/* 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![29, 8, -12, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [5, 5, -w + 2], [5, 5, -3/5*w^3 - w^2 + 26/5*w + 42/5], [9, 3, 2/5*w^3 - 19/5*w - 3/5], [9, 3, -1/5*w^3 + 2/5*w + 4/5], [11, 11, -2/5*w^3 - w^2 + 14/5*w + 28/5], [11, 11, 1/5*w^3 - w^2 - 7/5*w + 36/5], [16, 2, 2], [29, 29, -w], [29, 29, 1/5*w^3 - 12/5*w + 1/5], [41, 41, -2/5*w^3 - w^2 + 14/5*w + 38/5], [41, 41, -w^2 + 10], [41, 41, 11/5*w^3 + 3*w^2 - 102/5*w - 149/5], [41, 41, -1/5*w^3 + w^2 + 7/5*w - 26/5], [49, 7, 1/5*w^3 + w^2 - 12/5*w - 19/5], [49, 7, 1/5*w^3 + w^2 - 12/5*w - 44/5], [59, 59, -4/5*w^3 - w^2 + 33/5*w + 36/5], [59, 59, -2*w^3 - 3*w^2 + 17*w + 26], [61, 61, -1/5*w^3 + w^2 + 12/5*w - 51/5], [61, 61, 4/5*w^3 + w^2 - 28/5*w - 41/5], [71, 71, 1/5*w^3 + w^2 - 7/5*w - 49/5], [71, 71, -2*w^3 - 3*w^2 + 18*w + 28], [89, 89, -3/5*w^3 + 16/5*w + 7/5], [89, 89, 3/5*w^3 + w^2 - 26/5*w - 57/5], [121, 11, -3/5*w^3 + 21/5*w + 2/5], [131, 131, w^2 - w - 8], [131, 131, 2/5*w^3 + w^2 - 19/5*w - 23/5], [149, 149, 2/5*w^3 - w^2 - 14/5*w + 22/5], [149, 149, 3/5*w^3 + w^2 - 21/5*w - 42/5], [179, 179, -6/5*w^3 - w^2 + 57/5*w + 69/5], [179, 179, -2/5*w^3 - 2*w^2 + 14/5*w + 73/5], [179, 179, 2*w^2 - 11], [179, 179, -3/5*w^3 + 2*w^2 + 16/5*w - 48/5], [181, 181, -7/5*w^3 - 2*w^2 + 59/5*w + 78/5], [181, 181, -2/5*w^3 + 24/5*w - 27/5], [191, 191, -3/5*w^3 + 2*w^2 + 21/5*w - 78/5], [191, 191, 2/5*w^3 - 2*w^2 - 9/5*w + 57/5], [199, 199, -1/5*w^3 + 17/5*w - 16/5], [199, 199, 1/5*w^3 - 17/5*w - 14/5], [211, 211, 2/5*w^3 - w^2 + 1/5*w + 7/5], [211, 211, -1/5*w^3 + w^2 - 3/5*w - 26/5], [211, 211, 4/5*w^3 + w^2 - 38/5*w - 36/5], [211, 211, -1/5*w^3 - 2*w^2 + 7/5*w + 44/5], [239, 239, 3/5*w^3 + w^2 - 16/5*w - 37/5], [239, 239, 2/5*w^3 + w^2 - 14/5*w - 53/5], [251, 251, 2/5*w^3 - 24/5*w - 13/5], [251, 251, 2*w - 3], [251, 251, w^2 + w - 10], [251, 251, -4/5*w^3 - w^2 + 28/5*w + 46/5], [269, 269, 1/5*w^3 - 12/5*w - 24/5], [269, 269, w - 5], [271, 271, -3/5*w^3 + 31/5*w - 33/5], [271, 271, -8/5*w^3 - 2*w^2 + 66/5*w + 77/5], [281, 281, 4/5*w^3 - w^2 - 38/5*w + 39/5], [281, 281, -14/5*w^3 - 3*w^2 + 128/5*w + 