/* 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, -6, -7, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, w^3 - w^2 - 5*w - 2], [9, 3, w^3 - w^2 - 5*w - 1], [11, 11, -w^3 + w^2 + 6*w + 2], [11, 11, w - 1], [13, 13, w^3 - 2*w^2 - 4*w + 2], [13, 13, -w^2 + w + 4], [23, 23, w^2 - 2*w - 2], [23, 23, w^3 - w^2 - 6*w - 3], [23, 23, -w^2 + 2*w + 5], [23, 23, -w + 2], [37, 37, 2*w^3 - 2*w^2 - 12*w - 1], [37, 37, w^3 - 2*w^2 - 5*w + 2], [37, 37, w^3 - 2*w^2 - 5*w + 3], [37, 37, -w^3 + w^2 + 6*w - 2], [47, 47, w^2 - 2*w - 1], [47, 47, w^2 - 2*w - 6], [59, 59, 2*w - 1], [59, 59, -2*w^3 + 2*w^2 + 12*w + 3], [73, 73, -w^3 + w^2 + 7*w + 1], [83, 83, -w^3 + w^2 + 4*w + 3], [83, 83, 2*w^3 - 2*w^2 - 11*w - 4], [107, 107, w^3 - w^2 - 4*w - 5], [107, 107, -2*w^3 + 2*w^2 + 11*w + 6], [121, 11, 2*w^3 - 2*w^2 - 10*w - 3], [131, 131, 2*w^3 - 3*w^2 - 7*w + 2], [131, 131, 3*w^3 - 2*w^2 - 18*w - 6], [157, 157, -2*w^3 + 2*w^2 + 9*w], [157, 157, -2*w^3 + 4*w^2 + 8*w - 3], [167, 167, -2*w^3 + w^2 + 11*w + 10], [167, 167, -3*w^3 + 4*w^2 + 14*w + 4], [169, 13, -2*w^3 + 2*w^2 + 10*w + 7], [179, 179, -w^3 + 6*w + 2], [179, 179, 2*w^3 - 3*w^2 - 9*w + 4], [181, 181, 2*w^2 - w - 8], [181, 181, 3*w^3 - 5*w^2 - 14*w + 3], [181, 181, -3*w^3 + 3*w^2 + 18*w + 2], [181, 181, w^3 - 7*w - 9], [191, 191, -2*w^3 + 3*w^2 + 10*w + 1], [191, 191, w^2 - 6], [193, 193, w^3 - 3*w^2 - 4*w + 3], [193, 193, 2*w^3 - 4*w^2 - 9*w + 8], [229, 229, -2*w^3 + 2*w^2 + 13*w + 3], [229, 229, w^2 - w - 8], [239, 239, w^3 - 7*w - 2], [239, 239, -w^3 + 2*w^2 + 3*w - 5], [241, 241, -3*w^3 + 3*w^2 + 16*w + 2], [241, 241, 2*w^3 - 2*w^2 - 9*w - 1], [251, 251, -2*w^3 + 3*w^2 + 8*w - 3], [251, 251, -2*w^3 + w^2 + 12*w + 4], [251, 251, w^3 + w^2 - 7*w - 7], [251, 251, 2*w^3 - 3*w^2 - 10*w - 2], [263, 263, 3*w^3 - 4*w^2 - 16*w], [263, 263, w^2 + w - 4], [277, 277, 2*w^2 - w - 7], [277, 277, 3*w^3 - 5*w^2 - 14*w + 4], [313, 313, 2*w^3 - 4*w^2 - 9*w + 3], [313, 313, -w^3 + 3*w^2 + 4*w - 8], [337, 337, 2*w^3 - 2*w^2 - 13*w - 1], [337, 337, w^3 - w^2 - 8*w - 2], [347, 347, 4*w^3 - 4*w^2 - 22*w + 1], [347, 347, -2*w^3 + 5*w^2 + 6*w - 10], [347, 347, -4*w^3 + 5*w^2 + 20*w + 3], [347, 347, -3*w^2 + 4*w + 9], [349, 349, -4*w^3 + 5*w^2 + 19*w + 4], [349, 349, -w^3 + 4*w^2 + w - 11], [349, 349, 2*w^3 - 3*w^2 - 9*w - 4], [349, 349, w^3 - 6*w - 10], [361, 19, -w^3 + 3*w^2 + 4*w - 4], [361, 19, -2*w^3 + 4*w^2 + 9*w - 7], [397, 397, 3*w^3 - 3*w^2 - 14*w - 6], [397, 397, -2*w^3 + 5*w^2 + 6*w - 6], [397, 397, 3*w^2 - 4*w - 13], [397, 397, -4*w^3 + 4*w^2 + 21*w + 7], [409, 409, 2*w^3 - 4*w^2 - 10*w + 7], [409, 409, -2*w^3 + 4*w^2 + 10*w - 3], [421, 421, w^3 - w^2 - 6*w - 6], [421, 421, w - 5], [431, 431, -3*w^3 + 4*w^2 + 18*w - 4], [431, 431, 2*w^3 - 3*w^2 - 6*w + 3], [433, 433, 2*w^3 - 15*w - 12], [433, 433, w^3 - w^2 - 9*w - 5], [443, 443, -3*w^3 + 5*w^2 + 14*w - 9], [443, 443, 2*w^3 - 12*w - 13], [457, 457, -w^3 + 2*w^2 + 5*w - 8], [457, 457, -2*w^3 + 4*w^2 + 9*w - 4], [457, 457, 2*w^3 - 2*w^2 - 14*w - 5], [457, 457, -w^3 + 3*w^2 + 4*w - 7], [467, 467, -2*w^3 + 3*w^2 + 8*w - 4], [467, 467, -2*w^3 + w^2 + 12*w + 3], [479, 479, w^3 + w^2 - 8*w - 7], [479, 479, 2*w^3 - 14*w - 9], [479, 479, -2*w^3 + 4*w^2 + 6*w - 5], [479, 479, -2*w^3 + 4*w^2 + 7*w - 6], [491, 491, -2*w^3 + 3*w^2 + 11*w + 2], [491, 491, -w^3 + 2*w^2 + 6*w - 6], [541, 541, -4*w^2 + 6*w + 15], [541, 541, -2*w^3 + 6*w^2 + 4*w - 11], [563, 563, -2*w^3 + w^2 + 14*w], [563, 563, -w^2 + 4*w + 9], [577, 577, w^2 - 4*w - 4], [577, 577, -2*w^3 + w^2 + 14*w + 5], [587, 587, w^3 - 4*w^2 + 6], [587, 587, -w^3 + 3*w^2 + 6*w - 7], [587, 587, 4*w^3 - 6*w^2 - 21*w + 2], [587, 587, 3*w^3 - 5*w^2 - 14*w], [599, 599, 2*w^3 - 2*w^2 - 7*w - 3], [599, 599, 5*w^3 - 5*w^2 - 28*w - 6], [601, 601, w^3 + w^2 - 10*w - 10], [601, 601, 2*w^3 - 3*w^2 - 10*w - 5], [613, 613, w^2 - 4*w - 3], [613, 613, 2*w^3 - w^2 - 14*w - 6], [625, 5, -5], [647, 647, -3*w^3 + 4*w^2 + 17*w + 1], [647, 647, -w^3 + 2*w^2 + 7*w - 4], [659, 659, -5*w^3 + 7*w^2 + 23*w + 3], [659, 659, w^3 - 2*w^2 - w - 2], [659, 659, -w^3 + 2*w^2 + 2*w - 6], [659, 659, -2*w^3 + w^2 + 13*w + 2], [683, 683, -2*w^3 + 3*w^2 + 8*w - 5], [683, 