/* 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, 3, 4, -5, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3, 3, w^4 - w^3 - 5*w^2 + 3*w + 3], [9, 3, -w^4 + 5*w^2 - 3], [9, 3, -w^4 + w^3 + 5*w^2 - 3*w - 2], [13, 13, -w^4 + w^3 + 4*w^2 - 3*w - 1], [17, 17, w^4 - w^3 - 5*w^2 + 3*w + 1], [19, 19, -w^3 + w^2 + 4*w - 2], [23, 23, -w^2 + 3], [31, 31, w^3 - 4*w + 2], [32, 2, 2], [37, 37, w^3 - 3*w - 1], [53, 53, -2*w^4 + w^3 + 9*w^2 - 3*w - 2], [59, 59, -w^4 + 5*w^2 + w - 4], [61, 61, -w^4 + w^3 + 5*w^2 - 4*w], [67, 67, -w^4 + 6*w^2 + 2*w - 4], [71, 71, 2*w^4 - w^3 - 9*w^2 + 4*w + 5], [79, 79, 2*w^4 - w^3 - 10*w^2 + 2*w + 7], [83, 83, -w^4 + 2*w^3 + 5*w^2 - 7*w - 2], [83, 83, -w^4 + w^3 + 4*w^2 - 3*w + 3], [83, 83, w^4 - w^3 - 5*w^2 + 4*w - 1], [83, 83, -w^4 + w^3 + 4*w^2 - 4*w - 2], [83, 83, -2*w^4 + w^3 + 9*w^2 - 3*w - 5], [97, 97, -w^4 - w^3 + 6*w^2 + 4*w - 4], [101, 101, -w^4 + 4*w^2 + 2*w - 2], [107, 107, -2*w^4 + w^3 + 11*w^2 - 3*w - 7], [127, 127, w^3 - w^2 - 4*w], [131, 131, w^4 - w^3 - 4*w^2 + 2*w - 1], [137, 137, w^4 - w^3 - 6*w^2 + 3*w + 3], [139, 139, -2*w^4 + w^3 + 9*w^2 - 3*w + 1], [157, 157, 2*w^4 - w^3 - 9*w^2 + 4*w + 2], [163, 163, -w^2 - 2*w + 3], [167, 167, w^4 - 5*w^2 + 2*w + 1], [169, 13, -2*w^4 + 2*w^3 + 9*w^2 - 7*w - 5], [169, 13, w^4 - 2*w^3 - 5*w^2 + 8*w + 2], [191, 191, 2*w^4 - w^3 - 9*w^2 + 3*w + 4], [193, 193, 2*w^4 - 2*w^3 - 8*w^2 + 7*w], [199, 199, -w^4 + 2*w^3 + 5*w^2 - 9*w - 3], [211, 211, -3*w^4 + w^3 + 13*w^2 - 2*w - 1], [227, 227, 2*w^3 - w^2 - 9*w + 1], [233, 233, -w^3 + w^2 + 4*w + 1], [251, 251, -3*w^4 + 14*w^2 + w - 2], [257, 257, -2*w^4 + 2*w^3 + 10*w^2 - 6*w - 5], [257, 257, -w^4 + w^3 + 5*w^2 - 6*w - 1], [257, 257, 2*w^4 - 2*w^3 - 11*w^2 + 9*w + 9], [257, 257, -2*w^4 + 2*w^3 + 9*w^2 - 6*w - 3], [257, 257, 2*w^4 - w^3 - 10*w^2 + 5*w + 5], [269, 269, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2], [271, 271, -w^4 + 4*w^2 + w - 3], [277, 277, -2*w^4 + 9*w^2 + w - 4], [281, 281, w^4 - w^3 - 3*w^2 + 3*w - 4], [283, 283, 2*w^4 - 3*w^3 - 8*w^2 + 13*w], [289, 