161/5], [311, 311, 1/5*w^3 - 2*w^2 - 2/5*w + 51/5], [311, 311, 1/5*w^3 - w^2 - 17/5*w + 21/5], [331, 331, 1/5*w^3 + 3/5*w - 4/5], [331, 331, 3/5*w^3 - 31/5*w - 2/5], [359, 359, -w^3 + 6*w - 1], [359, 359, -6/5*w^3 + 47/5*w - 6/5], [359, 359, 3/5*w^3 + w^2 - 31/5*w - 27/5], [359, 359, 3/5*w^3 + w^2 - 26/5*w - 22/5], [361, 19, 4/5*w^3 - 28/5*w - 11/5], [361, 19, 1/5*w^3 - 7/5*w - 24/5], [379, 379, 9/5*w^3 + 3*w^2 - 78/5*w - 121/5], [379, 379, -4/5*w^3 + 3*w^2 + 28/5*w - 99/5], [389, 389, -3/5*w^3 - 2*w^2 + 26/5*w + 42/5], [389, 389, -2*w^2 + w + 17], [401, 401, 4/5*w^3 - w^2 - 23/5*w + 24/5], [401, 401, 3/5*w^3 + w^2 - 26/5*w - 7/5], [401, 401, -4/5*w^3 - w^2 + 38/5*w + 66/5], [401, 401, 6/5*w^3 + w^2 - 52/5*w - 39/5], [409, 409, -6/5*w^3 + 52/5*w + 19/5], [409, 409, 1/5*w^3 + 2*w^2 - 7/5*w - 39/5], [421, 421, -1/5*w^3 - w^2 + 17/5*w + 44/5], [421, 421, 2/5*w^3 + w^2 - 24/5*w - 18/5], [431, 431, -3/5*w^3 - w^2 + 36/5*w + 47/5], [431, 431, -13/5*w^3 - 4*w^2 + 116/5*w + 182/5], [431, 431, -2/5*w^3 + w^2 + 24/5*w - 47/5], [431, 431, 2/5*w^3 - w^2 + 6/5*w - 8/5], [439, 439, 6/5*w^3 + w^2 - 47/5*w - 49/5], [439, 439, -2/5*w^3 + 2*w^2 + 9/5*w - 42/5], [439, 439, 4/5*w^3 - w^2 - 23/5*w + 14/5], [439, 439, -w^3 - 2*w^2 + 8*w + 17], [461, 461, w^3 + w^2 - 10*w - 9], [461, 461, -2*w^3 - 3*w^2 + 17*w + 27], [461, 461, -1/5*w^3 + w^2 - 8/5*w - 16/5], [461, 461, -3*w^3 - 4*w^2 + 26*w + 35], [479, 479, 2*w - 1], [479, 479, 2/5*w^3 - 24/5*w - 3/5], [491, 491, w^3 - 2*w^2 - 6*w + 11], [491, 491, w^3 + w^2 - 10*w - 15], [499, 499, -1/5*w^3 + 2*w^2 + 2/5*w - 36/5], [499, 499, 4/5*w^3 + 2*w^2 - 33/5*w - 91/5], [509, 509, 8/5*w^3 + w^2 - 66/5*w - 42/5], [509, 509, 1/5*w^3 - 2*w^2 + 8/5*w + 16/5], [521, 521, w^3 - 9*w - 4], [521, 521, 3/5*w^3 + w^2 - 16/5*w - 42/5], [521, 521, 3/5*w^3 - w^2 - 26/5*w + 23/5], [521, 521, -3/5*w^3 + 11/5*w + 22/5], [529, 23, 3/5*w^3 + w^2 - 36/5*w - 62/5], [529, 23, -4/5*w^3 + 2*w^2 + 23/5*w - 64/5], [541, 541, -w^3 - w^2 + 9*w + 7], [541, 541, 1/5*w^3 + 2*w^2 - 7/5*w - 64/5], [541, 541, 3/5*w^3 + w^2 - 26/5*w - 12/5], [571, 571, -w^3 - w^2 + 6*w + 9], [571, 571, -w^3 + w^2 + 8*w - 4], [619, 