/* 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, -6, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3, 3, w + 1], [7, 7, w - 1], [8, 2, 2], [9, 3, -w^2 + 2*w + 4], [11, 11, -w^2 + 2*w + 2], [13, 13, -w^2 + w + 4], [19, 19, w + 3], [19, 19, -w^2 + 2*w + 5], [19, 19, -w^2 + 3*w + 2], [23, 23, w^2 - w - 3], [23, 23, -w^2 + 2], [23, 23, -w + 4], [31, 31, w^2 - 5], [43, 43, w^2 - 3*w - 3], [49, 7, w^2 - 6], [53, 53, 2*w - 5], [61, 61, 2*w^2 - 2*w - 9], [71, 71, 2*w - 3], [73, 73, 2*w^2 - 5*w - 5], [83, 83, w^2 - w - 9], [97, 97, 2*w^2 - 3*w - 7], [103, 103, w^2 - 3*w - 6], [109, 109, w^2 - 4*w - 3], [121, 11, 2*w^2 - w - 7], [125, 5, -5], [127, 127, w^2 + w - 5], [131, 131, w^2 - 11], [137, 137, 3*w^2 - 7*w - 8], [137, 137, 2*w^2 - 5*w - 6], [137, 137, -w^2 + w - 2], [139, 139, 3*w^2 - 2*w - 16], [139, 139, 3*w - 2], [139, 139, -w^2 + 3*w - 3], [151, 151, w^2 + w - 11], [157, 157, 2*w^2 - 3*w - 3], [163, 163, w^2 - w - 10], [169, 13, 2*w^2 - 3*w - 4], [173, 173, -3*w^2 + 6*w + 8], [191, 191, -4*w^2 + 7*w + 15], [193, 193, 2*w^2 - 3], [197, 197, w^2 - 3*w - 11], [199, 199, 3*w^2 - 5*w - 16], [211, 211, w^2 - 4*w - 9], [223, 223, 2*w^2 - w - 4], [227, 227, 3*w - 4], [229, 229, w^2 + w - 8], [241, 241, w^2 - 4*w - 6], [251, 251, w - 7], [257, 257, w^2 - 4*w - 7], [263, 263, 3*w^2 - 5*w - 10], [263, 263, 2*w^2 - 5*w - 15], [263, 263, -w^2 + 6*w - 1], [269, 269, -w^2 - w + 14], [277, 277, 2*w^2 - 5*w - 8], [277, 277, w^2 + 3*w - 3], [277, 277, 3*w^2 - 4*w - 12], [281, 281, -2*w - 7], [293, 293, 3*w^2 - 2*w - 12], [307, 307, 2*w^2 + w - 8], [313, 313, w^2 + 4*w - 2], [317, 317, 3*w^2 - 6*w - 5], [331, 331, 4*w^2 - 4*w - 19], [337, 337, -5*w^2 + 8*w + 21], [347, 347, 3*w^2 - 6*w - 13], [347, 347, 3*w^2 - 5*w - 22], [347, 347, 3*w^2 - 5*w - 9], [349, 349, w^2 - 5*w - 11], [353, 353, -w^2 + 6*w - 7], [359, 359, -4*w^2 + 8*w + 11], [367, 367, 3*w^2 - 4*w - 11], [373, 373, w^2 + 2*w - 6], [379, 379, 5*w^2 - 9*w - 18], [409, 409, -w^2 - 4], [439, 439, 3*w^2 - 6*w - 14], [443, 443, -w^2 - 2*w + 13], [443, 443, 2*w^2 - 15], [443, 443, 3*w^2 - 13], [449, 449, w^2 + 2*w - 7], [449, 449, 2*w^2 - 6*w - 7], [449, 449, 3*w^2 - 3*w - 11], [461, 461, -w^2 - 2*w - 5], [467, 467, 3*w^2 - 5*w - 7], [479, 479, -w^2 + 6*w - 4], [491, 491, 3*w^2 - 5*w - 6], [499, 499, w^2 - 5*w - 8], [509, 509, 3*w^2 - 6*w - 