17, -w^4 + w^3 + 7*w^2 - 5*w - 5], [289, 17, 4*w^4 - 3*w^3 - 19*w^2 + 11*w + 6], [293, 293, -2*w^4 + 11*w^2 - 7], [317, 317, w^3 - 6*w], [337, 337, -w^4 + 2*w^3 + 5*w^2 - 9*w - 1], [337, 337, w^4 - 2*w^3 - 5*w^2 + 6*w + 3], [337, 337, -w^4 + w^3 + 5*w^2 - 4*w - 6], [337, 337, -2*w^3 + 8*w + 1], [337, 337, w^2 + w - 5], [347, 347, -2*w^4 + w^3 + 9*w^2 - 5*w - 3], [349, 349, 3*w^4 - 2*w^3 - 13*w^2 + 8*w + 2], [353, 353, w^4 - 3*w^3 - 4*w^2 + 12*w + 2], [359, 359, 2*w^3 - w^2 - 7*w + 3], [361, 19, w^4 - 2*w^3 - 4*w^2 + 7*w + 3], [361, 19, w^4 - w^3 - 3*w^2 + 3*w - 3], [367, 367, -2*w^4 + w^3 + 9*w^2 - w - 2], [379, 379, -2*w^3 + w^2 + 7*w + 2], [379, 379, -2*w^4 + w^3 + 10*w^2 - 4*w], [379, 379, 2*w^3 - w^2 - 6*w - 2], [379, 379, -2*w^3 + 2*w^2 + 6*w - 3], [379, 379, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 6], [383, 383, 2*w^4 + w^3 - 10*w^2 - 7*w + 4], [383, 383, -2*w^3 + w^2 + 10*w], [383, 383, w^3 + w^2 - 5*w - 1], [383, 383, 3*w^4 + 3*w^3 - 15*w^2 - 15*w + 4], [383, 383, -2*w^4 + 3*w^3 + 10*w^2 - 10*w - 2], [389, 389, 3*w^4 - w^3 - 14*w^2 + 3*w + 3], [397, 397, w^4 - 6*w^2 - 2*w + 5], [397, 397, w^4 - 2*w^3 - 2*w^2 + 6*w - 5], [397, 397, 2*w^4 - 2*w^3 - 11*w^2 + 10*w + 6], [397, 397, w^4 - w^3 - 3*w^2 + 4*w - 6], [397, 397, w^3 + w^2 - 3*w - 4], [401, 401, w^3 - w^2 - 6*w + 3], [401, 401, -w^4 + 4*w^2 + 2*w - 3], [401, 401, -w^4 + 6*w^2 - w - 7], [421, 421, -w^4 - w^3 + 5*w^2 + 5*w - 4], [421, 421, -w^4 + w^3 + 6*w^2 - 3*w - 2], [421, 421, -w^4 - w^3 + 6*w^2 + 6*w - 5], [421, 421, 3*w^4 - w^3 - 15*w^2 + w + 10], [421, 421, 2*w^4 - 9*w^2 + 2*w + 4], [431, 431, 2*w^4 - 11*w^2 - 2*w + 11], [439, 439, 2*w^4 - 3*w^3 - 9*w^2 + 10*w + 6], [443, 443, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 2], [449, 449, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4], [461, 461, 3*w^4 - 4*w^3 - 14*w^2 + 14*w + 5], [463, 463, w^3 - 6*w - 1], [467, 467, 3*w^4 - 3*w^3 - 13*w^2 + 10*w], [487, 487, 3*w^4 - w^3 - 14*w^2 + w + 3], [487, 487, 2*w^2 - w - 4], [487, 487, -2*w^4 + 2*w^3 + 9*w^2 - 8*w - 5], [487, 487, 