/* 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![-3, -7, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3, 3, w], [3, 3, w + 1], [3, 3, w + 2], [8, 2, 2], [11, 11, -w^2 + 5], [13, 13, w^2 - w - 7], [17, 17, -w^2 + w + 4], [19, 19, -w^2 - w + 4], [29, 29, 2*w^2 - 2*w - 11], [31, 31, w^2 - 2*w - 4], [37, 37, -2*w^2 + 3*w + 7], [41, 41, w^2 - 2], [59, 59, 2*w^2 - 13], [61, 61, w^2 + w - 10], [67, 67, 2*w^2 - w - 11], [73, 73, -w^2 - 1], [83, 83, w^2 - w - 10], [89, 89, -w^2 + 4*w - 2], [97, 97, w^2 - 2*w - 7], [97, 97, -w^2 - 4*w - 5], [97, 97, 2*w^2 + w - 8], [101, 101, -4*w^2 + 2*w + 25], [103, 103, w^2 + w - 7], [107, 107, -3*w - 4], [109, 109, 5*w^2 - 3*w - 31], [121, 11, 2*w^2 - 3*w - 4], [125, 5, -5], [137, 137, -5*w^2 + 3*w + 34], [139, 139, w^2 + 2*w - 4], [151, 151, 3*w^2 - 23], [151, 151, w^2 - 3*w - 5], [151, 151, 3*w^2 - 3*w - 17], [157, 157, 4*w^2 - 9*w - 8], [157, 157, w^2 + 3*w - 2], [157, 157, 2*w^2 - w - 2], [167, 167, 2*w^2 + w - 11], [167, 167, -w^2 + 11], [167, 167, -3*w^2 + 19], [169, 13, w^2 - 4*w - 4], [173, 173, 2*w^2 - 3*w - 10], [179, 179, -3*w + 7], [191, 191, -w^2 + 5*w - 5], [193, 193, -w^2 + w - 2], [199, 199, 3*w - 2], [211, 211, w^2 - 3*w - 11], [223, 223, 3*w^2 - 20], [223, 223, w^2 - 5*w - 4], [223, 223, 2*w^2 - w - 8], [229, 229, 4*w^2 - 3*w - 26], [229, 229, 2*w^2 + w - 20], [229, 229, 2*w^2 - 2*w - 5], [239, 239, -6*w^2 + 3*w + 38], [241, 241, -w^2 + 4*w - 5], [257, 257, -2*w^2 + 5*w - 1], [263, 263, -2*w^2 - w + 17], [269, 269, -w^2 - w + 13], [269, 269, 3*w - 4], [269, 269, -4*w^2 + w + 31], [271, 271, 2*w^2 - 7], [271, 271, 3*w - 5], [271, 271, w^2 - 3*w - 8], [289, 17, 2*w^2 + w - 5], [307, 307, 2*w^2 - w - 5], [311, 311, 3*w^2 - 3*w - 19], [311, 311, -7*w^2 + 13*w + 19], [311, 311, w^2 - 2*w - 13], [313, 313, w^2 + 2*w - 7], [317, 317, -5*w^2 + w + 35], [331, 331, -w^2 + 3*w - 4], [337, 337, 3*w^2 - 3*w - 13], [343, 7, -7], [347, 347, 2*w^2 - 3*w - 13], [349, 349, 3*w^2 - 3*w - 20], [359, 359, -w^2 + 5*w - 2], [361, 19, -2*w^2 + 7*w - 1], [389, 389, -2*w^2 + w - 1], [397, 397, w^2 - 6*w + 10], [401, 401, 6*w^2 - 3*w - 44], [419, 