19], [521, 521, 3*w^2 - 2*w - 10], [523, 523, 4*w - 9], [541, 541, 3*w^2 - 3*w - 10], [547, 547, 2*w^2 + w - 20], [547, 547, w^2 - 6*w - 5], [547, 547, 6*w^2 - 11*w - 24], [557, 557, -5*w^2 + 11*w + 11], [563, 563, 3*w^2 - 6*w - 16], [569, 569, 4*w^2 - 5*w - 17], [587, 587, 5*w^2 - 8*w - 20], [593, 593, 5*w - 3], [593, 593, 3*w^2 - w - 17], [593, 593, w - 9], [613, 613, 2*w^2 - w - 19], [613, 613, 3*w^2 - 4*w - 5], [613, 613, 3*w^2 - w - 8], [617, 617, 3*w^2 - 2*w - 9], [617, 617, -w^2 + w - 5], [617, 617, 4*w^2 - 6*w - 15], [619, 619, -w^2 + 3*w - 6], [631, 631, 4*w^2 - 7*w - 21], [641, 641, -w^2 - 3*w + 20], [643, 643, 3*w^2 - 14], [643, 643, w^2 - w - 13], [643, 643, 2*w^2 - 9*w + 2], [647, 647, 2*w^2 - 6*w - 9], [653, 653, 3*w^2 - 5], [659, 659, 6*w^2 - 11*w - 21], [661, 661, w^2 - 6*w - 6], [661, 661, w^2 + 3*w - 6], [661, 661, 5*w^2 - 7*w - 22], [673, 673, w^2 - 6*w - 12], [677, 677, 4*w^2 - 7*w - 22], [677, 677, 3*w^2 - 7*w - 12], [677, 677, -2*w^2 - 3], [683, 683, -2*w - 9], [683, 683, 4*w^2 - 7*w - 12], [683, 683, 3*w^2 - 3*w - 8], [691, 691, 2*w^2 + w - 11], [701, 701, 3*w^2 - 3*w - 5], [709, 709, -w^2 - 2*w - 6], [727, 727, 3*w^2 - w - 6], [727, 727, 3*w^2 - w - 27], [727, 727, 3*w^2 - w - 5], [733, 733, -2*w^2 + 9*w - 5], [739, 739, 3*w^2 - 4*w - 23], [743, 743, 2*w^2 - 7*w - 7], [743, 743, 3*w^2 - w - 20], [743, 743, 3*w^2 - 2*w - 7], [757, 757, -w - 9], [757, 757, 4*w^2 - 2*w - 29], [757, 757, 2*w^2 - 3*w - 18], [761, 761, 3*w^2 - 2*w - 6], [761, 761, 3*w^2 - 2*w - 25], [773, 773, -5*w - 12], [787, 787, -3*w - 10], [809, 809, 4*w^2 - w - 19], [821, 821, 5*w^2 - 10*w - 19], [823, 823, -4*w - 11], [827, 827, -7*w - 5], [827, 827, 2*w^2 - 6*w - 13], [827, 827, -w^2 - 2*w + 18], [839, 839, 6*w^2 - 9*w - 26], [839, 839, 5*w^2 - 12*w - 13], [839, 839, w - 10], [857, 857, -w^2 + 4*w - 8], [859, 859, 3*w^2 - 8*w - 10], [859, 859, 4*w^2 - 9*w - 14], [859, 859, 3*w^2 - 26], [863, 863, 2*w^2 + 3*w - 7], [877, 877, -5*w^2 + 11*w + 9], [877, 877, -5*w^2 + 10*w + 13], [877, 877, -w^2 + w - 6], [881, 881, w^2 - 3*w - 14], [883, 883, -w^2 + 3*w - 7], [887, 887, -3*w^2 + 6*w - 2], [907, 907, w^2 + 3*w - 8], [911, 911, 4*w^2 - 6*w - 29], [937, 937, -6*w^2 + 11*w + 25], [953, 953, 6*w^2 - 6*w - 29], [961, 31, -w^2 + 7*w - 5], [967, 967, 2*w^2 + w - 14], [971, 971, 5*w^2 - 3*w - 27], [971, 971, 4*w - 15], [971, 