/* 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![8, -7, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -w + 2], [4, 2, -w^2 - w + 5], [5, 5, -w + 3], [7, 7, -w^2 - 2*w + 3], [11, 11, -w^2 + 5], [13, 13, w + 1], [23, 23, w^2 - 3], [25, 5, w^2 + 2*w - 1], [27, 3, -3], [29, 29, -3*w + 7], [43, 43, -3*w^2 - 2*w + 17], [49, 7, -2*w^2 + w + 11], [67, 67, -2*w^2 - 4*w + 3], [71, 71, -2*w^2 + w + 9], [73, 73, w^2 + 2*w - 7], [73, 73, -2*w^2 - 2*w + 11], [73, 73, w - 5], [89, 89, 2*w^2 + w - 9], [89, 89, -w^2 - 2*w + 9], [89, 89, -2*w - 1], [101, 101, 4*w^2 + 4*w - 19], [103, 103, 2*w + 3], [107, 107, w + 5], [109, 109, 3*w^2 + 2*w - 15], [113, 113, 2*w - 7], [121, 11, 2*w^2 + w - 7], [127, 127, 2*w^2 - 9], [127, 127, -4*w + 11], [127, 127, -2*w^2 + 2*w + 9], [131, 131, 3*w^2 + 4*w - 13], [137, 137, -w^2 + 2*w - 3], [149, 149, 2*w^2 - 6*w + 1], [151, 151, 3*w - 1], [157, 157, -3*w^2 - 6*w + 7], [157, 157, -2*w^2 + 3*w + 3], [157, 157, 2*w^2 + 2*w - 15], [163, 163, 3*w^2 + 2*w - 19], [163, 163, 4*w^2 + 5*w - 17], [163, 163, -5*w^2 - 6*w + 19], [167, 167, w^2 - 11], [167, 167, -3*w - 7], [167, 167, -w^2 - 2*w + 13], [169, 13, w^2 - 2*w - 5], [181, 181, -2*w - 7], [181, 181, 2*w^2 - 7], [181, 181, 2*w^2 - 2*w - 5], [193, 193, 2*w^2 + 2*w - 3], [197, 197, -4*w^2 - 7*w + 9], [197, 197, 2*w^2 - 2*w - 19], [197, 197, 2*w^2 + 3*w - 11], [199, 199, -6*w^2 + 37], [227, 227, 2*w^2 - 3*w - 7], [233, 233, w^2 + 4*w - 15], [251, 251, -4*w^2 + 23], [257, 257, w^2 - 2*w - 7], [269, 269, -4*w^2 - w + 23], [271, 271, 2*w^2 - w - 19], [277, 277, -3*w^2 - 2*w + 9], [281, 281, w^2 - 2*w - 9], [283, 283, 6*w^2 + 5*w - 31], [293, 293, 2*w^2 - 2*w - 21], [311, 311, -w^2 - 4*w - 1], [317, 317, 3*w^2 + 2*w - 11], [317, 317, 2*w^2 + 3*w - 13], [317, 317, 2*w^2 - 2*w - 11], [331, 331, 4*w^2 - w - 21], [331, 331, 3*w + 5], [331, 331, 4*w^2 + 2*w - 21], [337, 337, -2*w^2 + 4*w + 5], [349, 349, w^2 - 4*w - 1], [353, 353, -w^2 - 4*w - 5], [367, 367, 6*w^2 + 7*w - 23], [379, 379, 2*w - 9], [389, 389, 2*w^2 + 3*w - 15], [389, 389, w^2 + 4*w - 13], [389, 389, 2*w^2 + 4*w - 1], [397, 397, 4*w - 1], [409, 