619, -w^3 - 2*w^2 + 8*w + 11], [619, 619, -2/5*w^3 + 2*w^2 + 9/5*w - 72/5], [619, 619, 2/5*w^3 - 2*w^2 - 4/5*w + 57/5], [619, 619, -6/5*w^3 - 2*w^2 + 52/5*w + 69/5], [631, 631, 1/5*w^3 + w^2 + 3/5*w - 24/5], [631, 631, 2/5*w^3 - w^2 - 24/5*w + 42/5], [659, 659, -13/5*w^3 - 4*w^2 + 111/5*w + 177/5], [659, 659, -11/5*w^3 - 3*w^2 + 92/5*w + 119/5], [659, 659, -8/5*w^3 - w^2 + 61/5*w + 27/5], [659, 659, -8/5*w^3 + 71/5*w + 2/5], [661, 661, -w^3 - w^2 + 7*w + 11], [661, 661, 4/5*w^3 - w^2 - 28/5*w + 9/5], [691, 691, 1/5*w^3 - w^2 - 17/5*w + 41/5], [691, 691, 3/5*w^3 + 2*w^2 - 26/5*w - 67/5], [701, 701, 12/5*w^3 + 4*w^2 - 99/5*w - 158/5], [701, 701, -17/5*w^3 - 4*w^2 + 149/5*w + 193/5], [709, 709, -1/5*w^3 + w^2 + 12/5*w - 6/5], [709, 709, 1/5*w^3 + w^2 - 2/5*w - 59/5], [719, 719, 2/5*w^3 + w^2 - 29/5*w - 8/5], [719, 719, -w^3 + w^2 + 7*w - 4], [719, 719, 6/5*w^3 + w^2 - 42/5*w - 44/5], [719, 719, -2/5*w^3 - w^2 + 29/5*w + 53/5], [739, 739, 14/5*w^3 + 4*w^2 - 123/5*w - 171/5], [739, 739, -1/5*w^3 + 2*w^2 + 12/5*w - 56/5], [739, 739, -6/5*w^3 + 3*w^2 + 37/5*w - 96/5], [739, 739, 2/5*w^3 + 2*w^2 - 9/5*w - 73/5], [751, 751, -9/5*w^3 - 2*w^2 + 73/5*w + 86/5], [751, 751, -w^3 + 2*w^2 + 5*w - 8], [761, 761, 1/5*w^3 + w^2 - 22/5*w + 16/5], [761, 761, -16/5*w^3 - 4*w^2 + 147/5*w + 214/5], [769, 769, 13/5*w^3 + 3*w^2 - 121/5*w - 172/5], [769, 769, 3/5*w^3 - w^2 - 41/5*w + 78/5], [769, 769, -12/5*w^3 - 4*w^2 + 99/5*w + 153/5], [769, 769, -6/5*w^3 + 3*w^2 + 27/5*w - 61/5], [809, 809, 1/5*w^3 + 3*w^2 - 22/5*w - 104/5], [809, 809, w^3 + 3*w^2 - 10*w - 17], [811, 811, -5*w + 11], [811, 811, -16/5*w^3 - 5*w^2 + 137/5*w + 214/5], [821, 821, -1/5*w^3 + w^2 + 17/5*w - 36/5], [821, 821, w^2 + 2*w - 6], [829, 829, 3*w^2 + w - 17], [829, 829, 2/5*w^3 + 3*w^2 - 9/5*w - 108/5], [841, 29, w^3 - 7*w - 3], [859, 859, 6/5*w^3 + w^2 - 37/5*w - 34/5], [859, 859, 2/5*w^3 + w^2 - 14/5*w - 68/5], [881, 881, 1/5*w^3 + 2*w^2 - 17/5*w - 79/5], [881, 881, 3/5*w^3 + 2*w^2 - 31/5*w - 47/5], [911, 911, 4/5*w^3 - 3*w^2 - 28/5*w + 89/5], [911, 911, 1/5*w^3 - w^2 - 7/5*w + 61/5], [911, 911, w^3 - 6*w - 3], [911, 911, -11/5*w^3 - 3*w^2 + 92/5*w + 139/5], [919, 