683, -2*w^3 + w^2 + 12*w + 2], [709, 709, w^3 + w^2 - 10*w - 8], [709, 709, 2*w^2 - 5*w - 7], [719, 719, w^3 - 5*w - 10], [719, 719, -3*w^3 + 4*w^2 + 15*w + 5], [733, 733, -w^3 + w^2 + 4*w - 5], [733, 733, 2*w^2 - 5*w - 6], [733, 733, -2*w^3 + 2*w^2 + 11*w - 4], [733, 733, w^3 + w^2 - 10*w - 9], [743, 743, -w^3 + 3*w^2 + 2*w - 10], [743, 743, 2*w^2 - 3*w - 3], [757, 757, 2*w^3 - 3*w^2 - 8*w - 5], [757, 757, w^3 - 9*w - 5], [757, 757, -2*w^3 + w^2 + 12*w + 12], [757, 757, w^3 - 9*w - 4], [769, 769, -4*w^3 + 4*w^2 + 21*w + 6], [769, 769, 3*w^3 - 3*w^2 - 14*w - 5], [827, 827, -2*w^3 + w^2 + 12*w + 1], [827, 827, -4*w^3 + 4*w^2 + 22*w + 7], [827, 827, 2*w^3 - 2*w^2 - 8*w - 5], [827, 827, -2*w^3 + 3*w^2 + 8*w - 6], [829, 829, -3*w^3 + w^2 + 17*w + 11], [829, 829, -4*w^3 + 2*w^2 + 23*w + 16], [839, 839, w^2 + 2*w - 4], [839, 839, 2*w^3 + w^2 - 14*w - 13], [839, 839, -4*w^3 + 7*w^2 + 16*w - 6], [839, 839, -4*w^3 + 5*w^2 + 22*w + 1], [863, 863, 2*w^3 - 13*w - 8], [863, 863, 3*w^3 - 5*w^2 - 12*w + 5], [887, 887, 2*w^2 + w - 7], [887, 887, -5*w^3 + 7*w^2 + 26*w - 2], [911, 911, -3*w^3 + 3*w^2 + 17*w + 7], [911, 911, 2*w^2 - 9], [911, 911, w^3 - w^2 - 3*w - 5], [911, 911, 4*w^3 - 6*w^2 - 20*w + 1], [937, 937, -4*w^3 + 5*w^2 + 22*w - 7], [937, 937, 4*w^3 - 4*w^2 - 21*w - 2], [937, 937, -4*w^2 + 8*w + 13], [937, 937, 3*w^3 - 3*w^2 - 14*w - 1], [947, 947, 3*w^3 - 4*w^2 - 12*w - 6], [947, 947, 3*w^3 - 5*w^2 - 10*w + 4], [947, 947, -3*w^3 + 4*w^2 + 19*w + 3], [947, 947, -3*w^3 + 4*w^2 + 19*w - 4], [971, 971, -3*w^3 + 5*w^2 + 13*w - 7], [971, 971, w^3 + w^2 - 7*w - 5], [983, 983, -5*w^3 + 8*w^2 + 23*w - 2], [983, 983, -w^3 + w^2 + 3*w + 7], [1009, 1009, w^3 - w^2 - 6*w - 7], [1009, 1009, w - 6], [1033, 1033, 4*w^3 - 4*w^2 - 21*w - 5], [1033, 1033, -3*w^3 + 3*w^2 + 14*w + 4], [1069, 1069, -w^2 - 3], [1069, 1069, 2*w^3 - 3*w^2 - 10*w + 8], [1103, 1103, -w^3 + 2*w^2 + 7*w - 5], [1103, 1103, 3*w^3 - 3*w^2 - 19*w - 7], [1103, 1103, 2*w^3 - 11*w - 12], [1103, 1103, 3*w^3 - 4*w^2 - 17*w - 2], [1117, 1117, -3*w^3 + 3*w^2 + 14*w + 2], [1117, 1117, 