409, 2*w^2 + w - 19], [409, 409, -w^2 + 6*w - 11], [409, 409, 2*w^2 + 4*w - 21], [419, 419, 4*w^2 + 4*w - 21], [421, 421, -w^2 - 3], [421, 421, -4*w^2 - 6*w + 17], [421, 421, 4*w^2 + 3*w - 31], [439, 439, 5*w^2 - 29], [443, 443, w^2 - 13], [449, 449, 4*w^2 - w - 27], [457, 457, 2*w^2 - 19], [467, 467, 4*w + 9], [467, 467, 3*w^2 - 2*w - 17], [467, 467, 2*w^2 - 2*w - 13], [479, 479, -w^2 + 4*w + 19], [487, 487, 4*w^2 + 5*w - 19], [491, 491, 3*w - 11], [503, 503, 4*w^2 + 3*w - 17], [523, 523, -2*w^2 - 5*w - 1], [529, 23, -4*w^2 - w + 21], [547, 547, 5*w^2 + 4*w - 23], [557, 557, 4*w^2 - 2*w - 33], [563, 563, 2*w^2 + 2*w - 1], [563, 563, 3*w^2 - 8*w - 1], [563, 563, 3*w^2 + 4*w - 5], [571, 571, 5*w^2 + 10*w - 11], [577, 577, -4*w + 13], [587, 587, 3*w^2 + 4*w - 17], [593, 593, 3*w^2 - 11], [593, 593, 4*w^2 + 2*w - 19], [593, 593, w - 9], [601, 601, -4*w^2 - 10*w + 3], [607, 607, 3*w^2 + 4*w - 27], [607, 607, -4*w^2 - 2*w + 33], [607, 607, 2*w^2 - 7*w + 9], [617, 617, -2*w^2 + 9*w - 5], [619, 619, -4*w - 1], [631, 631, -8*w^2 - 10*w + 29], [641, 641, w^2 - 4*w - 3], [653, 653, 3*w^2 - 2*w - 9], [659, 659, 2*w^2 + 4*w - 19], [661, 661, 6*w^2 + 6*w - 25], [673, 673, 6*w^2 + 7*w - 27], [683, 683, 5*w^2 + 2*w - 27], [701, 701, 7*w - 13], [709, 709, 6*w^2 + 8*w - 25], [719, 719, -2*w^2 + 4*w + 7], [719, 719, -7*w^2 - 10*w + 27], [719, 719, 4*w^2 - w - 19], [727, 727, 3*w^2 + 8*w + 1], [733, 733, 3*w^2 - 2*w - 7], [733, 733, 3*w^2 - 2*w - 29], [739, 739, -w - 9], [751, 751, -w^2 - 4*w + 19], [757, 757, -4*w - 7], [761, 761, -5*w^2 - 4*w + 29], [787, 787, -2*w^2 - 6*w + 3], [809, 809, -w^2 + 4*w - 9], [809, 809, -4*w^2 + 3*w + 15], [809, 809, 2*w^2 - 9*w + 13], [821, 821, -4*w^2 - 9*w + 7], [821, 821, 5*w - 1], [821, 821, 8*w^2 + 11*w - 31], [827, 827, 8*w^2 + 8*w - 35], [829, 829, 2*w^2 - 5*w - 27], [839, 839, -w^2 - 6*w - 7], [841, 29, 3*w^2 + 4*w - 31], [863, 863, 4*w^2 - 19], [883, 883, -5*w^2 + 27], [887, 887, 5*w^2 + 4*w - 21], [907, 907, -6*w^2 + 2*w + 31], [907, 907, 2*w^2 - 5*w - 29], [907, 907, 2*w^2 + 5*w - 15], [911, 911, w^2 - 15], [919, 919, 8*w - 17], [941, 941, -4*w^2 + 