419, -3*w^2 + 6*w + 10], [419, 419, 4*w^2 - 2*w - 31], [419, 419, -4*w^2 + 4*w + 19], [421, 421, 6*w^2 - 12*w - 13], [431, 431, 4*w^2 - 7*w - 10], [433, 433, w^2 - 4*w - 7], [443, 443, 2*w^2 + 4*w - 5], [443, 443, 5*w^2 - 11*w - 11], [443, 443, 3*w^2 - 14], [449, 449, -2*w^2 + 2*w - 1], [457, 457, w^2 - 6*w - 5], [457, 457, w^2 + 3*w - 5], [457, 457, 6*w^2 - 3*w - 37], [463, 463, w^2 + 5*w - 1], [467, 467, w^2 - w - 13], [479, 479, 5*w^2 - 9*w - 16], [487, 487, -6*w - 5], [499, 499, 3*w^2 - 3*w - 4], [499, 499, 3*w^2 + 3*w - 1], [499, 499, w^2 - 5*w + 8], [503, 503, 4*w^2 - 5*w - 16], [509, 509, 2*w^2 - 5*w - 8], [521, 521, 4*w^2 - w - 22], [523, 523, w^2 - 5*w - 16], [547, 547, 5*w^2 - 2*w - 38], [547, 547, 8*w^2 - 3*w - 58], [547, 547, 4*w^2 - 29], [557, 557, 7*w^2 - 4*w - 43], [557, 557, 3*w - 11], [557, 557, 2*w^2 - 4*w - 11], [569, 569, 5*w^2 - 4*w - 32], [569, 569, 3*w^2 - 3*w - 11], [569, 569, w^2 + 2*w - 16], [571, 571, w^2 - 4*w - 10], [593, 593, 2*w^2 - 6*w - 7], [601, 601, -w^2 - 3*w + 14], [607, 607, 3*w^2 + 3*w - 7], [619, 619, 3*w^2 - 6*w - 11], [631, 631, -w^2 - w - 5], [641, 641, 3*w^2 - 3*w - 5], [647, 647, -w^2 + 6*w - 1], [653, 653, -5*w^2 + 4*w + 35], [661, 661, 3*w^2 - 3*w - 10], [673, 673, w^2 + 4*w - 4], [677, 677, 5*w^2 - 6*w - 25], [677, 677, w^2 - 14], [677, 677, 4*w^2 - 3*w - 20], [691, 691, 5*w^2 - 2*w - 29], [701, 701, 5*w^2 - 10*w - 14], [719, 719, -3*w - 11], [727, 727, -2*w^2 + 9*w - 8], [733, 733, -w^2 - 4*w - 8], [739, 739, w^2 + 3*w - 8], [743, 743, 2*w^2 - 4*w - 17], [757, 757, -w^2 + w - 5], [769, 769, 3*w^2 - 3*w - 7], [769, 769, -w^2 - 3*w - 7], [769, 769, 5*w^2 - 3*w - 28], [773, 773, 7*w^2 - 6*w - 38], [787, 787, 2*w^2 - w - 20], [797, 797, -5*w^2 + 14*w - 1], [821, 821, -w^2 + 6*w - 4], [823, 823, 7*w^2 - 6*w - 41], [839, 839, 4*w^2 - 6*w - 11], [841, 29, 5*w^2 - 4*w - 26], [857, 857, 3*w^2 - 11], [863, 863, 2*w^2 + 2*w - 17], [881, 881, 5*w^2 - 5*w - 29], [883, 883, -4*w^2 + 8*w + 13], [907, 907, 4*w^2 - 9*w - 11], [911, 911, 2*w^2 + 3*w - 10], [947, 947, 2*w^2 + 5*w - 5], [953, 953, 2*w^2 - 5*w - 11], [961, 31, w^2 - 5*w - 13], [967, 