4*w^3 - 4*w^2 - 21*w - 3], [1129, 1129, -w^3 + 2*w^2 + 5*w - 9], [1129, 1129, -2*w^3 + w^2 + 15*w + 2], [1151, 1151, -3*w^2 + 5*w + 8], [1151, 1151, 3*w^3 - 3*w^2 - 18*w - 11], [1153, 1153, 3*w^3 - 3*w^2 - 14*w - 3], [1153, 1153, -4*w^3 + 4*w^2 + 21*w + 4], [1163, 1163, -w^3 + 3*w^2 + w - 9], [1163, 1163, -w^3 - w^2 + 9*w + 5], [1187, 1187, -3*w^3 + 4*w^2 + 18*w - 6], [1187, 1187, -2*w^3 + 11*w + 5], [1187, 1187, 2*w^3 - 3*w^2 - 13*w - 4], [1187, 1187, 5*w^3 - 7*w^2 - 24*w + 6], [1201, 1201, -5*w^3 + 5*w^2 + 30*w + 4], [1201, 1201, -5*w - 1], [1213, 1213, -6*w^3 + 7*w^2 + 31*w - 2], [1213, 1213, 2*w^3 - 6*w^2 - 5*w + 14], [1249, 1249, 3*w^3 - 3*w^2 - 13*w - 1], [1249, 1249, 3*w^3 - 2*w^2 - 15*w - 4], [1249, 1249, -5*w^3 + 6*w^2 + 25*w - 1], [1249, 1249, -5*w^3 + 5*w^2 + 27*w + 3], [1283, 1283, -w^3 - 3*w^2 + 12*w + 15], [1283, 1283, -2*w^3 + 6*w^2 + 3*w - 12], [1297, 1297, 5*w^3 - 7*w^2 - 22*w - 2], [1297, 1297, 6*w^3 - 8*w^2 - 29*w + 2], [1307, 1307, -5*w^3 + 6*w^2 + 27*w + 2], [1307, 1307, -3*w^3 + 2*w^2 + 19*w + 2], [1319, 1319, 3*w - 4], [1319, 1319, -3*w^3 + 3*w^2 + 18*w + 7], [1321, 1321, -5*w^3 + 5*w^2 + 27*w + 5], [1321, 1321, 4*w^3 - 7*w^2 - 18*w + 7], [1367, 1367, -3*w^3 + 4*w^2 + 12*w - 4], [1367, 1367, 4*w^3 - 3*w^2 - 23*w - 4], [1381, 1381, -2*w^3 + 2*w^2 + 12*w - 5], [1381, 1381, 2*w^3 - 16*w - 13], [1381, 1381, -2*w - 7], [1381, 1381, 2*w^2 - 6*w - 3], [1427, 1427, 3*w^3 - 5*w^2 - 15*w - 1], [1427, 1427, -w^3 + 3*w^2 + 5*w - 11], [1429, 1429, -w^3 + 4*w^2 + 2*w - 10], [1429, 1429, 2*w^3 - 5*w^2 - 7*w + 8], [1439, 1439, 4*w^3 - 5*w^2 - 22*w - 2], [1439, 1439, w^2 + 2*w - 5], [1451, 1451, 3*w^3 - 5*w^2 - 12*w + 6], [1451, 1451, 3*w^3 - w^2 - 19*w - 9], [1451, 1451, 3*w^3 - 5*w^2 - 11*w + 5], [1451, 1451, 2*w^3 - 13*w - 7], [1453, 1453, 5*w^3 - 6*w^2 - 25*w - 2], [1453, 1453, 3*w^3 - 6*w^2 - 13*w + 10], [1487, 1487, 4*w^3 - 5*w^2 - 20*w - 6], [1487, 1487, 2*w^3 - w^2 - 10*w - 11], [1487, 1487, -3*w^3 + 4*w^2 + 16*w + 4], [1487, 1487, w^2 + w - 8], [1489, 1489, -4*w^3 + 3*w^2 + 22*w + 7], [1489, 1489, -4*w^3 + 5*w^2 + 