/* 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![11, 0, -7, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, -w^2 + w + 3], [5, 5, w + 1], [5, 5, -w^3 + w^2 + 4*w - 4], [11, 11, w], [29, 29, w^3 - 2*w^2 - 3*w + 7], [29, 29, -w^3 - 2*w^2 + 3*w + 7], [31, 31, -w^3 + w^2 + 4*w - 2], [31, 31, -w^3 - w^2 + 4*w + 2], [41, 41, w^3 + 2*w^2 - 4*w - 6], [41, 41, w^3 - 5*w + 2], [49, 7, -w^3 + w^2 + 4*w - 1], [49, 7, w^3 + w^2 - 4*w - 1], [59, 59, -3*w^2 - w + 10], [59, 59, -3*w^2 + w + 10], [61, 61, -2*w^3 + 2*w^2 + 7*w - 5], [61, 61, 2*w^3 + w^2 - 8*w - 6], [71, 71, 2*w^2 + w - 9], [71, 71, 2*w^2 - w - 9], [81, 3, -3], [101, 101, 2*w^2 - w - 10], [101, 101, 2*w^2 + w - 10], [109, 109, 2*w^3 + w^2 - 8*w - 3], [109, 109, -2*w^3 + w^2 + 8*w - 3], [121, 11, -w^2 + 7], [131, 131, 4*w^2 - 2*w - 15], [131, 131, 2*w^2 + 2*w - 3], [131, 131, 2*w^2 - 2*w - 3], [131, 131, -w^3 - w^2 + 3*w + 7], [139, 139, w^3 + 4*w^2 - 5*w - 16], [139, 139, w^3 + 3*w^2 - 6*w - 10], [139, 139, 2*w^3 - w^2 - 8*w + 1], [139, 139, -4*w^2 - w + 12], [149, 149, 2*w^3 - w^2 - 8*w + 4], [149, 149, 2*w^3 + w^2 - 8*w - 4], [151, 151, 3*w^2 - 3*w - 8], [151, 151, 2*w^3 - 4*w^2 - 8*w + 17], [151, 151, w^2 - 2*w - 7], [151, 151, w^3 + 2*w^2 - 6*w - 8], [179, 179, w^3 - w^2 - 5*w + 1], [179, 179, -w^3 - w^2 + 5*w + 1], [191, 191, 2*w^3 - w^2 - 7*w + 2], [191, 191, 3*w^3 - 2*w^2 - 10*w + 2], [199, 199, w^3 - 4*w^2 - 3*w + 13], [199, 199, -w^3 - 4*w^2 + 3*w + 13], [211, 211, -w^3 + w^2 + 6*w - 5], [211, 211, w^3 - 5*w - 5], [211, 211, -w^3 + 5*w - 5], [211, 211, w^3 + w^2 - 6*w - 5], [229, 229, w^3 - 4*w^2 - 2*w + 14], [229, 229, 2*w^3 + 3*w^2 - 7*w - 12], [229, 229, 2*w^3 - 3*w^2 - 7*w + 12], [229, 229, 2*w^3 + w^2 - 7*w - 8], [241, 241, w^3 + w^2 - 6*w - 4], [241, 241, -w^3 + w^2 + 6*w - 4], [251, 251, 2*w^2 - w - 2], [251, 251, 2*w^2 + w - 2], [271, 271, 2*w^3 - 2*w^2 - 8*w + 5], [271, 271, -4*w^2 - w + 14], [271, 271, -4*w^2 + w + 14], [271, 271, 2*w^3 + 2*w^2 - 8*w - 5], [281, 281, -2*w^3 - w^2 + 9*w + 2], [281, 281, 2*w^3 - w^2 - 9*w + 2], [311, 311, -w^3 + 3*w - 5], [311, 311, w^3 - 3*w - 5], [331, 331, -3*w^3 + 3*w^2 + 12*w - 13], [331, 331, 3*w^3 + 3*w^2 - 12*w - 13], [349, 