5*w + 15], [947, 947, 4*w^2 + 2*w - 13], [947, 947, 3*w^2 - 6*w - 7], [947, 947, -5*w^2 - 4*w + 15]]; primes := [ideal : I in primesArray]; heckePol := x^9 - 2*x^8 - 13*x^7 + 25*x^6 + 47*x^5 - 85*x^4 - 36*x^3 + 41*x^2 + 23*x + 3; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -e^8 + 2*e^7 + 13*e^6 - 25*e^5 - 47*e^4 + 86*e^3 + 35*e^2 - 48*e - 16, 2*e^8 - 5*e^7 - 24*e^6 + 62*e^5 + 70*e^4 - 206*e^3 + 5*e^2 + 86*e + 18, -4*e^8 + 9*e^7 + 49*e^6 - 112*e^5 - 150*e^4 + 374*e^3 + 15*e^2 - 155*e - 34, 1, 4*e^8 - 9*e^7 - 50*e^6 + 113*e^5 + 162*e^4 - 385*e^3 - 52*e^2 + 186*e + 47, 10*e^8 - 23*e^7 - 123*e^6 + 287*e^5 + 383*e^4 - 966*e^3 - 69*e^2 + 432*e + 105, -6*e^8 + 14*e^7 + 74*e^6 - 175*e^5 - 232*e^4 + 590*e^3 + 46*e^2 - 266*e - 61, -3*e^8 + 7*e^7 + 37*e^6 - 88*e^5 - 116*e^4 + 300*e^3 + 24*e^2 - 145*e - 32, -e^8 + 2*e^7 + 12*e^6 - 25*e^5 - 33*e^4 + 83*e^3 - 14*e^2 - 25*e - 3, 9*e^8 - 20*e^7 - 112*e^6 + 250*e^5 + 358*e^4 - 844*e^3 - 95*e^2 + 382*e + 92, -7*e^8 + 16*e^7 + 86*e^6 - 200*e^5 - 267*e^4 + 677*e^3 + 46*e^2 - 315*e - 70, e^7 - 2*e^6 - 11*e^5 + 24*e^4 + 24*e^3 - 77*e^2 + 30*e + 23, -3*e^8 + 7*e^7 + 38*e^6 - 88*e^5 - 130*e^4 + 303*e^3 + 73*e^2 - 164*e - 48, -8*e^8 + 18*e^7 + 98*e^6 - 224*e^5 - 301*e^4 + 750*e^3 + 38*e^2 - 323*e - 76, -15*e^8 + 34*e^7 + 185*e^6 - 423*e^5 - 582*e^4 + 1421*e^3 + 130*e^2 - 643*e - 151, -8*e^8 + 20*e^7 + 96*e^6 - 249*e^5 - 278*e^4 + 831*e^3 - 33*e^2 - 350*e - 67, 16*e^8 - 37*e^7 - 197*e^6 + 461*e^5 + 616*e^4 - 1548*e^3 - 121*e^2 + 689*e + 165, 4*e^8 - 9*e^7 - 50*e^6 + 113*e^5 + 162*e^4 - 382*e^3 - 53*e^2 + 166*e + 51, 12*e^8 - 27*e^7 - 149*e^6 + 337*e^5 + 476*e^4 - 1136*e^3 - 131*e^2 + 521*e + 129, -7*e^8 + 16*e^7 + 86*e^6 - 200*e^5 - 266*e^4 + 675*e^3 + 40*e^2 - 308*e - 72, -14*e^8 + 33*e^7 + 172*e^6 - 412*e^5 - 534*e^4 + 1385*e^3 + 90*e^2 - 616*e - 148, 9*e^8 - 22*e^7 - 110*e^6 + 275*e^5 + 336*e^4 - 927*e^3 - 32*e^2 + 422*e + 93, -13*e^8 + 30*e^7 + 160*e^6 - 375*e^5 - 500*e^4 + 1267*e^3 + 100*e^2 - 584*e - 139, e^8 - 15*e^6 + 71*e^4 - 7*e^3 - 112*e^2 + 28*e + 30, -e^5 + e^4 + 10*e^3 - 6*e^2 - 17*e - 1, -29*e^8 + 67*e^7 + 356*e^6 - 834*e^5 - 