18*w], [1499, 1499, 3*w^3 - 4*w^2 - 16*w - 6], [1499, 1499, w^2 + w - 10], [1511, 1511, -2*w^3 + w^2 + 9*w + 8], [1511, 1511, -5*w^3 + 6*w^2 + 26*w + 4], [1523, 1523, 2*w^3 - 5*w^2 - 4*w + 9], [1523, 1523, 6*w^3 - 6*w^2 - 33*w + 1], [1559, 1559, 2*w^3 - 14*w - 7], [1559, 1559, -2*w^3 + 4*w^2 + 6*w - 7], [1571, 1571, -4*w^3 + 6*w^2 + 23*w - 7], [1571, 1571, -4*w^3 + 8*w^2 + 15*w - 11], [1571, 1571, 5*w^3 - 7*w^2 - 23*w - 5], [1571, 1571, 4*w^3 - w^2 - 28*w - 13], [1597, 1597, w^2 - 5*w - 4], [1597, 1597, 5*w^3 - 6*w^2 - 25*w - 1], [1597, 1597, -3*w^3 + 2*w^2 + 20*w + 6], [1597, 1597, 3*w^3 - 2*w^2 - 15*w - 6], [1607, 1607, 3*w^3 - 4*w^2 - 13*w + 6], [1607, 1607, 4*w^3 - 3*w^2 - 24*w - 4], [1607, 1607, -3*w^3 + 2*w^2 + 17*w + 1], [1607, 1607, -2*w^3 + 3*w^2 + 6*w - 5], [1609, 1609, -2*w^3 + 2*w^2 + 11*w - 5], [1609, 1609, -w^3 + w^2 + 4*w - 6], [1609, 1609, w^3 - 6*w^2 + 5*w + 13], [1609, 1609, -4*w^3 + 3*w^2 + 21*w + 18], [1619, 1619, -3*w^3 + 4*w^2 + 17*w + 3], [1619, 1619, -w^3 + 2*w^2 + 7*w - 6], [1621, 1621, -w^3 + 2*w^2 + 3*w - 10], [1621, 1621, -w^3 + 7*w - 3], [1657, 1657, -4*w^3 + 3*w^2 + 22*w + 9], [1657, 1657, -4*w^3 + 5*w^2 + 18*w + 2], [1667, 1667, 2*w^3 + w^2 - 14*w - 12], [1667, 1667, 3*w^3 - 5*w^2 - 15*w - 3], [1667, 1667, -4*w^3 + 6*w^2 + 17*w - 5], [1667, 1667, 3*w^3 - w^2 - 18*w - 8], [1669, 1669, 2*w^3 - 3*w^2 - 15*w - 2], [1669, 1669, 2*w^3 - 5*w^2 - 8*w + 18], [1681, 41, 4*w^3 - 3*w^2 - 22*w - 8], [1681, 41, -4*w^3 + 5*w^2 + 18*w + 1], [1693, 1693, -4*w^3 + 4*w^2 + 21*w - 5], [1693, 1693, -3*w^3 + 3*w^2 + 14*w - 6], [1741, 1741, 2*w^3 - 5*w^2 - 11*w + 4], [1741, 1741, -5*w^2 + 7*w + 22], [1787, 1787, 4*w^3 - 6*w^2 - 19*w - 2], [1787, 1787, 4*w^3 - 4*w^2 - 22*w - 9], [1787, 1787, 2*w^3 - 2*w^2 - 8*w - 7], [1787, 1787, w^3 + w^2 - 6*w - 13], [1801, 1801, 4*w^2 - 4*w - 19], [1801, 1801, 3*w^3 - 2*w^2 - 19*w - 16], [1801, 1801, -w^3 + 4*w^2 - 2*w - 8], [1801, 1801, 4*w^3 - 8*w^2 - 16*w + 5], [1811, 1811, 2*w^3 + 2*w^2 - 16*w - 17], [1811, 1811, -4*w^3 + 8*w^2 + 14*w - 9], [1823, 1823, -3*w^3 + 8*w^2 + 7*w - 14], [1823, 