971, w^2 - 2*w - 14], [977, 977, 3*w^2 - 5*w - 24], [983, 983, w^2 - 4*w - 15], [991, 991, 6*w^2 - 10*w - 23], [991, 991, 5*w^2 - 6*w - 22], [991, 991, 3*w^2 + w - 13]]; primes := [ideal : I in primesArray]; heckePol := x^8 - 2*x^7 - 17*x^6 + 29*x^5 + 91*x^4 - 124*x^3 - 149*x^2 + 136*x + 17; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -291/5419*e^7 + 1372/5419*e^6 + 2470/5419*e^5 - 15908/5419*e^4 - 3164/5419*e^3 + 48100/5419*e^2 - 555/5419*e - 24438/5419, 144/5419*e^7 - 232/5419*e^6 - 1334/5419*e^5 + 2453/5419*e^4 - 1898/5419*e^3 - 7154/5419*e^2 + 25135/5419*e + 3769/5419, -13/5419*e^7 + 322/5419*e^6 - 858/5419*e^5 - 2517/5419*e^4 + 7773/5419*e^3 + 119/5419*e^2 - 10360/5419*e + 26115/5419, -131/5419*e^7 - 90/5419*e^6 + 2192/5419*e^5 + 64/5419*e^4 - 11294/5419*e^3 + 12454/5419*e^2 + 17739/5419*e - 29884/5419, -216/5419*e^7 + 348/5419*e^6 + 2001/5419*e^5 - 970/5419*e^4 - 2572/5419*e^3 - 10945/5419*e^2 - 7898/5419*e + 24151/5419, 409/5419*e^7 - 960/5419*e^6 - 5520/5419*e^5 + 7908/5419*e^4 + 27650/5419*e^3 - 11664/5419*e^2 - 54639/5419*e + 15409/5419, 658/5419*e^7 - 458/5419*e^6 - 10762/5419*e^5 + 5263/5419*e^4 + 52592/5419*e^3 - 20196/5419*e^2 - 73382/5419*e + 36264/5419, -522/5419*e^7 + 841/5419*e^6 + 8900/5419*e^5 - 12279/5419*e^4 - 45955/5419*e^3 + 48964/5419*e^2 + 66714/5419*e - 30597/5419, -391/5419*e^7 + 931/5419*e^6 + 6708/5419*e^5 - 12343/5419*e^4 - 34661/5419*e^3 + 47348/5419*e^2 + 38137/5419*e - 44065/5419, 1, -316/5419*e^7 - 93/5419*e^6 + 6239/5419*e^5 + 2595/5419*e^4 - 34069/5419*e^3 - 17116/5419*e^2 + 41632/5419*e + 15362/5419, 844/5419*e^7 - 2564/5419*e^6 - 9324/5419*e^5 + 26269/5419*e^4 + 28916/5419*e^3 - 61499/5419*e^2 - 28949/5419*e + 16521/5419, 119/5419*e^7 - 1697/5419*e^6 + 2435/5419*e^5 + 15537/5419*e^4 - 27384/5419*e^3 - 29018/5419*e^2 + 34808/5419*e + 27312/5419, -1090/5419*e^7 + 1154/5419*e^6 + 14764/5419*e^5 - 7203/5419*e^4 - 57736/5419*e^3 - 7113/5419*e^2 + 57586/5419*e + 28295/5419, -550/5419*e^7 + 284/5419*e^6 + 7052/5419*e^5 + 641/5419*e^4 - 18792/5419*e^3 - 14974/5419*e^2 - 20211/5419*e - 7697/5419, 870/5419*e^7 - 3208/5419*e^6 - 7608/5419*e^5 + 31303/5419*e^4 + 18789/5419*e^3 - 72575/5419*e^2 - 51581/5419*e + 56414/5419, -1427/5419*e^7 + 1998/5419*e^6 + 19617/5419*e^5 - 16594/5419*e^4 - 80916/5419*e^3 + 22650/5419*e^2 + 100407/5419*e + 17480/5419, -64/5419*e^7 - 499/5419*e^6 + 6614/5419*e^5 + 114/5419*e^4 - 67195/5419*e^3 + 21845/5419*e^2 + 162839/5419*e - 28168/5419, 517/5419*e^7 - 1134/5419*e^6 - 9230/5419*e^5 + 13812/5419*e^4 + 50612/5419*e^3 - 30577/5419*e^2 - 61528/5419*e - 42728/5419, 303/5419*e^7 + 415/5419*e^6 - 7097/5419*e^5 - 5112/5419*e^4 + 47261/5419*e^3 + 22654/5419*e^2 - 73668/5419*e - 27180/5419, 769/5419*e^7 - 1540/5419*e^6 - 8855/5419*e^5 + 16750/5419*e^4 + 22905/5419*e^3 - 56644/5419*e^2 + 5489/5419*e + 108826/5419, 1008/5419*e^7 - 1624/5419*e^6 - 14757/5419*e^5 + 17171/5419*e^4 + 62580/5419*e^3 - 55497/5419*e^2 - 62491/5419*e + 48059/5419, 1622/5419*e^7 - 1409/5419*e^6 - 28423/5419*e^5 + 21835/5419*e^4 + 143148/5419*e^3 - 94882/5419*e^2 - 167186/5419*e + 51410/5419, 252/5419*e^7 - 406/5419*e^6 + 375/5419*e^5 + 2938/5419*e^4 - 27707/5419*e^3 - 4391/5419*e^2 + 61598/5419*e - 16435/5419, -315/5419*e^7 - 2202/5419*e^6 + 11724/5419*e^5 + 20713/5419*e^4 - 91358/5419*e^3 - 22961/5419*e^2 + 174986/5419*e - 56677/5419, -439/5419*e^7 - 798/5419*e^6 + 8959/5419*e^5 + 12128/5419*e^4 - 59317/5419*e^3 - 40584/5419*e^2 + 123688/5419*e - 5582/5419, -2050/5419*e^7 + 4507/5419*e^6 + 27270/5419*e^5 - 43426/5419*e^4 - 117336/5419*e^3 + 82126/5419*e^2 + 175420/5419*e - 4057/5419, 1423/5419*e^7 - 4400/5419*e^6 - 14462/5419*e^5 + 47083/5419*e^4 + 28284/5419*e^3 - 118488/5419*e^2 + 27295/5419*e + 43078/5419, -548/5419*e^7 + 1485/5419*e^6 + 7184/5419*e^5 - 17313/5419*e^4 - 30409/5419*e^3 + 49202/5419*e^2 + 35156/5419*e - 43/5419, -488/5419*e^7 - 418/5419*e^6 + 16563/5419*e^5 - 8614/5419*e^4 - 118807/5419*e^3 + 72413/5419*e^2 + 184265/5419*e - 25116/5419, 303/5419*e^7 + 415/5419*e^6 - 1678/5419*e^5 - 10531/5419*e^4 - 23186/5419*e^3 + 55168/5419*e^2 + 137673/5419*e - 43437/5419, 227/5419*e^7 - 1871/5419*e^6 - 1275/5419*e^5 + 21441/5419*e^4 - 9841/5419*e^3 - 53350/5419*e^2 + 49595/5419*e + 12527/5419, 1336/5419*e^7 + 256/5419*e^6 - 25623/5419*e^5 - 1025/5419*e^4 + 146165/5419*e^3 - 21817/5419*e^2 - 243374/5419*e + 51526/5419, -839/5419*e^7 + 2857/5419*e^6 + 4235/5419*e^5 - 27802/5419*e^4 + 26036/5419*e^3 + 59369/5419*e^2 - 90036/5419*e + 2614/5419, -535/5419*e^7 + 1163/5419*e^6 + 8042/5419*e^5 - 14796/5419*e^4 - 27344/5419*e^3 + 54502/5419*e^2 - 19512/5419*e - 53253/5419, -1387/5419*e^7 + 4342/5419*e^6 + 22257/5419*e^5 - 55953/5419*e^4 - 112753/5419*e^3 + 184437/5419*e^2 + 156461/5419*e - 73295/5419, -24/5419*e^7 + 1845/5419*e^6 - 1584/5419*e^5 - 22988/5419*e^4 + 14767/5419*e^3 + 48157/5419*e^2 - 14124/5419*e + 81560/5419, -1284/5419*e^7 - 1544/5419*e^6 + 29055/5419*e^5 + 5674/5419*e^4 - 166419/5419*e^3 + 43017/5419*e^2 + 219786/5419*e - 31349/5419, -31/5419*e^7 + 351/5419*e^6 - 2046/5419*e^5 + 1918/5419*e^4 + 20203/5419*e^3 - 46403/5419*e^2 - 42629/5419*e + 119799/5419, 999/5419*e^7 + 1100/5419*e^6 - 20770/5419*e^5 - 15835/5419*e^4 + 122985/5419*e^3 + 67555/5419*e^2 - 189715/5419*e - 35155/5419, -70/5419*e^7 + 1317/5419*e^6 - 4620/5419*e^5 - 5633/5419*e^4 + 43522/5419*e^3 - 29789/5419*e^2 - 73709/5419*e + 106021/5419, -755/5419*e^7 + 4528/5419*e^6 + 4360/5419*e^5 - 50305/5419*e^4 + 14994/5419*e^3 + 142803/5419*e^2 - 62278/5419*e - 93181/5419, 868/5419*e^7 + 1010/5419*e^6 - 18578/5419*e^5 + 486/5419*e^4 + 95434/5419*e^3 - 60885/5419*e^2 - 101529/5419*e + 54179/5419, -540/5419*e^7 + 6289/5419*e^6 - 3126/5419*e^5 - 72872/5419*e^4 + 74855/5419*e^3 + 213783/5419*e^2 - 198572/5419*e - 126578/5419, 762/5419*e^7 + 2385/5419*e^6 - 20155/5419*e^5 - 23372/5419*e^4 + 125883/5419*e^3 + 43880/5419*e^2 - 174748/5419*e + 17009/5419, 1944/5419*e^7 - 3132/5419*e^6 - 34266/5419*e^5 + 35825/5419*e^4 + 191137/5419*e^3 - 101998/5419*e^2 - 319086/5419*e + 80686/5419, 2129/5419*e^7 - 3129/5419*e^6 - 32894/5419*e^5 + 33294/5419*e^4 + 143465/5419*e^3 - 67009/5419*e^2 - 153314/5419*e - 62102/5419, 49/5419*e^7 - 380/5419*e^6 - 2185/5419*e^5 + 4485/5419*e^4 + 26976/5419*e^3 + 802/5419*e^2 - 76834/5419*e - 18399/5419, -398/5419*e^7 - 563/5419*e^6 + 827/5419*e^5 + 17982/5419*e^4 + 35803/5419*e^3 - 95983/5419*e^2 - 163776/5419*e + 97135/5419, 130/5419*e^7 + 2199/5419*e^6 - 13096/5419*e^5 - 7344/5419*e^4 + 101097/5419*e^3 - 66218/5419*e^2 - 118579/5419*e + 101923/5419, 783/5419*e^7 - 3971/5419*e^6 - 2512/5419*e^5 + 31966/5419*e^4 - 25900/5419*e^3 - 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53133/5419*e^4 - 314382/5419*e^3 + 120167/5419*e^2 + 493389/5419*e - 71887/5419, 1796/5419*e^7 + 5536/5419*e^6 - 38615/5419*e^5 - 71614/5419*e^4 + 210850/5419*e^3 + 259095/5419*e^2 - 254452/5419*e - 160570/5419, -1445/5419*e^7 + 2027/5419*e^6 + 18429/5419*e^5 - 12159/5419*e^4 - 63067/5419*e^3 - 34710/5419*e^2 + 100652/5419*e + 138259/5419, -2979/5419*e^7 - 3329/5419*e^6 + 68917/5419*e^5 + 21394/5419*e^4 - 430156/5419*e^3 + 28103/5419*e^2 + 666442/5419*e - 161627/5419, 1731/5419*e^7 + 7146/5419*e^6 - 42905/5419*e^5 - 95037/5419*e^4 + 276810/5419*e^3 + 313880/5419*e^2 - 441727/5419*e - 100442/5419, 2788/5419*e^7 - 277/5419*e^6 - 43590/5419*e^5 - 8353/5419*e^4 + 187539/5419*e^3 + 58682/5419*e^2 - 201722/5419*e + 43017/5419, 2992/5419*e^7 - 7831/5419*e^6 - 40964/5419*e^5 + 84084/5419*e^4 + 189366/5419*e^3 - 217887/5419*e^2 - 341780/5419*e + 48808/5419, -1668/5419*e^7 + 881/5419*e^6 + 36225/5419*e^5 - 26156/5419*e^4 - 184840/5419*e^3 + 157830/5419*e^2 + 113020/5419*e - 227452/5419, -2944/5419*e^7 - 1278/5419*e^6 + 49551/5419*e^5 + 32339/5419*e^4 - 256833/5419*e^3 - 154796/5419*e^2 + 397123/5419*e + 91536/5419, 2303/5419*e^7 - 1603/5419*e^6 - 43086/5419*e^5 + 4873/5419*e^4 + 259938/5419*e^3 + 32275/5419*e^2 - 435664/5419*e + 72734/5419, 2556/5419*e^7 - 4118/5419*e^6 - 42645/5419*e^5 + 31348/5419*e^4 + 234551/5419*e^3 + 16620/5419*e^2 - 408701/5419*e - 129539/5419, -634/5419*e^7 - 1387/5419*e^6 + 17765/5419*e^5 + 12306/5419*e^4 - 89035/5419*e^3 - 71313/5419*e^2 - 4617/5419*e + 201897/5419, 140/5419*e^7 - 2634/5419*e^6 - 1598/5419*e^5 + 49199/5419*e^4 + 5079/5419*e^3 - 254724/5419*e^2 - 15152/5419*e + 145612/5419, 3701/5419*e^7 - 12887/5419*e^6 - 32103/5419*e^5 + 108392/5419*e^4 + 51395/5419*e^3 - 86401/5419*e^2 + 7729/5419*e - 277908/5419, 1770/5419*e^7 - 4658/5419*e^6 - 29493/5419*e^5 + 53408/5419*e^4 + 139692/5419*e^3 - 82064/5419*e^2 - 128859/5419*e - 189625/5419, -926/5419*e^7 + 7513/5419*e^6 - 1507/5419*e^5 - 81329/5419*e^4 + 73470/5419*e^3 + 237325/5419*e^2 - 127688/5419*e - 281564/5419, -3003/5419*e^7 + 9354/5419*e^6 + 34819/5419*e^5 - 104555/5419*e^4 - 128182/5419*e^3 + 298439/5419*e^2 + 251312/5419*e - 69229/5419, 2280/5419*e^7 - 1867/5419*e^6 - 22928/5419*e^5 + 10841/5419*e^4 + 11494/5419*e^3 + 4140/5419*e^2 + 192952/5419*e - 42382/5419, -1416/5419*e^7 - 4944/5419*e^6 + 47438/5419*e^5 + 30972/5419*e^4 - 364279/5419*e^3 + 50478/5419*e^2 + 711099/5419*e - 173440/5419, -524/5419*e^7 - 360/5419*e^6 + 14187/5419*e^5 + 5675/5419*e^4 - 104785/5419*e^3 - 42307/5419*e^2 + 195593/5419*e + 48453/5419, 977/5419*e^7 - 6692/5419*e^6 - 546/5419*e^5 + 67860/5419*e^4 - 74368/5419*e^3 - 150671/5419*e^2 + 198344/5419*e + 86573/5419]; 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;