1102*e^4 + 2793*e^3 + 171*e^2 - 1212*e - 286, 10*e^8 - 23*e^7 - 122*e^6 + 286*e^5 + 367*e^4 - 950*e^3 - 5*e^2 + 371*e + 74, -3*e^8 + 7*e^7 + 36*e^6 - 87*e^5 - 103*e^4 + 289*e^3 - 21*e^2 - 113*e - 13, -3*e^8 + 7*e^7 + 39*e^6 - 89*e^5 - 138*e^4 + 307*e^3 + 83*e^2 - 155*e - 51, -10*e^8 + 22*e^7 + 126*e^6 - 276*e^5 - 418*e^4 + 939*e^3 + 175*e^2 - 452*e - 126, -23*e^8 + 54*e^7 + 283*e^6 - 674*e^5 - 882*e^4 + 2267*e^3 + 160*e^2 - 1019*e - 240, 9*e^8 - 20*e^7 - 110*e^6 + 249*e^5 + 336*e^4 - 835*e^3 - 40*e^2 + 362*e + 89, 16*e^8 - 37*e^7 - 196*e^6 + 461*e^5 + 603*e^4 - 1544*e^3 - 77*e^2 + 657*e + 149, -16*e^8 + 37*e^7 + 196*e^6 - 460*e^5 - 606*e^4 + 1539*e^3 + 98*e^2 - 668*e - 154, -27*e^8 + 62*e^7 + 331*e^6 - 772*e^5 - 1020*e^4 + 2584*e^3 + 139*e^2 - 1105*e - 256, e^8 - e^7 - 15*e^6 + 13*e^5 + 72*e^4 - 53*e^3 - 115*e^2 + 57*e + 29, e^7 - 2*e^6 - 12*e^5 + 26*e^4 + 31*e^3 - 90*e^2 + 22*e + 32, -13*e^8 + 30*e^7 + 159*e^6 - 372*e^5 - 488*e^4 + 1239*e^3 + 58*e^2 - 523*e - 106, 15*e^8 - 35*e^7 - 184*e^6 + 436*e^5 + 568*e^4 - 1461*e^3 - 77*e^2 + 640*e + 138, 7*e^8 - 15*e^7 - 87*e^6 + 186*e^5 + 283*e^4 - 627*e^3 - 109*e^2 + 300*e + 78, 2*e^8 - 5*e^7 - 25*e^6 + 63*e^5 + 81*e^4 - 214*e^3 - 23*e^2 + 100*e + 21, -10*e^8 + 22*e^7 + 123*e^6 - 272*e^5 - 386*e^4 + 904*e^3 + 90*e^2 - 379*e - 100, 12*e^8 - 26*e^7 - 149*e^6 + 324*e^5 + 476*e^4 - 1092*e^3 - 136*e^2 + 493*e + 134, 20*e^8 - 46*e^7 - 245*e^6 + 573*e^5 + 754*e^4 - 1922*e^3 - 104*e^2 + 841*e + 203, 12*e^8 - 28*e^7 - 147*e^6 + 350*e^5 + 450*e^4 - 1176*e^3 - 46*e^2 + 511*e + 122, -2*e^8 + 4*e^7 + 25*e^6 - 49*e^5 - 80*e^4 + 156*e^3 + 18*e^2 - 37*e - 13, -6*e^8 + 13*e^7 + 76*e^6 - 163*e^5 - 257*e^4 + 556*e^3 + 127*e^2 - 275*e - 75, -18*e^8 + 41*e^7 + 223*e^6 - 514*e^5 - 704*e^4 + 1740*e^3 + 157*e^2 - 789*e - 198, 17*e^8 - 39*e^7 - 210*e^6 + 488*e^5 + 660*e^4 - 1652*e^3 - 140*e^2 + 768*e + 180, -2*e^8 + 4*e^7 + 25*e^6 - 49*e^5 - 82*e^4 + 160*e^3 + 32*e^2 - 55*e - 25, 29*e^8 - 68*e^7 - 355*e^6 + 849*e^5 + 1090*e^4 - 2856*e^3 - 135*e^2 + 1263*e + 291, 12*e^8 - 29*e^7 - 146*e^6 + 360*e^5 + 440*e^4 - 1196*e^3 - 17*e^2 + 490*e + 114, 3*e^8 - 9*e^7 - 