1823, -4*w^3 + 4*w^2 + 24*w + 7], [1823, 1823, 4*w - 3], [1823, 1823, -w^3 - 4*w^2 + 13*w + 19], [1847, 1847, 3*w^2 - 2*w - 19], [1847, 1847, -7*w^3 + 8*w^2 + 39*w + 2], [1861, 1861, -w^3 - 3*w^2 + 8*w + 17], [1861, 1861, -6*w^3 + 10*w^2 + 27*w - 6], [1861, 1861, -w^3 + 4*w^2 + 3*w - 8], [1861, 1861, -3*w^3 + 6*w^2 + 13*w - 9], [1871, 1871, 2*w^3 + 2*w^2 - 17*w - 17], [1871, 1871, w^3 - 5*w^2 + w + 9], [1871, 1871, -5*w^3 + 4*w^2 + 30*w + 12], [1871, 1871, -w^3 + 5*w^2 - w - 17], [1873, 1873, -4*w^3 + 7*w^2 + 22*w - 5], [1873, 1873, 4*w^3 - 2*w^2 - 24*w - 21], [1873, 1873, 4*w^2 - 5*w - 17], [1873, 1873, -4*w^3 + 6*w^2 + 16*w + 7], [1907, 1907, -4*w^3 + 6*w^2 + 15*w + 6], [1907, 1907, 3*w^3 - 2*w^2 - 14*w - 10], [1907, 1907, -6*w^3 + 9*w^2 + 28*w - 1], [1907, 1907, 6*w^3 - 7*w^2 - 31*w - 6], [1931, 1931, 6*w^3 - 7*w^2 - 34*w - 2], [1931, 1931, 3*w^3 - 20*w - 14], [1933, 1933, -3*w^3 + 6*w^2 + 13*w - 6], [1933, 1933, w^3 - 10*w - 6], [1933, 1933, -2*w^3 + w^2 + 15*w + 4], [1933, 1933, -w^3 + 4*w^2 + 3*w - 11], [1993, 1993, 6*w^3 - 7*w^2 - 31*w], [1993, 1993, 5*w^3 - 8*w^2 - 25*w + 9]]; primes := [ideal : I in primesArray]; heckePol := x^2 - x - 26; K := NumberField(heckePol); heckeEigenvaluesArray := [1, 1, e, e - 1, -e - 1, e - 2, -6, e - 2, 6, e + 1, -e + 6, -e - 5, e - 6, e + 5, 8, -8, e + 7, e - 8, -6, -2*e, -2*e + 2, -e - 4, -e + 5, 8, -12, 12, -e - 10, e - 11, -e - 9, -e + 10, -20, -e - 2, -e + 3, e + 10, -e + 11, -e + 5, e + 4, -4*e + 5, -4*e - 1, 3*e - 8, -3*e - 5, e + 2, -e + 3, -2*e - 4, -2*e + 6, 2*e + 12, -2*e + 14, -e + 11, -e - 10, -28, 28, -4, 4, 2*e - 24, -2*e - 22, 2*e, -2*e + 2, 18, 18, -e + 1, -3*e + 21, -e, -3*e - 18, -4*e + 2, 4*e - 2, -2*e + 16, 2*e + 14, 5*e - 12, -5*e - 7, -3*e + 6, 3*e - 2, -3*e + 1, 3*e + 3, -2*e + 26, 2*e + 24, 2*e + 16, -2*e + 18, 2*e - 4, 2*e + 2, -2*e, 2*e - 2, 3*e - 23, 3*e + 20, 3, e - 36, 3, -e - 35, -6*e, -6*e + 6, 2*e + 24, 4*e - 7, 4*e + 3, 2*e - 26, -12, 12, -3*e + 7, 3*e + 4, -e - 13, -e + 14, -4*e + 20, 4*e + 16, -12, 28, -28, 12, -6*e - 12, -6*e + 18, -8*e + 11, 8*e + 