919, 7/5*w^3 + 2*w^2 - 54/5*w - 88/5], [919, 919, 4/5*w^3 - 2*w^2 - 23/5*w + 39/5], [929, 929, -7/5*w^3 + w^2 + 54/5*w - 17/5], [929, 929, -1/5*w^3 + w^2 + 22/5*w - 71/5], [941, 941, 9/5*w^3 + w^2 - 73/5*w - 31/5], [941, 941, 1/5*w^3 - w^2 - 2/5*w + 66/5], [941, 941, 2/5*w^3 + 2*w^2 - 19/5*w - 103/5], [941, 941, -w^3 + w^2 + 5*w - 3], [961, 31, w^3 - 7*w - 2], [961, 31, 2/5*w^3 - 14/5*w - 33/5], [971, 971, -2/5*w^3 - 2*w^2 + 24/5*w + 93/5], [971, 971, 2/5*w^3 + 2*w^2 - 24/5*w - 33/5], [991, 991, -w^3 + 2*w^2 + 7*w - 10], [991, 991, -7/5*w^3 - 2*w^2 + 49/5*w + 78/5]]; primes := [ideal : I in primesArray]; heckePol := x^5 - x^4 - 13*x^3 + 8*x^2 + 41*x - 3; K := NumberField(heckePol); heckeEigenvaluesArray := [e, 0, -4/5*e^4 - 7/5*e^3 + 29/5*e^2 + 59/5*e + 7/5, -3/5*e^4 - 4/5*e^3 + 23/5*e^2 + 33/5*e - 11/5, e^4 + 2*e^3 - 8*e^2 - 15*e, 2/5*e^4 + 1/5*e^3 - 12/5*e^2 - 12/5*e - 21/5, 1/5*e^4 - 2/5*e^3 - 11/5*e^2 + 9/5*e + 22/5, -e^4 + 8*e^2 + e - 6, -3/5*e^4 - 9/5*e^3 + 23/5*e^2 + 68/5*e - 21/5, -1/5*e^4 + 7/5*e^3 + 16/5*e^2 - 44/5*e - 57/5, 2*e^4 + e^3 - 16*e^2 - 11*e + 9, e^3 - 2*e^2 - 7*e + 12, -8/5*e^4 - 19/5*e^3 + 63/5*e^2 + 143/5*e + 9/5, e^3 - 9*e - 2, -2/5*e^4 - 1/5*e^3 + 22/5*e^2 + 7/5*e - 34/5, 11/5*e^4 + 28/5*e^3 - 81/5*e^2 - 216/5*e - 48/5, 7/5*e^4 + 11/5*e^3 - 67/5*e^2 - 92/5*e + 69/5, -1/5*e^4 + 7/5*e^3 + 11/5*e^2 - 69/5*e - 37/5, 12/5*e^4 + 11/5*e^3 - 97/5*e^2 - 87/5*e + 59/5, 6/5*e^4 + 3/5*e^3 - 46/5*e^2 - 41/5*e - 3/5, 4/5*e^4 + 2/5*e^3 - 34/5*e^2 - 14/5*e + 18/5, -9/5*e^4 - 2/5*e^3 + 64/5*e^2 + 34/5*e + 27/5, -14/5*e^4 - 27/5*e^3 + 114/5*e^2 + 204/5*e - 33/5, 17/5*e^4 + 36/5*e^3 - 132/5*e^2 - 292/5*e - 1/5, 17/5*e^4 + 41/5*e^3 - 132/5*e^2 - 327/5*e + 9/5, -8/5*e^4 - 29/5*e^3 + 53/5*e^2 + 233/5*e + 69/5, 2*e^3 + e^2 - 18*e - 6, -26/5*e^4 - 53/5*e^3 + 216/5*e^2 + 406/5*e - 57/5, 4/5*e^4 + 7/5*e^3 - 19/5*e^2 - 54/5*e - 42/5, 2*e^4 + 3*e^3 - 19*e^2 - 25*e + 21, -2/5*e^4 - 1/5*e^3 + 17/5*e^2 + 12/5*e - 69/5, -12/5*e^4 - 1/5*e^3 + 87/5*e^2 + 32/5*e + 21/5, -3*e^4 - 7*e^3 + 27*e^2 + 56*e - 20, 13/5*e^4 + 9/5*e^3 - 123/5*e^2 - 63/5*e + 116/5, -11/5*e^4 - 8/5*e^3 + 101/5*e^2 + 66/5*e - 117/5, 