349, 2*w^3 + 2*w^2 - 7*w - 10], [349, 349, -2*w^3 + 2*w^2 + 7*w - 10], [361, 19, 4*w^2 - 15], [361, 19, -4*w^2 + 13], [379, 379, -w^3 + 3*w^2 + 3*w - 13], [379, 379, w^3 + 3*w^2 - 3*w - 13], [409, 409, 3*w^3 - 2*w^2 - 11*w + 2], [409, 409, w^3 + 3*w^2 - 6*w - 9], [419, 419, -w^3 + w^2 + 6*w + 2], [419, 419, -3*w^3 + 5*w^2 + 13*w - 19], [421, 421, -w^3 + 3*w^2 + 7*w - 13], [421, 421, 2*w^3 - 2*w^2 - 9*w + 6], [421, 421, -2*w^3 - 2*w^2 + 9*w + 6], [421, 421, w^3 + 3*w^2 - 7*w - 13], [439, 439, 3*w^2 + 2*w - 13], [439, 439, 3*w^3 + 2*w^2 - 12*w - 10], [439, 439, -3*w^3 + 2*w^2 + 12*w - 10], [439, 439, 3*w^2 - 2*w - 13], [449, 449, 2*w^3 - 4*w^2 - 9*w + 14], [449, 449, -3*w^3 + 3*w^2 + 11*w - 9], [449, 449, 3*w^3 + 3*w^2 - 11*w - 9], [449, 449, w^3 - 2*w^2 - 5*w + 2], [461, 461, -w - 5], [461, 461, w - 5], [499, 499, w^3 - 5*w^2 - w + 17], [499, 499, -w^3 - 5*w^2 + w + 17], [509, 509, w^3 + 5*w^2 - 4*w - 16], [509, 509, w^2 - 2*w - 10], [509, 509, w^2 + 2*w - 10], [509, 509, w^3 - w^2 - 5*w + 9], [541, 541, 3*w^3 - 10*w - 6], [541, 541, -2*w^3 + 5*w^2 + 5*w - 16], [569, 569, 3*w^3 + 2*w^2 - 13*w - 6], [569, 569, -2*w^3 + 4*w^2 + 7*w - 10], [599, 599, w^3 + 2*w^2 - 7*w - 10], [599, 599, -w^3 + 2*w^2 + 7*w - 10], [601, 601, -3*w^3 - w^2 + 12*w + 6], [601, 601, 3*w^3 - w^2 - 12*w + 6], [619, 619, 3*w^3 - 12*w + 2], [619, 619, -3*w^3 + 12*w + 2], [631, 631, -w^3 + w^2 + 2*w - 8], [631, 631, w^3 + w^2 - 2*w - 8], [641, 641, w^3 + w^2 - 7*w - 1], [641, 641, -3*w^3 - 2*w^2 + 12*w + 4], [641, 641, 3*w^3 - 2*w^2 - 12*w + 4], [641, 641, w^3 - w^2 - 7*w + 1], [659, 659, -2*w^3 + w^2 + 10*w - 1], [659, 659, -w^3 + 5*w^2 + 4*w - 19], [659, 659, w^3 + 5*w^2 - 4*w - 19], [659, 659, 2*w^3 + w^2 - 10*w - 1], [691, 691, 3*w^2 - 2*w - 14], [691, 691, 3*w^2 + 2*w - 14], [701, 701, w^3 + 4*w^2 - 7*w - 12], [701, 701, 3*w^3 - w^2 - 12*w - 1], [719, 719, -w^3 + 7*w - 3], [719, 719, w^3 - 7*w - 3], [739, 739, 3*w^3 + w^2 - 12*w - 3], [739, 739, w^3 + 2*w^2 - 3*w - 1], [739, 739, -w^3 + 2*w^2 + 3*w - 1], [739, 739, -3*w^3 + w^2 + 12*w - 3], [751, 751, -2*w^3 + 6*w^2 + 7*w - 20], [751, 751, 2*w^3 + 6*w^2 - 7*w - 20], [761, 761, 2*w^3 + 6*w^2 - 8*w - 21], [761, 