32*e^6 + 111*e^5 + 55*e^4 - 357*e^3 + 167*e^2 + 99*e - 9, 31*e^8 - 71*e^7 - 380*e^6 + 884*e^5 + 1170*e^4 - 2959*e^3 - 155*e^2 + 1272*e + 294, 18*e^8 - 40*e^7 - 223*e^6 + 499*e^5 + 706*e^4 - 1678*e^3 - 168*e^2 + 737*e + 183, -16*e^8 + 39*e^7 + 192*e^6 - 486*e^5 - 552*e^4 + 1621*e^3 - 88*e^2 - 658*e - 124, 25*e^8 - 58*e^7 - 308*e^6 + 724*e^5 + 964*e^4 - 2441*e^3 - 196*e^2 + 1120*e + 266, 11*e^8 - 24*e^7 - 140*e^6 + 301*e^5 + 476*e^4 - 1024*e^3 - 237*e^2 + 496*e + 153, -4*e^8 + 9*e^7 + 47*e^6 - 111*e^5 - 126*e^4 + 365*e^3 - 58*e^2 - 133*e - 13, -15*e^8 + 37*e^7 + 179*e^6 - 460*e^5 - 504*e^4 + 1526*e^3 - 134*e^2 - 599*e - 96, 9*e^8 - 20*e^7 - 112*e^6 + 251*e^5 + 358*e^4 - 853*e^3 - 98*e^2 + 398*e + 99, -16*e^8 + 36*e^7 + 197*e^6 - 448*e^5 - 615*e^4 + 1502*e^3 + 114*e^2 - 656*e - 144, 29*e^8 - 66*e^7 - 359*e^6 + 824*e^5 + 1140*e^4 - 2781*e^3 - 294*e^2 + 1279*e + 306, -14*e^8 + 33*e^7 + 170*e^6 - 411*e^5 - 506*e^4 + 1370*e^3 - 11*e^2 - 552*e - 105, -5*e^8 + 11*e^7 + 63*e^6 - 137*e^5 - 208*e^4 + 457*e^3 + 83*e^2 - 185*e - 79, 4*e^8 - 7*e^7 - 53*e^6 + 88*e^5 + 198*e^4 - 306*e^3 - 163*e^2 + 173*e + 62, 9*e^8 - 20*e^7 - 112*e^6 + 248*e^5 + 364*e^4 - 828*e^3 - 133*e^2 + 364*e + 104, e^8 - 3*e^7 - 12*e^6 + 37*e^5 + 36*e^4 - 119*e^3 - 3*e^2 + 38*e + 5, 18*e^8 - 41*e^7 - 221*e^6 + 508*e^5 + 688*e^4 - 1695*e^3 - 128*e^2 + 741*e + 170, -2*e^8 + 7*e^7 + 21*e^6 - 86*e^5 - 34*e^4 + 276*e^3 - 117*e^2 - 83*e, 28*e^8 - 63*e^7 - 344*e^6 + 785*e^5 + 1064*e^4 - 2634*e^3 - 157*e^2 + 1144*e + 257, -31*e^8 + 72*e^7 + 379*e^6 - 894*e^5 - 1160*e^4 + 2977*e^3 + 126*e^2 - 1245*e - 280, -7*e^8 + 16*e^7 + 84*e^6 - 200*e^5 - 238*e^4 + 667*e^3 - 58*e^2 - 246*e - 42, 18*e^8 - 42*e^7 - 222*e^6 + 525*e^5 + 694*e^4 - 1771*e^3 - 131*e^2 + 812*e + 195, 24*e^8 - 55*e^7 - 292*e^6 + 683*e^5 + 875*e^4 - 2268*e^3 - 7*e^2 + 897*e + 183, -18*e^8 + 42*e^7 + 220*e^6 - 522*e^5 - 674*e^4 + 1746*e^3 + 80*e^2 - 758*e - 166, 20*e^8 - 44*e^7 - 250*e^6 + 551*e^5 + 812*e^4 - 1872*e^3 - 278*e^2 + 890*e + 239, 13*e^8 - 29*e^7 - 160*e^6 + 361*e^5 + 502*e^4 - 1215*e^3 - 119*e^2 + 552*e + 155, -33*e^8 + 76*e^7 + 404*e^6 - 947*e^5 - 1238*e^4 + 3173*e^3 + 136*e^2 - 1356*e - 313, 21*e^8 - 49*e^7 - 258*e^6 + 611*e^5 + 800*e^4 - 2052*e^3 - 126*e^2 + 912*e + 213, 21*e^8 - 50*e^7 - 260*e^6 + 626*e^5 + 826*e^4 - 2119*e^3 - 212*e^2 + 996*e + 260, 17*e^8 - 39*e^7 - 210*e^6 + 487*e^5 + 665*e^4 - 1645*e^3 - 177*e^2 + 767*e + 209, -2*e^8 + 6*e^7 + 21*e^6 - 72*e^5 - 30*e^4 + 216*e^3 - 150*e^2 - 9*e + 44, -2*e^8 + 2*e^7 + 28*e^6 - 23*e^5 - 122*e^4 + 77*e^3 + 171*e^2 - 54*e - 49, -23*e^8 + 53*e^7 + 283*e^6 - 659*e^5 - 882*e^4 + 2201*e^3 + 163*e^2 - 939*e - 249, -10*e^8 + 23*e^7 + 126*e^6 - 289*e^5 - 422*e^4 + 992*e^3 + 199*e^2 - 522*e - 135, -18*e^8 + 43*e^7 + 216*e^6 - 531*e^5 - 625*e^4 + 1750*e^3 - 81*e^2 - 679*e - 115, 21*e^8 - 45*e^7 - 262*e^6 + 562*e^5 + 850*e^4 - 1907*e^3 - 295*e^2 + 914*e + 246, -2*e^8 + 4*e^7 + 25*e^6 - 49*e^5 - 82*e^4 + 162*e^3 + 32*e^2 - 65*e - 21, -12*e^8 + 27*e^7 + 148*e^6 - 336*e^5 - 463*e^4 + 1128*e^3 + 85*e^2 - 501*e - 108, -20*e^8 + 46*e^7 + 248*e^6 - 575*e^5 - 784*e^4 + 1935*e^3 + 171*e^2 - 856*e - 201, -25*e^8 + 60*e^7 + 305*e^6 - 747*e^5 - 929*e^4 + 2497*e^3 + 88*e^2 - 1080*e - 253, -5*e^8 + 12*e^7 + 58*e^6 - 146*e^5 - 151*e^4 + 465*e^3 - 88*e^2 - 121*e, -48*e^8 + 110*e^7 + 591*e^6 - 1370*e^5 - 1851*e^4 + 4604*e^3 + 390*e^2 - 2068*e - 510, -5*e^8 + 11*e^7 + 60*e^6 - 134*e^5 - 174*e^4 + 429*e^3 - 23*e^2 - 126*e - 10, -26*e^8 + 61*e^7 + 316*e^6 - 759*e^5 - 952*e^4 + 2537*e^3 + 50*e^2 - 1078*e - 241, 18*e^8 - 41*e^7 - 221*e^6 + 509*e^5 + 688*e^4 - 1705*e^3 - 134*e^2 + 761*e + 209, -24*e^8 + 52*e^7 + 301*e^6 - 652*e^5 - 988*e^4 + 2224*e^3 + 380*e^2 - 1087*e - 294, -19*e^8 + 44*e^7 + 237*e^6 - 553*e^5 - 763*e^4 + 1881*e^3 + 222*e^2 - 894*e - 219, 20*e^8 - 49*e^7 - 240*e^6 + 609*e^5 + 692*e^4 - 2022*e^3 + 101*e^2 + 806*e + 135, 32*e^8 - 75*e^7 - 392*e^6 + 936*e^5 + 1207*e^4 - 3150*e^3 - 159*e^2 + 1413*e + 312, -44*e^8 + 99*e^7 + 544*e^6 - 1234*e^5 - 1721*e^4 + 4158*e^3 + 427*e^2 - 1907*e - 478, 2*e^8 - 5*e^7 - 27*e^6 + 66*e^5 + 102*e^4 - 242*e^3 - 81*e^2 + 163*e + 68, -62*e^8 + 144*e^7 + 763*e^6 - 1798*e^5 - 2375*e^4 + 6048*e^3 + 410*e^2 - 2682*e - 636, -2*e^7 + 5*e^6 + 23*e^5 - 60*e^4 - 62*e^3 + 