9/5*e^4 + 22/5*e^3 - 79/5*e^2 - 164/5*e - 12/5, 14/5*e^4 + 27/5*e^3 - 109/5*e^2 - 229/5*e + 23/5, -2/5*e^4 - 11/5*e^3 + 7/5*e^2 + 87/5*e + 71/5, 24/5*e^4 + 42/5*e^3 - 184/5*e^2 - 344/5*e + 8/5, -1/5*e^4 + 2/5*e^3 + 11/5*e^2 - 39/5*e - 7/5, -7/5*e^4 - 16/5*e^3 + 42/5*e^2 + 137/5*e + 26/5, 9/5*e^4 + 17/5*e^3 - 54/5*e^2 - 139/5*e - 82/5, 2*e^4 + 3*e^3 - 19*e^2 - 19*e + 21, 18/5*e^4 + 29/5*e^3 - 163/5*e^2 - 223/5*e + 111/5, 19/5*e^4 + 22/5*e^3 - 159/5*e^2 - 189/5*e + 63/5, -5*e^4 - 4*e^3 + 43*e^2 + 38*e - 33, e^4 + 3*e^3 - 7*e^2 - 23*e - 12, -26/5*e^4 - 38/5*e^3 + 226/5*e^2 + 291/5*e - 147/5, -6/5*e^4 - 3/5*e^3 + 51/5*e^2 + 1/5*e - 72/5, -16/5*e^4 - 48/5*e^3 + 151/5*e^2 + 366/5*e - 102/5, -e^4 - 4*e^3 + 6*e^2 + 33*e + 19, 6/5*e^4 - 12/5*e^3 - 56/5*e^2 + 109/5*e + 122/5, 7/5*e^4 + 6/5*e^3 - 62/5*e^2 - 37/5*e + 39/5, -5*e^4 - 8*e^3 + 44*e^2 + 62*e - 27, -2/5*e^4 - 6/5*e^3 + 7/5*e^2 + 22/5*e + 66/5, 3*e^4 + 6*e^3 - 23*e^2 - 48*e + 6, -7/5*e^4 - 21/5*e^3 + 52/5*e^2 + 187/5*e + 31/5, -6/5*e^4 - 3/5*e^3 + 56/5*e^2 - 4/5*e - 107/5, 6/5*e^4 + 23/5*e^3 - 86/5*e^2 - 171/5*e + 147/5, -16/5*e^4 - 23/5*e^3 + 131/5*e^2 + 166/5*e - 57/5, -7/5*e^4 - 21/5*e^3 + 77/5*e^2 + 142/5*e - 84/5, 18/5*e^4 + 34/5*e^3 - 153/5*e^2 - 238/5*e + 96/5, -21/5*e^4 - 23/5*e^3 + 166/5*e^2 + 211/5*e - 62/5, -18/5*e^4 - 9/5*e^3 + 133/5*e^2 + 73/5*e - 41/5, 21/5*e^4 + 18/5*e^3 - 166/5*e^2 - 151/5*e + 52/5, -9/5*e^4 - 42/5*e^3 + 59/5*e^2 + 329/5*e + 112/5, -33/5*e^4 - 54/5*e^3 + 288/5*e^2 + 418/5*e - 171/5, 19/5*e^4 + 52/5*e^3 - 144/5*e^2 - 419/5*e - 72/5, -29/5*e^4 - 42/5*e^3 + 199/5*e^2 + 364/5*e + 117/5, 7/5*e^4 + 6/5*e^3 - 52/5*e^2 - 32/5*e + 54/5, 12/5*e^4 + 21/5*e^3 - 112/5*e^2 - 217/5*e + 159/5, 6/5*e^4 + 8/5*e^3 - 16/5*e^2 - 51/5*e - 168/5, -3/5*e^4 + 11/5*e^3 + 43/5*e^2 - 117/5*e - 161/5, -5*e^4 - 12*e^3 + 41*e^2 + 95*e - 13, 4/5*e^4 + 22/5*e^3 - 19/5*e^2 - 174/5*e - 107/5, -37/5*e^4 - 51/5*e^3 + 297/5*e^2 + 412/5*e - 119/5, -4*e^4 - 11*e^3 + 38*e^2 + 85*e - 30, -18/5*e^4 - 34/5*e^3 + 158/5*e^2 + 273/5*e - 81/5, -9/5*e^4 - 2/5*e^3 + 69/5*e^2 + 34/5*e - 108/5, -14/5*e^4 - 42/5*e^3 + 139/5*e^2 + 324/5*e - 168/5, 13/5*e^4 + 14/5*e^3 - 83/5*e^2 - 