2*w^4 - 10*w^2 - 3*w + 4], [487, 487, -2*w^4 + w^3 + 8*w^2 - 3*w - 1], [499, 499, -w^4 + 2*w^3 + 4*w^2 - 8*w - 2], [499, 499, -2*w^4 + w^3 + 10*w^2 - w - 9], [499, 499, -w^3 - 2*w^2 + 2*w + 5], [499, 499, -w^4 + 4*w^2 + 4*w], [499, 499, -2*w^4 + w^3 + 12*w^2 - 4*w - 9], [509, 509, 3*w^4 - 2*w^3 - 16*w^2 + 6*w + 9], [521, 521, -2*w^4 + w^3 + 10*w^2 - 5*w - 4], [523, 523, w - 4], [529, 23, -w^3 + 2*w^2 + 4*w - 4], [529, 23, w^4 + 2*w^3 - 6*w^2 - 11*w + 4], [569, 569, -2*w^4 + 11*w^2 - 10], [571, 571, -2*w^4 + w^3 + 10*w^2 - 3*w - 2], [593, 593, -2*w^4 + 3*w^3 + 10*w^2 - 11*w - 8], [613, 613, -2*w^3 + 2*w^2 + 9*w - 5], [617, 617, -2*w^4 + 10*w^2 - w - 7], [631, 631, -2*w^4 + 2*w^3 + 7*w^2 - 8*w + 3], [641, 641, -2*w^4 + 2*w^3 + 8*w^2 - 9*w + 3], [643, 643, -3*w^4 + w^3 + 13*w^2 - w + 1], [643, 643, w^4 + w^3 - 3*w^2 - 6*w - 2], [643, 643, -w^4 - w^3 + 6*w^2 + 3*w - 4], [643, 643, 2*w^4 - w^3 - 10*w^2 + 4*w + 1], [643, 643, 2*w^4 - 2*w^3 - 8*w^2 + 6*w + 5], [647, 647, -2*w^3 + w^2 + 6*w - 4], [659, 659, 4*w^4 - 3*w^3 - 19*w^2 + 11*w + 10], [661, 661, 2*w^4 + 3*w^3 - 11*w^2 - 14*w + 6], [673, 673, w^4 - 3*w^3 - 3*w^2 + 10*w - 1], [683, 683, w^4 - 2*w^3 - 4*w^2 + 6*w - 4], [701, 701, w^3 - w^2 - 2*w + 4], [727, 727, -w^4 + 6*w^2 + 2*w - 7], [743, 743, w^4 - w^3 - 7*w^2 + 5*w + 9], [769, 769, w^4 + 2*w^3 - 5*w^2 - 9*w + 4], [787, 787, w^4 + w^3 - 7*w^2 - 3*w + 5], [797, 797, -2*w^4 - w^3 + 10*w^2 + 4*w - 4], [797, 797, w^4 - 3*w^3 - 5*w^2 + 12*w + 5], [797, 797, -3*w^4 + 2*w^3 + 13*w^2 - 8*w], [797, 797, -w^4 + w^3 + 3*w^2 - 5*w - 1], [797, 797, 2*w^4 - 11*w^2 + 2*w + 7], [809, 809, -4*w^4 + 2*w^3 + 18*w^2 - 6*w + 1], [809, 809, 3*w^4 - 3*w^3 - 14*w^2 + 11*w + 8], [809, 809, w^4 - 2*w^3 - 2*w^2 + 8*w - 4], [809, 809, 2*w^4 - 8*w^2 - 2*w + 3], [809, 809, 2*w^4 - 3*w^3 - 9*w^2 + 13*w + 4], [821, 821, -2*w^4 + 2*w^3 + 8*w^2 - 6*w + 3], [823, 823, -2*w^4 + 3*w^3 + 8*w^2 - 12*w - 1], [829, 829, 2*w^4 - 3*w^3 - 10*w^2 + 13*w + 4], [839, 839, -2*w^4 + 12*w^2 - w - 10], [863, 863, 2*w^4 - w^3 - 9*w^2 + 5], [877, 877, -w^4 + w^3 + 5*w^2 - 5*w - 7], [881, 881, 2*w^4 - w^3 - 9*w^2 + w - 1], [887, 887, 3*w^4 - 3*w^3 - 15*w^2 + 11*w + 6], [907, 907, -3*w^4 + w^3 + 13*w^2 - w - 5], [919, 919, w^4 + w^3 - 3*w^2 - 5*w - 3], [929, 929, 2*w^4 - 3*w^3 - 10*w^2 + 9*w + 4], [937, 937, -2*w^4 + 10*w^2 + 3*w - 7], [941, 941, -w^4 + 2*w^3 + 3*w^2 - 6*w - 2], [961, 31, 2*w^4 + 2*w^3 - 10*w^2 - 12*w + 5], [961, 31, -w^4 + 5*w^2 - w - 7], [967, 967, -2*w^4 + w^3 + 9*w^2 - 3*w + 2], [991, 991, -3*w^4 + w^3 + 15*w^2 - 4*w - 5], [997, 997, -3*w^4 + 15*w^2 - 5], [997, 997, 2*w^4 - w^3 - 9*w^2 + 6*w + 2], [997, 997, 2*w^4 - w^3 - 11*w^2 + w + 7], [997, 997, 2*w^3 - 5*w - 1], [997, 997, 2*w^4 - 3*w^3 - 6*w^2 + 5*w + 1]]; primes := [ideal : I in primesArray]; heckePol := x^4 + 5*x^3 - 8*x^2 - 19*x + 20; K := NumberField(heckePol); heckeEigenvaluesArray := [-1, e, -1, -1/7*e^3 - 2/7*e^2 + 3*e - 2/7, -2/7*e^3 - 11/7*e^2 + e - 4/7, -2/7*e^3 - 11/7*e^2 + 38/7, -3/7*e^3 - 13/7*e^2 + 2*e + 8/7, 3/7*e^3 + 20/7*e^2 + e - 78/7, -4/7*e^3 - 29/7*e^2 - 2*e + 83/7, 2/7*e^3 + 11/7*e^2 - e - 38/7, 8/7*e^3 + 44/7*e^2 - 3*e - 110/7, 4/7*e^3 + 29/7*e^2 + e - 118/7, 4/7*e^3 + 22/7*e^2 - 3*e - 90/7, 4/7*e^3 + 22/7*e^2 + e - 6/7, 5/7*e^3 + 24/7*e^2 - 7*e - 60/7, -e^2 - 6*e, -8/7*e^3 - 51/7*e^2 + e + 138/7, e^2 + 4*e - 8, -2/7*e^3 - 4/7*e^2 + 5*e + 52/7, -3/7*e^3 - 20/7*e^2 - 3*e + 50/7, -1/7*e^3 - 2/7*e^2 + 5*e + 12/7, -6/7*e^3 - 40/7*e^2 + e + 114/7, -1/7*e^3 - 16/7*e^2 - 2*e + 124/7, 5/7*e^3 + 31/7*e^2 - 2*e - 60/7, 3/7*e^3 + 13/7*e^2 - 4*e - 8/7, 6/7*e^3 + 33/7*e^2 - 2*e - 100/7, 4/7*e^3 + 22/7*e^2 - 3*e - 90/7, 9/7*e^3 + 53/7*e^2 - e - 192/7, 10/7*e^3 + 55/7*e^2 - 4*e - 176/7, 6/7*e^3 + 26/7*e^2 - 7*e - 114/7, -11/7*e^3 - 85/7*e^2 - 8*e + 272/7, -3/7*e^3 - 6/7*e^2 + 7*e - 48/7, 8/7*e^3 + 44/7*e^2 - 5*e - 54/7, 2/7*e^3 + 18/7*e^2 - 108/7, -1/7*e^3 - 16/7*e^2 - 6*e + 40/7, 4/7*e^3 + 22/7*e^2 - 5*e - 62/7, -3/7*e^3 - 20/7*e^2 - 3*e + 120/7, 9/7*e^3 + 60/7*e^2 - 136/7, -5/7*e^3 - 17/7*e^2 + 4*e - 66/7, 3/7*e^3 + 20/7*e^2 + 48/7, 8/7*e^3 + 30/7*e^2 - 10*e - 40/7, -6/7*e^3 - 