/14*e^7 + 3/4*e^5 + 35/2*e^3 - 699/14*e, -29/56*e^7 + 51/4*e^5 - 175/4*e^3 + 162/7*e, 3/28*e^7 - 3/2*e^5 - 35/2*e^3 + 386/7*e, 15/8*e^6 - 181/4*e^4 + 547/4*e^2 - 48, 3/2*e^6 - 75/2*e^4 + 141*e^2 - 131, 23/8*e^6 - 283/4*e^4 + 959/4*e^2 - 131, -e^6 + 26*e^4 - 114*e^2 + 104, -e^6 + 25*e^4 - 94*e^2 + 84, -17/56*e^7 + 27/4*e^5 - 35/4*e^3 - 172/7*e, -1/4*e^4 + 11/2*e^2 - 21/2, 9/56*e^7 - 4*e^5 + 57/4*e^3 - 81/14*e, 19/28*e^7 - 69/4*e^5 + 71*e^3 - 1161/14*e, -9/8*e^6 + 28*e^4 - 403/4*e^2 + 129/2, -45/56*e^7 + 75/4*e^5 - 175/4*e^3 - 130/7*e, 0, -3*e^6 + 75*e^4 - 282*e^2 + 214, 2*e^6 - 50*e^4 + 188*e^2 - 152, -3/2*e^6 + 75/2*e^4 - 141*e^2 + 81, 1/7*e^7 - 15/4*e^5 + 37/2*e^3 - 513/14*e, -3/4*e^6 + 75/4*e^4 - 69*e^2 + 93/2, 1/8*e^6 - 3/2*e^4 - 97/4*e^2 + 121/2, -73/56*e^7 + 33*e^5 - 529/4*e^3 + 1917/14*e, 1/4*e^6 - 7*e^4 + 79/2*e^2 - 47, 47/28*e^7 - 165/4*e^5 + 140*e^3 - 993/14*e, -37/56*e^7 + 33/2*e^5 - 245/4*e^3 + 585/14*e, 5/8*e^6 - 33/2*e^4 + 307/4*e^2 - 151/2, 29/56*e^7 - 27/2*e^5 + 245/4*e^3 - 877/14*e, -7/4*e^6 + 42*e^4 - 245/2*e^2 + 35, -2, -59/56*e^7 + 53/2*e^5 - 407/4*e^3 + 1161/14*e, -11/8*e^7 + 139/4*e^5 - 551/4*e^3 + 135*e, 1/28*e^7 - 3/4*e^5 - e^3 + 297/14*e, -27/28*e^7 + 24*e^5 - 175/2*e^3 + 397/7*e, 103/56*e^7 - 185/4*e^5 + 709/4*e^3 - 999/7*e, -67/56*e^7 + 121/4*e^5 - 481/4*e^3 + 837/7*e, 3/4*e^5 - 35/2*e^3 + 79/2*e, -17/56*e^7 + 6*e^5 + 35/4*e^3 - 897/14*e, -13/8*e^7 + 41*e^5 - 643/4*e^3 + 297/2*e, 3*e^6 - 75*e^4 + 282*e^2 - 214, -25/8*e^6 + 153/2*e^4 - 1007/4*e^2 + 251/2, 55/28*e^7 - 99/2*e^5 + 385/2*e^3 - 1188/7*e, -11/28*e^7 + 39/4*e^5 - 34*e^3 + 135/14*e, -23/8*e^6 + 71*e^4 - 981/4*e^2 + 283/2, 5/4*e^6 - 30*e^4 + 175/2*e^2 - 25, -3/2*e^6 + 75/2*e^4 - 141*e^2 + 111, -e^6 + 25*e^4 - 94*e^2 + 104, 41/56*e^7 - 73/4*e^5 + 263/4*e^3 - 216/7*e, 27/56*e^7 - 49/4*e^5 + 201/4*e^3 - 405/7*e, 41/56*e^7 - 73/4*e^5 + 263/4*e^3 - 216/7*e, -23/8*e^6 + 281/4*e^4 - 915/4*e^2 + 110, -99/56*e^7 + 87/2*e^5 - 595/4*e^3 + 1087/14*e, -39/56*e^7 + 