3, e - 24, -e - 23, -29, -2*e + 13, -2*e - 11, -5*e + 15, -5*e - 10, e - 23, e + 22, e - 17, e + 16, 5*e - 10, -5*e - 5, 2*e - 16, 2*e + 14, -e + 37, -e - 18, e + 36, e - 19, 31, -31, 5*e - 17, -5*e - 7, -5*e - 12, 5*e - 12, -8*e + 14, 8*e + 6, 4*e - 4, -3*e - 14, -3*e + 17, 4*e, -8*e + 14, 8*e + 6, -3*e + 14, 4*e - 2, 4*e - 2, -3*e - 11, -7*e + 25, -7*e - 18, -4*e + 15, -4*e - 11, -4*e, 6*e + 19, -4*e + 4, 6*e - 25, 4*e - 9, 37, -4*e - 5, 37, 4*e + 16, 4*e - 20, -3*e - 24, -3*e + 27, -12, 12, 46, -46, 2*e + 24, -2*e + 26, -2*e + 32, 2*e + 30, 3*e + 31, -3*e + 34, 2*e + 38, -5*e - 14, -5*e + 19, 2*e - 40, -3*e + 36, 3*e + 33, -4*e - 38, 4*e - 42, 4*e + 30, 4*e - 34, 2*e - 20, -2*e - 18, e - 7, e + 6, 5*e + 38, 3*e + 39, 5*e - 43, 3*e - 42, 2, 2, 5*e - 14, -5*e - 9, 6*e - 13, -2*e - 44, 2*e - 46, -6*e - 7, -e - 13, -e + 14, -6*e + 25, 6*e + 19, -3*e + 46, -3*e - 43, 3*e + 51, 3*e - 54, 4*e - 24, -4*e - 20, -2*e + 8, -2*e - 6, e + 29, -5*e + 7, -e + 30, 5*e + 2, 6*e + 24, 6*e - 30, 11*e - 18, -11*e - 7, 4*e - 12, 4*e + 8, -9*e + 15, -e + 30, -e - 29, -9*e - 6, 6*e - 32, -6*e - 26, 2*e - 4, 2*e + 2, -8*e + 16, -8*e - 8, -12*e + 16, 12*e + 4, -12*e - 4, -12*e + 16, -8*e - 14, -8*e + 22, 8*e + 32, 8*e - 40, 2*e + 9, 2*e - 11, -5*e + 2, -5*e + 3, 4*e + 36, 4*e - 40, 2*e + 42, 3*e - 37, -2*e + 44, -3*e - 34, 5*e - 43, 3*e - 6, 5*e + 38, 3*e + 3, 5*e - 45, -5*e - 40, -8*e - 26, 8*e - 34, 3*e + 21, 3*e - 24, 9*e + 10, -9*e + 19, -38, -38, -72, 72, -2*e - 32, -2*e + 34, 10*e, -10*e + 10, -10*e + 8, 10*e - 2, -e + 2, e + 1, -5*e - 22, 5*e - 27, 3*e - 27, -e + 50, -e - 49, 3*e + 24, 8*e - 6, -8*e + 31, 8*e + 23, -8*e + 2, 2*e - 34, 2*e + 32, 4*e + 4, -4*e + 23, -4*e - 19, 4*e - 8, 12*e - 28, 12*e + 16, -e + 36, e + 35, -2*e + 34, 2*e + 32, -6*e + 15, 4*e + 40, -6*e - 9, 4*e - 44, 9*e - 36, -8*e + 8, -9*e - 27, 8*e, -2*e + 64, -3*e + 17, -2*e - 62, -3*e - 14, 5*e - 22, 5*e + 17, 7*e - 45, e - 9, -e - 8, -7*e - 38, -e - 56, e - 57]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := -1; 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;