47/7*e^2 - 4*e + 142/7, -20/7*e^3 - 117/7*e^2 + 3*e + 310/7, -11/7*e^3 - 50/7*e^2 + 9*e + 146/7, 4/7*e^3 + 29/7*e^2 + 6*e - 34/7, 3/7*e^3 + 6/7*e^2 - 13*e - 64/7, 3/7*e^3 + 20/7*e^2 + 4*e - 8/7, e^3 + 6*e^2 - 3*e - 22, -11/7*e^3 - 92/7*e^2 - 13*e + 314/7, 11/7*e^3 + 57/7*e^2 - 4*e - 132/7, -12/7*e^3 - 66/7*e^2 + 10*e + 158/7, 9/7*e^3 + 46/7*e^2 - e + 4/7, -2/7*e^3 - 4/7*e^2 + 5*e - 18/7, 9/7*e^3 + 67/7*e^2 + 5*e - 150/7, e^3 + 4*e^2 - 6*e + 10, -12/7*e^3 - 80/7*e^2 + e + 228/7, 1/7*e^3 + 9/7*e^2 - 2*e - 124/7, 6/7*e^3 + 19/7*e^2 - 4*e + 82/7, -2*e^3 - 14*e^2 - 10*e + 50, -3/7*e^3 + 1/7*e^2 + 3*e - 216/7, -8/7*e^3 - 65/7*e^2 - 6*e + 166/7, 13/7*e^3 + 103/7*e^2 + 12*e - 366/7, -19/7*e^3 - 108/7*e^2 + 10*e + 270/7, 5/7*e^3 + 17/7*e^2 - 11*e - 46/7, -6/7*e^3 - 19/7*e^2 + 13*e + 30/7, -18/7*e^3 - 99/7*e^2 + 13*e + 230/7, 20/7*e^3 + 124/7*e^2 - e - 296/7, 4/7*e^3 + 50/7*e^2 + 15*e - 174/7, -12/7*e^3 - 66/7*e^2 + 8*e + 130/7, 11/7*e^3 + 50/7*e^2 - 15*e - 76/7, 17/7*e^3 + 90/7*e^2 - 7*e - 176/7, -2*e^3 - 13*e^2 - 4*e + 30, 1/7*e^3 + 9/7*e^2 + 2*e + 72/7, 6/7*e^3 + 54/7*e^2 + 7*e - 128/7, -20/7*e^3 - 110/7*e^2 + 3*e + 240/7, -2*e^2 - 7*e + 28, 5/7*e^3 + 10/7*e^2 - 10*e + 52/7, -5/7*e^3 - 31/7*e^2 - e - 52/7, 2*e^3 + 12*e^2 - e - 48, 2*e^3 + 10*e^2 - 7*e - 22, -12/7*e^3 - 59/7*e^2 + 10*e + 172/7, -8/7*e^3 - 51/7*e^2 - e + 26/7, 1/7*e^3 + 2/7*e^2 - 2*e - 54/7, e^3 + 10*e^2 + 14*e - 30, -13/7*e^3 - 96/7*e^2 - 7*e + 310/7, e^3 + 3*e^2 - 18*e - 10, 15/7*e^3 + 100/7*e^2 + 6*e - 418/7, 2/7*e^3 + 25/7*e^2 + 4*e - 220/7, 1/7*e^3 + 2/7*e^2 + e + 16/7, 11/7*e^3 + 50/7*e^2 - 9*e - 202/7, 2/7*e^3 + 11/7*e^2 - 4*e + 60/7, 2/7*e^3 + 25/7*e^2 + 9*e - 80/7, -e^3 - 6*e^2 + 7*e + 24, -3/7*e^3 - 6/7*e^2 + 5*e - 118/7, -20/7*e^3 - 117/7*e^2 + 3*e + 212/7, -18/7*e^3 - 127/7*e^2 - 12*e + 454/7, -2/7*e^3 - 11/7*e^2 - 7*e - 4/7, -6/7*e^3 - 47/7*e^2 - 9*e + 58/7, 4/7*e^3 + 22/7*e^2 - 2*e + 8/7, -15/7*e^3 - 86/7*e^2 + 3*e + 152/7, 4/7*e^3 + 50/7*e^2 + 12*e - 216/7, -10/7*e^3 - 69/7*e^2 - 9*e + 148/7, 8/7*e^3 + 51/7*e^2 + 9*e - 68/7, 5/7*e^3 + 31/7*e^2 - 7*e - 144/7, 12/7*e^3 + 59/7*e^2 - 9*e - 88/7, 9/7*e^3 + 53/7*e^2 - 10*e - 220/7, 13/7*e^3 + 61/7*e^2 - 10*e - 170/7, 6/7*e^3 + 61/7*e^2 + 10*e - 338/7, 9/7*e^3 + 39/7*e^2 - 7*e - 136/7, e^3 + 6*e^2 + e - 16, -8/7*e^3 - 58/7*e^2 - 5*e + 110/7, -6/7*e^3 - 47/7*e^2 + 4*e + 338/7, -8/7*e^3 - 65/7*e^2 - 9*e + 194/7, 27/7*e^3 + 152/7*e^2 - 11*e - 296/7, 3/7*e^3 + 27/7*e^2 + 8*e - 92/7, -2*e^3 - 10*e^2 + 11*e + 6, 8/7*e^3 + 37/7*e^2 - 13*e + 2/7, 4/7*e^3 + 1/7*e^2 - 14*e + 50/7, -e^3 - 11*e^2 - 18*e + 44, -2*e^3 - 15*e^2 - 3*e + 56, 10/7*e^3 + 62/7*e^2 - e - 92/7, -24/7*e^3 - 132/7*e^2 + 10*e + 302/7, e^3 + 6*e^2 - e + 4, 24/7*e^3 + 118/7*e^2 - 18*e - 288/7, e^3 + 11*e^2 + 17*e - 44, -25/7*e^3 - 141/7*e^2 + 10*e + 300/7, 8/7*e^3 + 58/7*e^2 + 9*e - 138/7, -e^2 + 2*e + 6, -13/7*e^3 - 61/7*e^2 + 7*e + 16/7, 17/7*e^3 + 104/7*e^2 - 7*e - 302/7, 12/7*e^3 + 80/7*e^2 + 4*e - 116/7, e^3 + e^2 - 20*e, 2/7*e^3 + 25/7*e^2 + 6*e - 66/7, 3/7*e^3 + 6/7*e^2 - 12*e - 36/7, -25/7*e^3 - 134/7*e^2 + 11*e + 258/7, 17/7*e^3 + 118/7*e^2 + 2*e - 470/7, -5/7*e^3 - 45/7*e^2 - 14*e + 102/7, 12/7*e^3 + 80/7*e^2 - 2*e - 88/7, 10/7*e^3 + 34/7*e^2 - 21*e - 134/7, e^2 - 3*e - 28, -9/7*e^3 - 74/7*e^2 - 12*e + 206/7, -10/7*e^3 - 41/7*e^2 + 19*e + 134/7, 11/7*e^3 + 36/7*e^2 - 24*e - 202/7, 3*e^2 + 21*e, 12/7*e^3 + 80/7*e^2 + 6*e - 452/7, -4/7*e^3 - 22/7*e^2 + 5*e - 92/7, 3/7*e^3 + 48/7*e^2 + 19*e - 190/7, 23/7*e^3 + 130/7*e^2 - 11*e - 444/7, -4/7*e^3 - 43/7*e^2 - 2*e + 244/7, -10/7*e^3 - 62/7*e^2 + 2*e + 36/7, -26/7*e^3 - 150/7*e^2 - e + 382/7, -17/7*e^3 - 97/7*e^2 + 6*e + 8/7, -2/7*e^3 + 10/7*e^2 + 11*e - 116/7, -13/7*e^3 - 82/7*e^2 + 6*e + 184/7, 3*e^3 + 21*e^2 + 11*e - 82, -4/7*e^3 - 36/7*e^2 - 2*e + 118/7, -3*e^3 - 16*e^2 + 10*e + 58, -10/7*e^3 - 62/7*e^2 + 3*e + 8/7, -4*e^3 - 20*e^2 + 28*e + 54, 5/7*e^3 + 45/7*e^2 + 11*e - 214/7, 4*e^2 + 14*e - 32, -16/7*e^3 - 95/7*e^2 + 4*e + 66/7, -2/7*e^3 - 11/7*e^2 + e + 94/7, 5*e^2 + 18*e - 36, -3*e^2 - 12*e - 30, 12/7*e^3 + 45/7*e^2 - 18*e + 66/7]; 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;