761, -2*w^3 + 6*w^2 + 8*w - 21], [769, 769, -2*w^3 + 6*w^2 + 8*w - 23], [769, 769, w^2 + 2*w + 3], [809, 809, 2*w^3 + 3*w^2 - 7*w - 14], [809, 809, -2*w^3 + 3*w^2 + 7*w - 14], [821, 821, -3*w^3 - w^2 + 11*w + 1], [821, 821, 3*w^3 - w^2 - 11*w + 1], [829, 829, 3*w^3 - w^2 - 12*w + 4], [829, 829, w^3 - 4*w^2 - 6*w + 12], [829, 829, -w^3 - 4*w^2 + 6*w + 12], [829, 829, -3*w^3 - w^2 + 12*w + 4], [839, 839, -w^3 + 3*w - 6], [839, 839, w^3 - 3*w - 6], [841, 29, 5*w^2 - 19], [859, 859, w^3 + 2*w^2 - 7*w - 9], [859, 859, -w^3 + 2*w^2 + 7*w - 9], [911, 911, -2*w^3 + 4*w^2 + 9*w - 13], [911, 911, 2*w^3 + 4*w^2 - 9*w - 13], [919, 919, 2*w^3 + 4*w^2 - 7*w - 17], [919, 919, 3*w^3 + 2*w^2 - 11*w - 6], [919, 919, 3*w^3 - 2*w^2 - 11*w + 6], [919, 919, 2*w^3 - 4*w^2 - 7*w + 17], [941, 941, -2*w^3 + 2*w^2 + 7*w - 15], [941, 941, 2*w^3 + 2*w^2 - 7*w - 15], [961, 31, -2*w^2 + 13], [971, 971, w^3 + 2*w^2 - 7*w - 7], [971, 971, -w^3 + 2*w^2 + 7*w - 7], [991, 991, 3*w^3 + 2*w^2 - 11*w - 7], [991, 991, -3*w^3 + 2*w^2 + 11*w - 7]]; primes := [ideal : I in primesArray]; heckePol := x^6 - 4*x^5 - 5*x^4 + 24*x^3 + 5*x^2 - 34*x - 1; K := NumberField(heckePol); heckeEigenvaluesArray := [e, 2*e^5 - 4*e^4 - 19*e^3 + 13*e^2 + 41*e + 5, -e^5 + 2*e^4 + 9*e^3 - 5*e^2 - 19*e - 4, -3*e^5 + 6*e^4 + 29*e^3 - 22*e^2 - 62*e + 2, 6*e^5 - 13*e^4 - 54*e^3 + 47*e^2 + 111*e - 3, -2*e^5 + 4*e^4 + 19*e^3 - 14*e^2 - 38*e - 1, 7*e^5 - 16*e^4 - 59*e^3 + 54*e^2 + 118*e + 4, -4*e^5 + 9*e^4 + 34*e^3 - 30*e^2 - 69*e, -1, 5*e^5 - 10*e^4 - 47*e^3 + 34*e^2 + 98*e + 4, e^4 - 3*e^3 - 5*e^2 + 8*e + 11, -e^5 + 2*e^4 + 10*e^3 - 8*e^2 - 25*e + 6, -2*e^5 + 2*e^4 + 25*e^3 - 4*e^2 - 58*e - 13, -16*e^5 + 35*e^4 + 141*e^3 - 119*e^2 - 293*e - 8, 13*e^5 - 28*e^4 - 117*e^3 + 99*e^2 + 241*e + 6, -15*e^5 + 34*e^4 + 129*e^3 - 118*e^2 - 264*e, 14*e^5 - 31*e^4 - 122*e^3 + 104*e^2 + 248*e + 15, 2*e^4 - 6*e^3 - 11*e^2 + 20*e + 13, -13*e^5 + 28*e^4 + 116*e^3 - 97*e^2 - 236*e - 8, -4*e^5 + 7*e^4 + 42*e^3 - 27*e^2 - 94*e + 6, 12*e^5 - 25*e^4 - 111*e^3 + 85*e^2 + 239*e + 10, 8*e^5 - 16*e^4 - 76*e^3 + 54*e^2 + 166*e + 4, 11*e^5 - 24*e^4 - 97*e^3 + 82*e^2 + 198*e + 12, e^5 - e^4 - 13*e^3 + 2*e^2 + 33*e - 2, 7*e^5 - 15*e^4 - 