138/5*e - 139/5, -12/5*e^4 - 1/5*e^3 + 107/5*e^2 + 52/5*e - 119/5, -28/5*e^4 - 69/5*e^3 + 228/5*e^2 + 533/5*e - 61/5, 5*e^4 + 9*e^3 - 42*e^2 - 68*e + 14, -3*e^3 - 7*e^2 + 18*e + 39, 14/5*e^4 + 37/5*e^3 - 114/5*e^2 - 309/5*e + 18/5, e^4 - 2*e^3 - 7*e^2 + 8*e - 18, -6/5*e^4 - 8/5*e^3 + 36/5*e^2 + 91/5*e + 108/5, -41/5*e^4 - 108/5*e^3 + 336/5*e^2 + 826/5*e - 27/5, -8*e^4 - 8*e^3 + 67*e^2 + 63*e - 45, -28/5*e^4 - 24/5*e^3 + 223/5*e^2 + 213/5*e - 96/5, -39/5*e^4 - 67/5*e^3 + 314/5*e^2 + 499/5*e - 18/5, -2/5*e^4 + 4/5*e^3 + 22/5*e^2 - 13/5*e - 109/5, -6/5*e^4 - 23/5*e^3 + 21/5*e^2 + 191/5*e + 113/5, -24/5*e^4 - 17/5*e^3 + 219/5*e^2 + 139/5*e - 183/5, 14/5*e^4 + 47/5*e^3 - 89/5*e^2 - 359/5*e - 117/5, 6*e^4 + 8*e^3 - 47*e^2 - 66*e, 11/5*e^4 + 23/5*e^3 - 91/5*e^2 - 191/5*e + 72/5, -2/5*e^4 - 11/5*e^3 + 7/5*e^2 + 142/5*e + 66/5, 36/5*e^4 + 38/5*e^3 - 281/5*e^2 - 346/5*e + 102/5, 21/5*e^4 + 8/5*e^3 - 161/5*e^2 - 111/5*e + 7/5, 26/5*e^4 + 63/5*e^3 - 201/5*e^2 - 491/5*e - 68/5, 3*e^4 + 2*e^3 - 20*e^2 - 22*e - 7, 43/5*e^4 + 89/5*e^3 - 363/5*e^2 - 673/5*e + 101/5, -17/5*e^4 - 21/5*e^3 + 137/5*e^2 + 167/5*e - 29/5, -37/5*e^4 - 66/5*e^3 + 297/5*e^2 + 497/5*e - 19/5, -2/5*e^4 - 21/5*e^3 - 33/5*e^2 + 167/5*e + 206/5, -13/5*e^4 + 6/5*e^3 + 118/5*e^2 - 2/5*e - 136/5, -21/5*e^4 - 23/5*e^3 + 161/5*e^2 + 186/5*e + 68/5, -34/5*e^4 - 67/5*e^3 + 229/5*e^2 + 534/5*e + 202/5, -7*e^4 - 12*e^3 + 55*e^2 + 95*e + 5, 22/5*e^4 + 46/5*e^3 - 162/5*e^2 - 417/5*e - 26/5, 6*e^4 + 9*e^3 - 49*e^2 - 72*e + 26, -13/5*e^4 - 64/5*e^3 + 128/5*e^2 + 448/5*e - 111/5, -18/5*e^4 - 19/5*e^3 + 173/5*e^2 + 148/5*e - 186/5, 21/5*e^4 + 73/5*e^3 - 181/5*e^2 - 556/5*e + 12/5, -22/5*e^4 - 51/5*e^3 + 182/5*e^2 + 377/5*e - 69/5, -7/5*e^4 - 11/5*e^3 + 37/5*e^2 + 97/5*e + 91/5, -10*e^4 - 23*e^3 + 86*e^2 + 175*e - 37, -4/5*e^4 - 17/5*e^3 + 44/5*e^2 + 109/5*e - 223/5, e^4 - 3*e^3 - 8*e^2 + 19*e - 2, e^4 + 11*e^3 - 7*e^2 - 77*e - 30, 37/5*e^4 + 46/5*e^3 - 327/5*e^2 - 392/5*e + 264/5, 7*e^4 + 18*e^3 - 54*e^2 - 142*e - 1, -6*e^4 - 12*e^3 + 49*e^2 + 97*e - 16, -54/5*e^4 - 67/5*e^3 + 419/5*e^2 + 539/5*e - 18/5, 39/5*e^4 + 57/5*e^3 - 324/5*e^2 - 414/5*e + 78/5, -21/5*e^4 - 