69/4*e^5 - 245/4*e^3 + 256/7*e, 55/56*e^7 - 51/2*e^5 + 455/4*e^3 - 1587/14*e, 19/56*e^7 - 17/2*e^5 + 127/4*e^3 - 297/14*e, -65/56*e^7 + 57/2*e^5 - 385/4*e^3 + 669/14*e, 31/8*e^6 - 95*e^4 + 1261/4*e^2 - 323/2, 1/2*e^6 - 25/2*e^4 + 47*e^2 - 17, 7/4*e^6 - 173/4*e^4 + 150*e^2 - 175/2, 13/4*e^6 - 319/4*e^4 + 266*e^2 - 277/2, -9/2*e^6 + 225/2*e^4 - 423*e^2 + 343, 23/8*e^6 - 139/2*e^4 + 849/4*e^2 - 157/2, 5/28*e^7 - 6*e^5 + 105/2*e^3 - 647/7*e, 11/56*e^7 - 15/4*e^5 - 35/4*e^3 + 339/7*e, 3/8*e^6 - 33/4*e^4 + 39/4*e^2 + 24, -2*e^7 + 195/4*e^5 - 315/2*e^3 + 127/2*e, -2/7*e^7 + 33/4*e^5 - 105/2*e^3 + 1075/14*e, -17/28*e^7 + 61/4*e^5 - 58*e^3 + 621/14*e, 5/2*e^6 - 125/2*e^4 + 235*e^2 - 173, -23/56*e^7 + 21/2*e^5 - 179/4*e^3 + 837/14*e, 5*e^6 - 125*e^4 + 470*e^2 - 366, -23/8*e^6 + 283/4*e^4 - 959/4*e^2 + 131, -5/8*e^6 + 65/4*e^4 - 285/4*e^2 + 65, -5/8*e^6 + 15*e^4 - 175/4*e^2 + 25/2, -19/8*e^6 + 231/4*e^4 - 731/4*e^2 + 79, 1/2*e^6 - 25/2*e^4 + 47*e^2 - 49, -2*e^6 + 50*e^4 - 188*e^2 + 162, -2*e^6 + 50*e^4 - 188*e^2 + 134, 143/56*e^7 - 257/4*e^5 + 989/4*e^3 - 1431/7*e, -3/8*e^6 + 27/4*e^4 + 93/4*e^2 - 87, -3*e^6 + 149/2*e^4 - 265*e^2 + 165, -e^6 + 25*e^4 - 94*e^2 + 108, -2, -1/2*e^7 + 57/4*e^5 - 175/2*e^3 + 249/2*e, 13/8*e^6 - 41*e^4 + 631/4*e^2 - 233/2, 3/4*e^6 - 20*e^4 + 193/2*e^2 - 99, -45/28*e^7 + 81/2*e^5 - 315/2*e^3 + 846/7*e, -21/8*e^6 + 267/4*e^4 - 1065/4*e^2 + 210, -31/28*e^7 + 105/4*e^5 - 70*e^3 - 51/14*e, -9/4*e^6 + 58*e^4 - 491/2*e^2 + 213, -21/8*e^6 + 265/4*e^4 - 1021/4*e^2 + 189, -1/2*e^6 + 25/2*e^4 - 47*e^2 + 15, -2*e^6 + 50*e^4 - 188*e^2 + 168, 43/28*e^7 - 75/2*e^5 + 245/2*e^3 - 366/7*e, -5/28*e^7 + 17/4*e^5 - 10*e^3 - 351/14*e, -129/56*e^7 + 233/4*e^5 - 927/4*e^3 + 1620/7*e, -15/56*e^7 + 27/4*e^5 - 105/4*e^3 + 141/7*e, 33/14*e^7 - 231/4*e^5 + 385/2*e^3 - 1265/14*e, -17/56*e^7 + 27/4*e^5 - 35/4*e^3 - 172/7*e, 4*e^6 - 100*e^4 + 376*e^2 - 314, -19/56*e^7 + 15/2*e^5 - 35/4*e^3 - 417/14*e, 11/8*e^6 - 69/2*e^4 + 517/4*e^2 - 181/2, -137/56*e^7 + 247/4*e^5 - 971/4*e^3 + 1593/7*e, -9/8*e^7 + 57/2*e^5 - 455/4*e^3 + 185/2*e, -e^6 + 49/2*e^4 - 81*e^2 + 41, 12/7*e^7 - 173/4*e^5 + 339/2*e^3 - 2187/14*e, -11/14*e^7 + 20*e^5 - 83*e^3 + 702/7*e, 22, -3*e^6 + 75*e^4 - 282*e^2 + 210, 5/14*e^7 - 9*e^5 + 35*e^3 - 188/7*e, -65/56*e^7 + 117/4*e^5 - 455/4*e^3 + 702/7*e, -24, -7/4*e^6 + 171/4*e^4 - 139*e^2 + 133/2, 3/2*e^6 - 73/2*e^4 + 116*e^2 - 51, -5/4*e^6 + 123/4*e^4 - 104*e^2 + 113/2, 2*e^6 - 199/4*e^4 + 357/2*e^2 - 227/2, -15/8*e^6 + 187/4*e^4 - 679/4*e^2 + 111, -23/8*e^6 + 291/4*e^4 - 1135/4*e^2 + 215, -4*e^6 + 100*e^4 - 376*e^2 + 322, 73/56*e^7 - 131/4*e^5 + 499/4*e^3 - 675/7*e, -9/28*e^7 + 9*e^5 - 105/2*e^3 + 501/7*e, -41/28*e^7 + 147/4*e^5 - 140*e^3 + 1431/14*e, -13/14*e^7 + 24*e^5 - 105*e^3 + 710/7*e, -2*e^6 + 50*e^4 - 188*e^2 + 172, 155/56*e^7 - 279/4*e^5 + 1085/4*e^3 - 1674/7*e, 27/56*e^7 - 45/4*e^5 + 105/4*e^3 + 78/7*e, -7/4*e^7 + 171/4*e^5 - 140*e^3 + 121/2*e, -3/2*e^6 + 75/2*e^4 - 141*e^2 + 81, 15/56*e^7 - 13/2*e^5 + 75/4*e^3 + 243/14*e, 0, -17/28*e^7 + 63/4*e^5 - 73*e^3 + 1755/14*e, -11/4*e^6 + 137/2*e^4 - 495/2*e^2 + 160, 2*e^6 - 50*e^4 + 188*e^2 - 142, 2*e^6 - 50*e^4 + 188*e^2 - 144, -2*e^6 + 95/2*e^4 - 129*e^2 + 19, -9/2*e^6 + 441/4*e^4 - 729/2*e^2 + 369/2, -17/56*e^7 + 8*e^5 - 161/4*e^3 + 1161/14*e, -9/2*e^6 + 225/2*e^4 - 423*e^2 + 353, 23/14*e^7 - 159/4*e^5 + 245/2*e^3 - 513/14*e, 59/56*e^7 - 105/4*e^5 + 377/4*e^3 - 297/7*e, -3/2*e^6 + 38*e^4 - 149*e^2 + 114, -25/8*e^6 + 301/4*e^4 - 897/4*e^2 + 73, -17/56*e^7 + 8*e^5 - 161/4*e^3 + 1161/14*e, -5/4*e^6 + 29*e^4 - 131/2*e^2 - 17, 11/56*e^7 - 9/2*e^5 + 35/4*e^3 + 125/14*e, 1/8*e^7 - 3/2*e^5 - 105/4*e^3 + 155/2*e, 57/56*e^7 - 99/4*e^5 + 315/4*e^3 - 204/7*e, -1/2*e^6 + 47/4*e^4 - 59/2*e^2 - 1/2, 11/8*e^6 - 125/4*e^4 + 231/4*e^2 + 46, -61/56*e^7 + 111/4*e^5 - 455/4*e^3 + 684/7*e, 47/28*e^7 - 87/2*e^5 + 385/2*e^3 - 1326/7*e, e^6 - 25*e^4 + 94*e^2 - 78, 9/7*e^7 - 65/2*e^5 + 129*e^3 - 891/7*e, 13/8*e^7 - 41*e^5 + 643/4*e^3 - 297/2*e, 