65*e^3 + 58*e^2 + 135*e - 12, 20*e^5 - 44*e^4 - 177*e^3 + 156*e^2 + 359*e, -18*e^5 + 38*e^4 + 165*e^3 - 132*e^2 - 351*e - 4, -7*e^5 + 14*e^4 + 67*e^3 - 47*e^2 - 144*e - 13, 8*e^5 - 17*e^4 - 74*e^3 + 60*e^2 + 164*e + 3, -5*e^5 + 8*e^4 + 54*e^3 - 28*e^2 - 118*e - 1, -16*e^5 + 33*e^4 + 149*e^3 - 115*e^2 - 314*e - 9, e^4 - 2*e^3 - 8*e^2 + 2*e + 11, -14*e^5 + 32*e^4 + 119*e^3 - 114*e^2 - 237*e + 12, -4*e^5 + 10*e^4 + 32*e^3 - 36*e^2 - 62*e - 2, -16*e^5 + 33*e^4 + 147*e^3 - 111*e^2 - 306*e - 13, 7*e^5 - 16*e^4 - 57*e^3 + 51*e^2 + 106*e + 9, -14*e^5 + 28*e^4 + 132*e^3 - 92*e^2 - 286*e - 18, e^4 - 4*e^3 - 2*e^2 + 11*e - 6, -7*e^5 + 14*e^4 + 65*e^3 - 42*e^2 - 140*e - 14, -18*e^5 + 41*e^4 + 154*e^3 - 144*e^2 - 309*e - 2, 8*e^5 - 18*e^4 - 67*e^3 + 54*e^2 + 137*e + 20, 5*e^5 - 8*e^4 - 53*e^3 + 23*e^2 + 119*e + 20, 13*e^5 - 30*e^4 - 110*e^3 + 107*e^2 + 216*e - 4, -22*e^5 + 47*e^4 + 200*e^3 - 169*e^2 - 416*e + 6, 15*e^5 - 35*e^4 - 126*e^3 + 126*e^2 + 249*e - 7, -9*e^5 + 20*e^4 + 82*e^3 - 77*e^2 - 178*e + 16, -4*e^5 + 10*e^4 + 30*e^3 - 32*e^2 - 54*e - 12, -3*e^5 + 4*e^4 + 34*e^3 - 6*e^2 - 89*e - 26, 11*e^5 - 23*e^4 - 102*e^3 + 82*e^2 + 217*e - 19, -4*e^5 + 6*e^4 + 46*e^3 - 20*e^2 - 112*e - 8, -9*e^5 + 20*e^4 + 79*e^3 - 67*e^2 - 168*e - 3, -2*e^5 + 5*e^4 + 14*e^3 - 12*e^2 - 26*e - 23, -12*e^5 + 24*e^4 + 113*e^3 - 81*e^2 - 235*e - 13, 4*e^5 - 13*e^4 - 22*e^3 + 50*e^2 + 31*e - 16, 14*e^5 - 30*e^4 - 127*e^3 + 108*e^2 + 260*e - 7, -8*e^5 + 16*e^4 + 74*e^3 - 45*e^2 - 164*e - 33, 29*e^5 - 61*e^4 - 265*e^3 + 207*e^2 + 558*e + 22, 4*e^5 - 7*e^4 - 41*e^3 + 23*e^2 + 93*e - 2, -6*e^5 + 11*e^4 + 58*e^3 - 25*e^2 - 129*e - 35, 6*e^5 - 14*e^4 - 51*e^3 + 51*e^2 + 101*e + 5, -9*e^5 + 20*e^4 + 81*e^3 - 76*e^2 - 166*e + 14, 13*e^5 - 28*e^4 - 114*e^3 + 92*e^2 + 224*e + 27, 24*e^5 - 55*e^4 - 202*e^3 + 187*e^2 + 402*e + 8, 14*e^5 - 28*e^4 - 135*e^3 + 100*e^2 + 296*e + 5, -6*e^5 + 16*e^4 + 46*e^3 - 62*e^2 - 94*e + 18, 6*e^5 - 12*e^4 - 57*e^3 + 41*e^2 + 123*e - 1, 17*e^5 - 32*e^4 - 168*e^3 + 105*e^2 + 368*e + 42, -18*e^5 + 40*e^4 + 158*e^3 - 142*e^2 - 322*e, 34*e^5 - 75*e^4 - 299*e^3 + 263*e^2 + 615*e + 2, -4*e^5 + 12*e^4 + 24*e^3 - 44*e^2 - 34*e + 20, 8*e^5 - 18*e^4 - 74*e^3 + 74*e^2 + 164*e - 28, 17*e^5 - 36*e^4 - 150*e^3 + 109*e^2 + 308*e + 46, 9*e^5 - 16*e^4 - 95*e^3 + 60*e^2 + 218*e + 12, -16*e^5 + 34*e^4 + 142*e^3 - 106*e^2 - 302*e - 40, -11*e^5 + 20*e^4 + 108*e^3 - 60*e^2 - 231*e - 30, 27*e^5 - 59*e^4 - 237*e^3 + 193*e^2 + 496*e + 48, 26*e^5 - 57*e^4 - 230*e^3 + 196*e^2 + 485*e + 8, 31*e^5 - 66*e^4 - 281*e^3 + 224*e^2 + 594*e + 24, -12*e^5 + 26*e^4 + 106*e^3 - 88*e^2 - 212*e + 6, -25*e^5 + 58*e^4 + 209*e^3 - 200*e^2 - 414*e + 2, 21*e^5 - 43*e^4 - 195*e^3 + 140*e^2 + 419*e + 28, 23*e^5 - 46*e^4 - 217*e^3 + 149*e^2 + 477*e + 44, 11*e^5 - 24*e^4 - 95*e^3 + 72*e^2 + 192*e + 36, -12*e^5 + 23*e^4 + 119*e^3 - 81*e^2 - 261*e - 6, 30*e^5 - 64*e^4 - 273*e^3 + 226*e^2 + 575*e - 8, 16*e^5 - 36*e^4 - 137*e^3 + 118*e^2 + 289*e + 30, 13*e^5 - 29*e^4 - 113*e^3 + 101*e^2 + 226*e - 2, -35*e^5 + 79*e^4 + 303*e^3 - 281*e^2 - 622*e + 20, 29*e^5 - 62*e^4 - 261*e^3 + 211*e^2 + 540*e + 19, 40*e^5 - 86*e^4 - 360*e^3 + 299*e^2 + 740*e + 25, 16*e^5 - 35*e^4 - 146*e^3 + 134*e^2 + 312*e - 19, -4*e^5 + 8*e^4 + 39*e^3 - 30*e^2 - 84*e - 3, 15*e^5 - 37*e^4 - 116*e^3 + 128*e^2 + 213*e - 13, 10*e^5 - 24*e^4 - 84*e^3 + 94*e^2 + 172*e - 22, 2*e^5 - e^4 - 26*e^3 - 6*e^2 + 52*e + 27, -8*e^5 + 18*e^4 + 70*e^3 - 68*e^2 - 132*e + 16, -16*e^5 + 36*e^4 + 142*e^3 - 136*e^2 - 294*e + 14, -36*e^5 + 78*e^4 + 324*e^3 - 280*e^2 - 680*e + 18, 26*e^5 - 59*e^4 - 227*e^3 + 221*e^2 + 460*e - 29, e^5 - 3*e^4 - 10*e^3 + 20*e^2 + 37*e - 31, -37*e^5 + 78*e^4 + 339*e^3 - 271*e^2 - 721*e - 10, 15*e^5 - 32*e^4 - 131*e^3 + 97*e^2 + 271*e + 26, -2*e^4 + 7*e^3 + 10*e^2 - 21*e - 22, -9*e^5 + 19*e^4 + 81*e^3 - 65*e^2 - 156*e - 10, 31*e^5 - 66*e^4 - 283*e^3 + 233*e^2 + 597*e + 8, -11*e^5 + 20*e^4 + 113*e^3 - 70*e^2 - 254*e - 20, 22*e^5 - 53*e^4 - 180*e^3 + 196*e^2 + 340*e - 39, 20*e^5 - 43*e^4 - 180*e^3 + 147*e^2 + 377*e + 21, 12*e^5 - 28*e^4 - 101*e^3 + 100*e^2 + 209*e - 2, 21*e^5 - 44*e^4 - 191*e^3 + 141*e^2 + 405*e + 36, -21*e^5 + 42*e^4 + 197*e^3 - 136*e^2 - 410*e - 46, -13*e^5 + 32*e^4 + 105*e^3 - 122*e^2 - 210*e + 34, -6*e^5 + 13*e^4 + 50*e^3 - 34*e^2 - 83*e - 42, -11*e^5 + 27*e^4 + 90*e^3 - 106*e^2 - 177*e + 29, -18*e^5 + 37*e^4 + 166*e^3 - 116*e^2 - 358*e - 55, -21*e^5 + 42*e^4 + 197*e^3 - 133*e^2 - 425*e - 28, -14*e^5 + 27*e^4 + 136*e^3 - 89*e^2 - 304*e - 24, 26*e^5 - 56*e^4 - 232*e^3 + 186*e^2 + 486*e + 36, e^5 + 2*e^4 - 21*e^3 - 14*e^2 + 54*e + 12, -21*e^5 + 46*e^4 + 185*e^3 - 154*e^2 - 382*e - 18, -27*e^5 + 54*e^4 + 253*e^3 - 178*e^2 - 522*e - 44, 14*e^5 - 26*e^4 - 137*e^3 + 76*e^2 + 303*e + 34, -9*e^5 + 26*e^4 + 57*e^3 - 96*e^2 - 88*e + 28, -24*e^5 + 53*e^4 + 212*e^3 - 191*e^2 - 433*e + 11, -36*e^5 + 75*e^4 + 328*e^3 - 245*e^2 - 698*e - 48, 7*e^5 - 15*e^4 - 62*e^3 + 52*e^2 + 127*e + 5, -37*e^5 + 79*e^4 + 332*e^3 - 260*e^2 - 701*e - 39, -50*e^5 + 114*e^4 + 429*e^3 - 406*e^2 - 862*e + 23, 22*e^5 - 47*e^4 - 196*e^3 + 154*e^2 + 412*e + 11, -16*e^5 + 35*e^4 + 136*e^3 - 105*e^2 - 260*e - 50, 20*e^5 - 35*e^4 - 204*e^3 + 104*e^2 + 450*e + 59, -20*e^5 + 46*e^4 + 173*e^3 - 178*e^2 - 342*e + 43, -8*e^5 + 20*e^4 + 67*e^3 - 84*e^2 - 136*e + 15, 10*e^5 - 20*e^4 - 90*e^3 + 52*e^2 + 192*e + 38, -32*e^5 + 73*e^4 + 270*e^3 - 240*e^2 - 543*e - 36, -35*e^5 + 78*e^4 + 308*e^3 - 282*e^2 - 627*e + 22, -26*e^5 + 56*e^4 + 235*e^3 - 194*e^2 - 495*e - 30, 14*e^5 - 29*e^4 - 125*e^3 + 91*e^2 + 256*e + 15, e^5 - 8*e^4 + 7*e^3 + 50*e^2 - 38*e - 56, 14*e^5 - 34*e^4 - 114*e^3 + 126*e^2 + 234*e - 22, -24*e^5 + 55*e^4 + 203*e^3 - 187*e^2 - 403*e - 26, 38*e^5 - 83*e^4 - 334*e^3 + 280*e^2 + 688*e + 37, -44*e^5 + 91*e^4 + 405*e^3 - 303*e^2 - 851*e - 70, 24*e^5 - 52*e^4 - 216*e^3 + 186*e^2 + 450*e - 8, 18*e^5 - 38*e^4 - 164*e^3 + 132*e^2 + 348*e + 4, -28*e^5 + 60*e^4 + 251*e^3 - 211*e^2 - 511*e - 3, 15*e^5 - 33*e^4 - 137*e^3 + 125*e^2 + 296*e - 8, -58*e^5 + 127*e^4 + 510*e^3 - 430*e^2 - 1054*e - 53, -33*e^5 + 70*e^4 + 297*e^3 - 232*e^2 - 628*e - 48, -10*e^5 + 22*e^4 + 91*e^3 - 83*e^2 - 207*e + 33, 8*e^5 - 20*e^4 - 62*e^3 + 66*e^2 + 124*e + 4, -28*e^5 + 57*e^4 + 260*e^3 - 187*e^2 - 554*e - 28, 36*e^5 - 75*e^4 - 332*e^3 + 258*e^2 + 702*e + 23, 19*e^5 - 45*e^4 - 155*e^3 + 152*e^2 + 297*e, -25*e^5 + 51*e^4 + 233*e^3 - 177*e^2 - 488*e - 14, -43*e^5 + 96*e^4 + 377*e^3 - 336*e^2 - 782*e - 12, 43*e^5 - 93*e^4 - 388*e^3 + 336*e^2 + 801*e - 19, 51*e^5 - 108*e^4 - 465*e^3 + 372*e^2 + 982*e + 2]; 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;