182*e^2 - 17*e - 39, -39*e^8 + 89*e^7 + 478*e^6 - 1110*e^5 - 1472*e^4 + 3735*e^3 + 203*e^2 - 1668*e - 378, -35*e^8 + 77*e^7 + 436*e^6 - 960*e^5 - 1410*e^4 + 3239*e^3 + 469*e^2 - 1498*e - 384, 20*e^8 - 46*e^7 - 248*e^6 + 576*e^5 + 788*e^4 - 1951*e^3 - 201*e^2 + 912*e + 230, 28*e^8 - 66*e^7 - 342*e^6 + 821*e^5 + 1052*e^4 - 2753*e^3 - 157*e^2 + 1230*e + 299, -44*e^8 + 104*e^7 + 541*e^6 - 1299*e^5 - 1680*e^4 + 4370*e^3 + 276*e^2 - 1951*e - 469, 11*e^8 - 24*e^7 - 142*e^6 + 305*e^5 + 504*e^4 - 1069*e^3 - 342*e^2 + 614*e + 203, 49*e^8 - 114*e^7 - 601*e^6 + 1422*e^5 + 1852*e^4 - 4767*e^3 - 244*e^2 + 2043*e + 474, 36*e^8 - 85*e^7 - 441*e^6 + 1059*e^5 + 1361*e^4 - 3552*e^3 - 203*e^2 + 1566*e + 383, -9*e^8 + 22*e^7 + 110*e^6 - 277*e^5 - 330*e^4 + 933*e^3 - 12*e^2 - 388*e - 61, -36*e^8 + 85*e^7 + 442*e^6 - 1061*e^5 - 1372*e^4 + 3576*e^3 + 233*e^2 - 1636*e - 381, 53*e^8 - 122*e^7 - 652*e^6 + 1524*e^5 + 2030*e^4 - 5137*e^3 - 366*e^2 + 2316*e + 546, 25*e^8 - 56*e^7 - 312*e^6 + 701*e^5 + 1012*e^4 - 2381*e^3 - 342*e^2 + 1160*e + 285, -8*e^8 + 19*e^7 + 103*e^6 - 241*e^5 - 370*e^4 + 848*e^3 + 283*e^2 - 531*e - 187, 29*e^8 - 65*e^7 - 357*e^6 + 810*e^5 + 1110*e^4 - 2717*e^3 - 185*e^2 + 1179*e + 260, 31*e^8 - 68*e^7 - 386*e^6 + 847*e^5 + 1246*e^4 - 2857*e^3 - 410*e^2 + 1334*e + 363, -35*e^8 + 82*e^7 + 430*e^6 - 1022*e^5 - 1336*e^4 + 3428*e^3 + 235*e^2 - 1510*e - 366, 47*e^8 - 110*e^7 - 572*e^6 + 1367*e^5 + 1728*e^4 - 4565*e^3 - 106*e^2 + 1950*e + 431, 28*e^8 - 61*e^7 - 350*e^6 + 763*e^5 + 1132*e^4 - 2580*e^3 - 357*e^2 + 1180*e + 321, 23*e^8 - 51*e^7 - 289*e^6 + 640*e^5 + 948*e^4 - 2176*e^3 - 344*e^2 + 1037*e + 264, 30*e^8 - 70*e^7 - 368*e^6 + 872*e^5 + 1142*e^4 - 2926*e^3 - 204*e^2 + 1294*e + 330, -16*e^8 + 37*e^7 + 193*e^6 - 460*e^5 - 570*e^4 + 1542*e^3 - 11*e^2 - 671*e - 124, -13*e^8 + 24*e^7 + 168*e^6 - 303*e^5 - 588*e^4 + 1047*e^3 + 354*e^2 - 530*e - 181, 17*e^8 - 42*e^7 - 203*e^6 + 520*e^5 + 580*e^4 - 1723*e^3 + 106*e^2 + 705*e + 110, 6*e^8 - 13*e^7 - 74*e^6 + 162*e^5 + 228*e^4 - 537*e^3 - 26*e^2 + 192*e + 74, -31*e^8 + 71*e^7 + 384*e^6 - 888*e^5 - 1218*e^4 + 2999*e^3 + 307*e^2 - 1366*e - 