3/2*e^6 - 75/2*e^4 + 141*e^2 - 171, 99/56*e^7 - 177/4*e^5 + 665/4*e^3 - 820/7*e, -e^6 + 25*e^4 - 94*e^2 + 66, -36, 39/56*e^7 - 18*e^5 + 315/4*e^3 - 1065/14*e, -2*e^6 + 50*e^4 - 188*e^2 + 148, -35/8*e^7 + 441/4*e^5 - 1715/4*e^3 + 378*e, 59/28*e^7 - 201/4*e^5 + 140*e^3 - 117/14*e, 9/4*e^6 - 233/4*e^4 + 251*e^2 - 447/2, 1/8*e^7 - 3/2*e^5 - 105/4*e^3 + 155/2*e, 127/56*e^7 - 225/4*e^5 + 805/4*e^3 - 862/7*e, -7*e^6 + 175*e^4 - 658*e^2 + 528, -19/56*e^7 + 17/2*e^5 - 127/4*e^3 + 297/14*e, -8, 2*e^6 - 50*e^4 + 188*e^2 - 118, -73/28*e^7 + 263/4*e^5 - 257*e^3 + 3267/14*e, 3/2*e^6 - 75/2*e^4 + 141*e^2 - 83, -14, 3*e^6 - 75*e^4 + 282*e^2 - 256, 9/2*e^6 - 225/2*e^4 + 423*e^2 - 327, -3/8*e^7 + 21/2*e^5 - 245/4*e^3 + 167/2*e, -31/8*e^6 + 96*e^4 - 1349/4*e^2 + 407/2, -69/56*e^7 + 125/4*e^5 - 507/4*e^3 + 972/7*e, -129/56*e^7 + 231/4*e^5 - 867/4*e^3 + 1053/7*e, -31/14*e^7 + 56*e^5 - 223*e^3 + 1566/7*e, 13/8*e^7 - 153/4*e^5 + 385/4*e^3 + 20*e, 9/4*e^7 - 54*e^5 + 315/2*e^3 - 27*e, -6, -2*e^6 + 50*e^4 - 188*e^2 + 192, -29/8*e^6 + 361/4*e^4 - 1301/4*e^2 + 209, -1/8*e^6 + 5*e^4 - 211/4*e^2 + 173/2, 7/2*e^6 - 175/2*e^4 + 322*e^2 - 217, 19/8*e^6 - 239/4*e^4 + 907/4*e^2 - 163, -65/56*e^7 + 27*e^5 - 245/4*e^3 - 437/14*e, 15/8*e^6 - 91/2*e^4 + 569/4*e^2 - 117/2, 21/8*e^6 - 255/4*e^4 + 801/4*e^2 - 84, 61/28*e^7 - 213/4*e^5 + 175*e^3 - 1077/14*e, -237/56*e^7 + 427/4*e^5 - 1671/4*e^3 + 2673/7*e, 157/56*e^7 - 283/4*e^5 + 1111/4*e^3 - 1809/7*e, e^7 - 93/4*e^5 + 105/2*e^3 + 55/2*e, -2*e^6 + 50*e^4 - 188*e^2 + 112, -4*e^6 + 100*e^4 - 376*e^2 + 294, -17/28*e^7 + 27/2*e^5 - 35/2*e^3 - 344/7*e, -15/56*e^7 + 27/4*e^5 - 105/4*e^3 + 162/7*e, 5*e^6 - 125*e^4 + 470*e^2 - 360, -27/56*e^7 + 49/4*e^5 - 201/4*e^3 + 405/7*e, -113/56*e^7 + 101/2*e^5 - 749/4*e^3 + 1647/14*e, -25/14*e^7 + 93/2*e^5 - 210*e^3 + 1493/7*e, -e^6 + 25*e^4 - 94*e^2 + 94, -19/14*e^7 + 137/4*e^5 - 269/2*e^3 + 1755/14*e, -3/2*e^6 + 81/2*e^4 - 204*e^2 + 219]; 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;