53/5*e^3 + 156/5*e^2 + 411/5*e + 48/5, 7/5*e^4 - 9/5*e^3 - 67/5*e^2 + 28/5*e + 69/5, -2*e^4 - 4*e^3 + 22*e^2 + 28*e - 40, -3/5*e^4 - 24/5*e^3 + 23/5*e^2 + 113/5*e + 59/5, 7/5*e^4 + 6/5*e^3 - 92/5*e^2 - 62/5*e + 94/5, 5*e^4 + 4*e^3 - 42*e^2 - 38*e + 46, -2*e^4 + 22*e^2 - 8*e - 32, -34/5*e^4 - 82/5*e^3 + 289/5*e^2 + 619/5*e - 148/5, e^4 - 4*e^3 - 13*e^2 + 36*e + 30, -6/5*e^4 - 43/5*e^3 + 66/5*e^2 + 306/5*e - 132/5, 5*e^4 + 11*e^3 - 44*e^2 - 91*e + 25, -21/5*e^4 - 73/5*e^3 + 181/5*e^2 + 561/5*e - 32/5, -46/5*e^4 - 93/5*e^3 + 381/5*e^2 + 741/5*e - 92/5, -21/5*e^4 + 2/5*e^3 + 206/5*e^2 + 36/5*e - 232/5, -e^4 - 4*e^3 + 2*e^2 + 29*e + 36, -9*e^3 - e^2 + 67*e + 27, -44/5*e^4 - 47/5*e^3 + 369/5*e^2 + 369/5*e - 278/5, 31/5*e^4 + 103/5*e^3 - 231/5*e^2 - 781/5*e - 203/5, 49/5*e^4 + 77/5*e^3 - 409/5*e^2 - 579/5*e + 63/5, 12/5*e^4 + 21/5*e^3 - 112/5*e^2 - 167/5*e - 36/5, 16/5*e^4 + 43/5*e^3 - 81/5*e^2 - 331/5*e - 218/5, 4*e^4 + 7*e^3 - 38*e^2 - 47*e + 32, 17/5*e^4 + 61/5*e^3 - 142/5*e^2 - 417/5*e + 14/5, -49/5*e^4 - 82/5*e^3 + 384/5*e^2 + 679/5*e + 2/5, 17/5*e^4 + 56/5*e^3 - 112/5*e^2 - 392/5*e - 211/5, 14/5*e^4 + 12/5*e^3 - 109/5*e^2 - 69/5*e + 18/5, 32/5*e^4 + 36/5*e^3 - 287/5*e^2 - 232/5*e + 234/5, -31/5*e^4 - 48/5*e^3 + 241/5*e^2 + 411/5*e + 33/5, 9/5*e^4 - 23/5*e^3 - 59/5*e^2 + 111/5*e + 3/5, -32/5*e^4 - 96/5*e^3 + 232/5*e^2 + 722/5*e + 141/5, -14/5*e^4 - 62/5*e^3 + 119/5*e^2 + 499/5*e - 3/5, 8/5*e^4 + 4/5*e^3 - 53/5*e^2 - 68/5*e - 49/5, -62/5*e^4 - 86/5*e^3 + 517/5*e^2 + 687/5*e - 244/5, 6*e^4 + 16*e^3 - 53*e^2 - 116*e + 30, 11/5*e^4 + 58/5*e^3 - 71/5*e^2 - 426/5*e - 183/5, 14/5*e^4 + 42/5*e^3 - 99/5*e^2 - 369/5*e - 102/5, -23/5*e^4 - 44/5*e^3 + 143/5*e^2 + 413/5*e + 219/5, -16/5*e^4 + 7/5*e^3 + 116/5*e^2 - 79/5*e - 27/5, 44/5*e^4 + 67/5*e^3 - 354/5*e^2 - 499/5*e + 33/5, -49/5*e^4 - 87/5*e^3 + 389/5*e^2 + 694/5*e - 173/5, 47/5*e^4 + 56/5*e^3 - 342/5*e^2 - 507/5*e - 116/5, 4*e^4 + 3*e^3 - 24*e^2 - 25*e - 33, -23/5*e^4 - 39/5*e^3 + 173/5*e^2 + 278/5*e + 84/5, 29/5*e^4 + 77/5*e^3 - 239/5*e^2 - 574/5*e + 68/5, -27/5*e^4 - 26/5*e^3 + 197/5*e^2 + 232/5*e + 56/5]; 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;