967, 2*w^2 - 3*w - 22], [967, 967, 5*w^2 - 9*w - 10], [967, 967, 3*w^2 - 10], [977, 977, 5*w^2 - 37], [983, 983, 7*w^2 - 5*w - 40], [997, 997, -7*w^2 + 5*w + 49]]; primes := [ideal : I in primesArray]; heckePol := x^6 + 3*x^5 - 9*x^4 - 28*x^3 + 15*x^2 + 63*x + 26; K := NumberField(heckePol); heckeEigenvaluesArray := [e, 1/3*e^4 + 1/3*e^3 - 7/3*e^2 - 4/3*e + 1/3, -1/3*e^5 - 2/3*e^4 + 3*e^3 + 14/3*e^2 - 6*e - 19/3, 1/3*e^4 + 4/3*e^3 - 1/3*e^2 - 19/3*e - 23/3, -1, -1/3*e^5 - e^4 + 5/3*e^3 + 6*e^2 - 2/3*e - 20/3, 2/3*e^4 + 5/3*e^3 - 11/3*e^2 - 26/3*e - 1/3, 1/3*e^5 + 4/3*e^4 - 7/3*e^3 - 28/3*e^2 + 10/3*e + 11, 1/3*e^5 + e^4 - 5/3*e^3 - 6*e^2 - 1/3*e + 5/3, -e^2 + 4, -1/3*e^5 - 2/3*e^4 + 4*e^3 + 17/3*e^2 - 12*e - 34/3, -e^4 - 2*e^3 + 5*e^2 + 8*e, 1/3*e^5 + 2/3*e^4 - 3*e^3 - 17/3*e^2 + 6*e + 40/3, e^5 + 5/3*e^4 - 31/3*e^3 - 41/3*e^2 + 82/3*e + 86/3, 1/3*e^5 + e^4 - 2/3*e^3 - 4*e^2 - 22/3*e + 2/3, -e^5 - 7/3*e^4 + 23/3*e^3 + 37/3*e^2 - 41/3*e - 19/3, -2/3*e^5 - 2*e^4 + 16/3*e^3 + 12*e^2 - 31/3*e - 25/3, e^5 + 1/3*e^4 - 38/3*e^3 - 13/3*e^2 + 110/3*e + 52/3, 5/3*e^4 + 11/3*e^3 - 29/3*e^2 - 50/3*e - 4/3, -e^5 - 4/3*e^4 + 35/3*e^3 + 31/3*e^2 - 110/3*e - 76/3, 4/3*e^5 + 4/3*e^4 - 40/3*e^3 - 25/3*e^2 + 85/3*e + 7, -1/3*e^5 - e^4 - 1/3*e^3 + 5*e^2 + 43/3*e - 17/3, e^5 + 4/3*e^4 - 26/3*e^3 - 16/3*e^2 + 50/3*e + 16/3, -e^5 - 4/3*e^4 + 23/3*e^3 + 13/3*e^2 - 38/3*e + 29/3, -e^5 - 4/3*e^4 + 44/3*e^3 + 46/3*e^2 - 152/3*e - 133/3, 2*e^5 + 4*e^4 - 17*e^3 - 26*e^2 + 37*e + 32, e^5 + 4/3*e^4 - 38/3*e^3 - 40/3*e^2 + 104/3*e + 73/3, 1/3*e^5 + 2/3*e^4 - 4*e^3 - 23/3*e^2 + 14*e + 43/3, 2/3*e^5 + 5/3*e^4 - 23/3*e^3 - 47/3*e^2 + 65/3*e + 32, -7/3*e^5 - 8/3*e^4 + 26*e^3 + 71/3*e^2 - 71*e - 169/3, -1/3*e^5 - 4/3*e^4 - 5/3*e^3 + 7/3*e^2 + 59/3*e + 8, 1/3*e^5 + 4/3*e^4 - 7/3*e^3 - 13/3*e^2 + 22/3*e - 12, 1/3*e^5 - e^4 - 14/3*e^3 + 6*e^2 + 23/3*e - 19/3, -1/3*e^5 + 5/3*e^3 - e^2 + 25/3*e + 16/3, -2/3*e^5 - 8/3*e^4 + 5/3*e^3 + 53/3*e^2 + 31/3*e - 15, -2/3*e^5 - 4/3*e^4 + 7*e^3 + 31/3*e^2 - 17*e - 35/3, -1/3*e^5 - 5/3*e^4 + 6*e^3 + 44/3*e^2 - 31*e - 76/3, e^5 + 8/3*e^4 - 19/3*e^3 - 47/3*e^2 + 19/3*e + 26/3, -e^4 + e^3 + 11*e^2 - 10*e - 19, 1/3*e^5 - 4/3*e^4 - 10*e^3 + 13/3*e^2 + 43*e + 34/3, e^4 + 4*e^3 - 2*e^2 - 22*e - 12, -e^5 - 8/3*e^4 + 19/3*e^3 + 47/3*e^2 - 25/3*e - 32/3, 1/3*e^5 + 8/3*e^4 - e^3 - 62/3*e^2 + 5*e + 94/3, e^5 + 2*e^4 - 12*e^3 - 21*e^2 + 39*e + 42, e^5 - 2/3*e^4 - 53/3*e^3 - 25/3*e^2 + 182/3*e + 178/3, 4/3*e^4 + 16/3*e^3 - 7/3*e^2 - 67/3*e - 56/3, 2/3*e^5 - 49/3*e^3 - 14*e^2 + 193/3*e + 181/3, 2/3*e^5 + 4/3*e^4 - 7*e^3 - 22/3*e^2 + 23*e + 26/3, 1/3*e^5 + e^4 - 2/3*e^3 - 2*e^2 - 31/3*e - 64/3, 2/3*e^5 + 2/3*e^4 - 35/3*e^3 - 44/3*e^2 + 137/3*e + 52, 2*e^5 + 5*e^4 - 15*e^3 - 31*e^2 + 23*e + 36, -1/3*e^5 - 5/3*e^4 - e^3 + 35/3*e^2 + 14*e - 67/3, -2/3*e^5 - 8/3*e^4 + 23/3*e^3 + 65/3*e^2 - 74/3*e - 28, 8/3*e^5 + 4*e^4 - 82/3*e^3 - 30*e^2 + 181/3*e + 124/3, 1/3*e^5 - e^4 - 14/3*e^3 + 7*e^2 + 29/3*e + 20/3, -3*e^5 - 5*e^4 + 27*e^3 + 32*e^2 - 61*e - 49, 4/3*e^5 + 11/3*e^4 - 15*e^3 - 104/3*e^2 + 43*e + 166/3, -8/3*e^5 - 5*e^4 + 55/3*e^3 + 24*e^2 - 64/3*e - 34/3, 2/3*e^5 - 4/3*e^4 - 26/3*e^3 + 28/3*e^2 + 50/3*e - 8, 4/3*e^5 + 8/3*e^4 - 10*e^3 - 41/3*e^2 + 14*e + 25/3, -2/3*e^5 + 2*e^4 + 43/3*e^3 - 11*e^2 - 151/3*e - 4/3, 2*e^4 + 3*e^3 - 6*e^2 - 7*e - 20, 1/3*e^5 + 7/3*e^4 + 5/3*e^3 - 37/3*e^2 - 35/3*e - 1, -1/3*e^5 - 2*e^4 - 13/3*e^3 + 5*e^2 + 106/3*e + 67/3, e^5 + 4/3*e^4 - 20/3*e^3 + 5/3*e^2 + 26/3*e - 86/3, 1/3*e^5 - 2/3*e^4 - 25/3*e^3 - 1/3*e^2 + 118/3*e + 11, -11/3*e^4 - 32/3*e^3 + 59/3*e^2 + 143/3*e - 20/3, -5/3*e^5 - 11/3*e^4 + 50/3*e^3 + 86/3*e^2 - 119/3*e - 45, 5/3*e^5 + 14/3*e^4 - 53/3*e^3 - 116/3*e^2 + 140/3*e + 65, -5/3*e^5 - 7/3*e^4 + 18*e^3 + 67/3*e^2 - 47*e - 146/3, -2/3*e^5 - 16/3*e^4 + 3*e^3 + 118/3*e^2 - 5*e - 143/3, e^5 + 11/3*e^4 - 22/3*e^3 - 71/3*e^2 + 55/3*e + 32/3, -5/3*e^5 - 10/3*e^4 + 18*e^3 + 82/3*e^2 - 52*e - 194/3, 14/3*e^5 + 26/3*e^4 - 122/3*e^3 - 176/3*e^2 + 239/3*e + 82, -2/3*e^5 - 5/3*e^4 + 23/3*e^3 + 47/3*e^2 - 59/3*e - 48, 2/3*e^5 + 14/3*e^4 + 13/3*e^3 - 62/3*e^2 - 103/3*e - 10, -2/3*e^5 + 1/3*e^4 + 29/3*e^3 + 17/3*e^2 - 83/3*e - 35, -7/3*e^5 - 6*e^4 + 50/3*e^3 + 37*e^2 - 83/3*e - 92/3, 2*e^5 + 11/3*e^4 - 46/3*e^3 - 62/3*e^2 + 58/3*e + 44/3, 5/3*e^5 + 