346, 11*e^8 - 21*e^7 - 145*e^6 + 266*e^5 + 540*e^4 - 934*e^3 - 454*e^2 + 553*e + 206, 13*e^8 - 29*e^7 - 161*e^6 + 359*e^5 + 518*e^4 - 1203*e^3 - 175*e^2 + 561*e + 159, -51*e^8 + 121*e^7 + 621*e^6 - 1507*e^5 - 1878*e^4 + 5045*e^3 + 117*e^2 - 2173*e - 487, 26*e^8 - 64*e^7 - 312*e^6 + 796*e^5 + 904*e^4 - 2654*e^3 + 100*e^2 + 1110*e + 216, -e^8 - 2*e^7 + 22*e^6 + 24*e^5 - 163*e^4 - 49*e^3 + 424*e^2 - 127*e - 114, 17*e^8 - 38*e^7 - 211*e^6 + 473*e^5 + 672*e^4 - 1588*e^3 - 169*e^2 + 717*e + 153, 8*e^8 - 16*e^7 - 102*e^6 + 199*e^5 + 346*e^4 - 670*e^3 - 156*e^2 + 304*e + 63, -24*e^8 + 58*e^7 + 292*e^6 - 724*e^5 - 878*e^4 + 2428*e^3 + 22*e^2 - 1054*e - 210, -20*e^8 + 45*e^7 + 249*e^6 - 562*e^5 - 805*e^4 + 1902*e^3 + 267*e^2 - 898*e - 246, -27*e^8 + 63*e^7 + 331*e^6 - 781*e^5 - 1026*e^4 + 2599*e^3 + 177*e^2 - 1101*e - 279, 21*e^8 - 45*e^7 - 263*e^6 + 565*e^5 + 858*e^4 - 1929*e^3 - 307*e^2 + 937*e + 233, -29*e^8 + 68*e^7 + 356*e^6 - 848*e^5 - 1108*e^4 + 2854*e^3 + 205*e^2 - 1292*e - 306, -20*e^8 + 48*e^7 + 248*e^6 - 603*e^5 - 786*e^4 + 2044*e^3 + 190*e^2 - 948*e - 265, -2*e^8 + 8*e^7 + 14*e^6 - 94*e^5 + 46*e^4 + 270*e^3 - 350*e^2 + 34*e + 78, 49*e^8 - 116*e^7 - 597*e^6 + 1442*e^5 + 1810*e^4 - 4811*e^3 - 130*e^2 + 2031*e + 458, 53*e^8 - 122*e^7 - 648*e^6 + 1517*e^5 + 1984*e^4 - 5071*e^3 - 228*e^2 + 2176*e + 507, -30*e^8 + 66*e^7 + 374*e^6 - 825*e^5 - 1211*e^4 + 2800*e^3 + 408*e^2 - 1347*e - 355, 33*e^8 - 76*e^7 - 404*e^6 + 949*e^5 + 1230*e^4 - 3183*e^3 - 84*e^2 + 1352*e + 287, -45*e^8 + 101*e^7 + 553*e^6 - 1259*e^5 - 1712*e^4 + 4223*e^3 + 263*e^2 - 1817*e - 439, 21*e^8 - 51*e^7 - 256*e^6 + 634*e^5 + 776*e^4 - 2107*e^3 - 49*e^2 + 862*e + 198, -3*e^8 + 7*e^7 + 33*e^6 - 85*e^5 - 70*e^4 + 275*e^3 - 113*e^2 - 101*e + 17, -17*e^8 + 42*e^7 + 206*e^6 - 522*e^5 - 616*e^4 + 1736*e^3 + 13*e^2 - 718*e - 162, 31*e^8 - 72*e^7 - 377*e^6 + 897*e^5 + 1123*e^4 - 2985*e^3 + 34*e^2 + 1170*e + 219, -3*e^8 + 5*e^7 + 40*e^6 - 66*e^5 - 150*e^4 + 251*e^3 + 125*e^2 - 190*e - 48, 23*e^8 - 57*e^7 - 282*e^6 + 714*e^5 + 872*e^4 - 2417*e^3 - 135*e^2 + 1138*e + 288]; 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;