3*e^4 - 52/3*e^3 - 24*e^2 + 133/3*e + 154/3, 3*e^5 + 7/3*e^4 - 113/3*e^3 - 88/3*e^2 + 329/3*e + 256/3, -3*e^5 - 5*e^4 + 34*e^3 + 46*e^2 - 97*e - 94, -4/3*e^5 - 8/3*e^4 + 16*e^3 + 89/3*e^2 - 48*e - 172/3, -e^5 + 2*e^4 + 26*e^3 + 4*e^2 - 100*e - 65, 2/3*e^5 + 2/3*e^4 - 5/3*e^3 - 2/3*e^2 - 37/3*e - 1, -5/3*e^5 - 5*e^4 + 16/3*e^3 + 19*e^2 + 62/3*e + 44/3, 1/3*e^5 + 1/3*e^4 - 13/3*e^3 + 11/3*e^2 + 55/3*e - 17, -4/3*e^5 - 4/3*e^4 + 37/3*e^3 + 34/3*e^2 - 70/3*e - 29, 2*e^5 + 2*e^4 - 21*e^3 - 16*e^2 + 48*e + 30, -5/3*e^4 - 32/3*e^3 - 1/3*e^2 + 170/3*e + 133/3, 7/3*e^5 + 13/3*e^4 - 52/3*e^3 - 79/3*e^2 + 79/3*e + 42, 13/3*e^4 + 52/3*e^3 - 37/3*e^2 - 259/3*e - 116/3, -4/3*e^5 - e^4 + 59/3*e^3 + 20*e^2 - 185/3*e - 173/3, 1/3*e^5 + 8/3*e^4 + 10*e^3 - 8/3*e^2 - 55*e - 134/3, -5*e^4 - 16*e^3 + 15*e^2 + 79*e + 40, -7/3*e^5 - 6*e^4 + 62/3*e^3 + 43*e^2 - 164/3*e - 170/3, 10/3*e^5 + 17/3*e^4 - 31*e^3 - 119/3*e^2 + 64*e + 196/3, -4/3*e^5 + 4/3*e^4 + 19*e^3 - 25/3*e^2 - 62*e - 22/3, 2*e^5 + 4/3*e^4 - 59/3*e^3 - 10/3*e^2 + 128/3*e + 19/3, -2/3*e^5 - 7/3*e^4 + 11*e^3 + 88/3*e^2 - 47*e - 206/3, -2/3*e^5 - e^4 - 11/3*e^3 - 8*e^2 + 119/3*e + 152/3, 5/3*e^5 - 2/3*e^4 - 24*e^3 - 4/3*e^2 + 73*e + 56/3, 11/3*e^5 + 4*e^4 - 115/3*e^3 - 26*e^2 + 283/3*e + 124/3, -2*e^5 - 23/3*e^4 + 49/3*e^3 + 170/3*e^2 - 142/3*e - 227/3, -1/3*e^5 + 2/3*e^4 + 19/3*e^3 - 2/3*e^2 - 88/3*e - 22, e^5 - 1/3*e^4 - 55/3*e^3 - 41/3*e^2 + 193/3*e + 194/3, 1/3*e^5 + 7/3*e^3 + 5*e^2 - 94/3*e - 37/3, 2*e^5 + 19/3*e^4 - 41/3*e^3 - 124/3*e^2 + 89/3*e + 115/3, 10/3*e^5 + 10/3*e^4 - 94/3*e^3 - 43/3*e^2 + 205/3*e + 8, -2*e^5 - 7/3*e^4 + 92/3*e^3 + 103/3*e^2 - 311/3*e - 316/3, 5/3*e^5 + 17/3*e^4 - 23/3*e^3 - 77/3*e^2 + 11/3*e - 24, 5/3*e^5 + 10/3*e^4 - 24*e^3 - 121/3*e^2 + 85*e + 278/3, -e^5 - 1/3*e^4 + 35/3*e^3 - 8/3*e^2 - 89/3*e + 50/3, -5/3*e^5 - 10/3*e^4 + 14*e^3 + 58/3*e^2 - 22*e - 11/3, -1/3*e^5 + 2*e^4 + 29/3*e^3 - 9*e^2 - 116/3*e - 41/3, -16/3*e^4 - 19/3*e^3 + 106/3*e^2 + 67/3*e - 82/3, -11/3*e^5 - 5*e^4 + 112/3*e^3 + 34*e^2 - 286/3*e - 211/3, 2/3*e^5 + 5/3*e^4 - 17/3*e^3 - 41/3*e^2 + 38/3*e + 56, -e^4 - 2*e^3 + 6*e^2 + 9*e + 20, 7/3*e^4 + 19/3*e^3 - 25/3*e^2 - 61/3*e - 26/3, 7/3*e^5 + 7/3*e^4 - 76/3*e^3 - 64/3*e^2 + 226/3*e + 66, 4*e^5 + 5/3*e^4 - 166/3*e^3 - 92/3*e^2 + 508/3*e + 383/3, 4/3*e^5 + 13/3*e^4 - 7/3*e^3 - 64/3*e^2 - 89/3*e + 20, 2*e^3 + 6*e^2 - 9*e - 15, -10/3*e^5 - 8/3*e^4 + 37*e^3 + 77/3*e^2 - 86*e - 190/3, -3*e^5 - 5*e^4 + 29*e^3 + 35*e^2 - 58*e - 59, 8/3*e^4 + 32/3*e^3 - 44/3*e^2 - 179/3*e + 8/3, -1/3*e^4 - 10/3*e^3 + 13/3*e^2 + 58/3*e - 4/3, 2/3*e^5 + 4/3*e^4 - 2*e^3 - 28/3*e^2 - 2*e + 128/3, -3*e^5 - 5*e^4 + 27*e^3 + 24*e^2 - 62*e - 8, 10/3*e^5 + 10*e^4 - 44/3*e^3 - 46*e^2 - 31/3*e - 37/3, 2*e^5 + 10/3*e^4 - 92/3*e^3 - 127/3*e^2 + 344/3*e + 343/3, 5/3*e^5 + 14/3*e^4 - 20/3*e^3 - 47/3*e^2 - 34/3*e - 14, 2/3*e^5 + 2*e^4 - 7/3*e^3 - 16*e^2 - 17/3*e + 130/3, 4/3*e^5 - 1/3*e^4 - 11*e^3 + 31/3*e^2 + 12*e - 143/3, 7/3*e^5 + 2*e^4 - 77/3*e^3 - 18*e^2 + 215/3*e + 173/3, 5/3*e^5 + 3*e^4 - 49/3*e^3 - 25*e^2 + 97/3*e + 172/3, 5/3*e^5 - 25/3*e^3 + 16*e^2 - 53/3*e - 182/3, 1/3*e^5 + e^4 - 20/3*e^3 - 13*e^2 + 74/3*e + 35/3, -1/3*e^5 + 20/3*e^4 + 82/3*e^3 - 74/3*e^2 - 373/3*e - 39, -2/3*e^5 - 5/3*e^4 + 26/3*e^3 + 41/3*e^2 - 98/3*e - 24, e^5 - 10/3*e^4 - 67/3*e^3 + 46/3*e^2 + 250/3*e + 35/3, -1/3*e^4 - 37/3*e^3 - 44/3*e^2 + 226/3*e + 218/3, -4/3*e^5 - 11/3*e^4 + 12*e^3 + 65/3*e^2 - 31*e - 34/3, 5/3*e^5 + 7*e^4 - 19/3*e^3 - 47*e^2 - 26/3*e + 187/3, 13/3*e^5 + 7/3*e^4 - 145/3*e^3 - 40/3*e^2 + 397/3*e + 49, 7/3*e^5 + 32/3*e^4 - 12*e^3 - 200/3*e^2 + 16*e + 160/3, 1/3*e^5 - 7/3*e^4 + 88/3*e^2 - 12*e - 215/3, 3*e^5 + 11/3*e^4 - 94/3*e^3 - 92/3*e^2 + 223/3*e + 170/3, -11/3*e^5 - 17/3*e^4 + 101/3*e^3 + 89/3*e^2 - 230/3*e - 41, -1/3*e^4 - 22/3*e^3 - 47/3*e^2 + 106/3*e + 170/3, 5/3*e^5 + 4/3*e^4 - 11*e^3 - 7/3*e^2 - 2*e - 37/3, -5*e^5 - 26/3*e^4 + 130/3*e^3 + 161/3*e^2 - 250/3*e - 206/3, -13/3*e^5 - 9*e^4 + 125/3*e^3 + 70*e^2 - 284/3*e - 374/3, -5/3*e^5 + 2*e^4 + 58/3*e^3 - 17*e^2 - 118/3*e + 56/3, -11/3*e^5 - 9*e^4 + 121/3*e^3 + 76*e^2 - 316/3*e - 343/3, 4*e^5 + 25/3*e^4 - 125/3*e^3 - 187/3*e^2 + 305/3*e + 232/3, -e^5 - 22/3*e^4 - 10/3*e^3 + 127/3*e^2 + 112/3*e - 70/3]; 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;