/* This code can be loaded, or copied and paste using cpaste, into Sage. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. */ P. = PolynomialRing(QQ) g = P([11, 6, -8, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([31,31,2/3*w^3 - 7/3*w + 2/3]) primes_array = [ [9, 3, 1/3*w^3 - 8/3*w - 2/3],\ [9, 3, -w + 1],\ [11, 11, w],\ [11, 11, -1/3*w^3 + 2/3*w + 5/3],\ [11, 11, 1/3*w^3 - 8/3*w + 1/3],\ [11, 11, -2/3*w^3 + 13/3*w + 4/3],\ [16, 2, 2],\ [25, 5, 2/3*w^3 - 10/3*w - 1/3],\ [29, 29, -w - 3],\ [29, 29, -1/3*w^3 + 8/3*w - 10/3],\ [31, 31, w^3 - 6*w + 1],\ [31, 31, w^3 - 6*w - 2],\ [41, 41, 2/3*w^3 + w^2 - 13/3*w - 10/3],\ [41, 41, 2/3*w^3 - 13/3*w + 5/3],\ [59, 59, 2/3*w^3 - 13/3*w + 8/3],\ [59, 59, w^3 + w^2 - 6*w - 4],\ [79, 79, 2/3*w^3 + w^2 - 10/3*w - 19/3],\ [79, 79, 1/3*w^3 + w^2 - 2/3*w - 17/3],\ [89, 89, 4/3*w^3 - 23/3*w - 5/3],\ [89, 89, 1/3*w^3 + 2*w^2 - 5/3*w - 32/3],\ [101, 101, 5/3*w^3 + w^2 - 31/3*w - 16/3],\ [101, 101, 2/3*w^3 - w^2 - 4/3*w + 8/3],\ [109, 109, -w^3 - 2*w^2 + 7*w + 7],\ [109, 109, -1/3*w^3 + 2*w^2 + 2/3*w - 16/3],\ [131, 131, w^3 - w^2 - 7*w + 5],\ [131, 131, -2/3*w^3 - w^2 + 4/3*w + 13/3],\ [139, 139, -2/3*w^3 - w^2 + 13/3*w + 4/3],\ [139, 139, w^2 - w - 7],\ [151, 151, -5/3*w^3 + 28/3*w - 5/3],\ [151, 151, -1/3*w^3 + w^2 + 2/3*w - 16/3],\ [151, 151, w^3 + w^2 - 6*w - 3],\ [151, 151, -w^3 + w^2 + 7*w - 6],\ [179, 179, 5/3*w^3 + w^2 - 28/3*w - 19/3],\ [179, 179, -1/3*w^3 + w^2 + 11/3*w - 4/3],\ [181, 181, -4/3*w^3 - 2*w^2 + 20/3*w + 29/3],\ [181, 181, 1/3*w^3 + w^2 - 2/3*w - 20/3],\ [181, 181, w^3 + w^2 - 5*w - 7],\ [181, 181, 5/3*w^3 + w^2 - 28/3*w - 16/3],\ [191, 191, 2*w - 1],\ [191, 191, 2/3*w^3 - 16/3*w - 1/3],\ [199, 199, 1/3*w^3 - 11/3*w + 4/3],\ [199, 199, 1/3*w^3 - 11/3*w - 2/3],\ [211, 211, -2/3*w^3 - w^2 + 19/3*w + 16/3],\ [211, 211, 2/3*w^3 + w^2 - 19/3*w - 7/3],\ [229, 229, 1/3*w^3 - 11/3*w + 1/3],\ [239, 239, -5/3*w^3 - w^2 + 25/3*w + 19/3],\ [239, 239, -4/3*w^3 + w^2 + 20/3*w - 7/3],\ [251, 251, -2/3*w^3 + w^2 + 16/3*w - 14/3],\ [251, 251, -1/3*w^3 - w^2 - 1/3*w + 14/3],\ [269, 269, 2/3*w^3 + w^2 - 7/3*w - 19/3],\ [269, 269, 1/3*w^3 + 2*w^2 - 2/3*w - 32/3],\ [281, 281, w^3 - w^2 - 5*w + 1],\ [281, 281, 4/3*w^3 + w^2 - 20/3*w - 23/3],\ [289, 17, 2/3*w^3 + w^2 - 16/3*w - 4/3],\ [289, 17, 1/3*w^3 + w^2 - 11/3*w - 20/3],\ [331, 331, 1/3*w^3 + 2*w^2 - 8/3*w - 23/3],\ [331, 331, 2/3*w^3 + 2*w^2 - 13/3*w - 28/3],\ [349, 349, 4/3*w^3 - 17/3*w + 1/3],\ [349, 349, -5/3*w^3 + 28/3*w - 2/3],\ [361, 19, -1/3*w^3 + 5/3*w + 14/3],\ [361, 19, 4/3*w^3 - 20/3*w - 5/3],\ [379, 379, 4/3*w^3 + 2*w^2 - 23/3*w - 41/3],\ [379, 379, -5/3*w^3 + 31/3*w - 5/3],\ [389, 389, 5/3*w^3 + w^2 - 25/3*w - 16/3],\ [389, 389, -4/3*w^3 + w^2 + 20/3*w - 10/3],\ [401, 401, -1/3*w^3 - 3*w^2 + 5/3*w + 26/3],\ [401, 401, 1/3*w^3 + w^2 - 11/3*w - 23/3],\ [401, 401, -2/3*w^3 - 3*w^2 + 10/3*w + 52/3],\ [401, 401, 2/3*w^3 + w^2 - 16/3*w - 1/3],\ [409, 409, -2*w^3 - w^2 + 11*w + 2],\ [409, 409, -5/3*w^3 - 2*w^2 + 28/3*w + 37/3],\ [409, 409, 1/3*w^3 - 11/3*w - 17/3],\ [409, 409, 5/3*w^3 + 3*w^2 - 25/3*w - 37/3],\ [419, 419, -1/3*w^3 + 3*w^2 - 1/3*w - 37/3],\ [419, 419, 2*w^3 + 3*w^2 - 12*w - 13],\ [421, 421, -1/3*w^3 + 3*w^2 + 2/3*w - 31/3],\ [421, 421, -5/3*w^3 - w^2 + 25/3*w + 13/3],\ [421, 421, -w^3 + 7*w - 4],\ [421, 421, -4/3*w^3 + w^2 + 20/3*w - 13/3],\ [439, 439, -4/3*w^3 + w^2 + 14/3*w - 7/3],\ [439, 439, 4/3*w^3 - 17/3*w - 5/3],\ [439, 439, -7/3*w^3 - w^2 + 41/3*w + 17/3],\ [439, 439, 5/3*w^3 - 28/3*w - 4/3],\ [449, 449, w^3 + 3*w^2 - 6*w - 17],\ [449, 449, 1/3*w^3 + 3*w^2 - 8/3*w - 26/3],\ [449, 449, -4/3*w^3 + 26/3*w + 17/3],\ [449, 449, 7/3*w^3 + w^2 - 38/3*w - 8/3],\ [461, 461, -5/3*w^3 + w^2 + 28/3*w - 14/3],\ [461, 461, -2/3*w^3 - 2*w^2 + 10/3*w + 43/3],\ [461, 461, -2*w^3 - w^2 + 12*w + 4],\ [461, 461, -w^3 + 8*w - 5],\ [479, 479, w^3 + w^2 - 7*w - 2],\ [479, 479, -4/3*w^3 - w^2 + 23/3*w + 5/3],\ [499, 499, -4/3*w^3 - 2*w^2 + 23/3*w + 17/3],\ [499, 499, -4/3*w^3 - w^2 + 29/3*w + 8/3],\ [521, 521, 2*w^3 - 11*w + 2],\ [521, 521, 4/3*w^3 - w^2 - 14/3*w + 1/3],\ [569, 569, w^3 + 3*w^2 - 6*w - 9],\ [569, 569, 4/3*w^3 + w^2 - 26/3*w - 32/3],\ [569, 569, -w^3 + 6*w - 6],\ [569, 569, -1/3*w^3 - 3*w^2 + 8/3*w + 50/3],\ [571, 571, 2/3*w^3 + 3*w^2 - 10/3*w - 46/3],\ [571, 571, 2*w^2 - 15],\ [599, 599, -w^3 + 2*w^2 + 4*w - 10],\ [599, 599, -w^2 - 3*w + 8],\ [601, 601, 5/3*w^3 + 3*w^2 - 28/3*w - 34/3],\ [601, 601, 1/3*w^3 - 3*w^2 - 2/3*w + 43/3],\ [619, 619, 2/3*w^3 - w^2 - 13/3*w + 2/3],\ [619, 619, 2/3*w^3 + w^2 - 7/3*w - 25/3],\ [619, 619, 2*w^3 + w^2 - 11*w - 6],\ [619, 619, -2/3*w^3 + 13/3*w - 17/3],\ [631, 631, 2/3*w^3 + 2*w^2 - 19/3*w - 31/3],\ [631, 631, w^3 + 2*w^2 - 8*w - 6],\ [631, 631, 2/3*w^3 + 2*w^2 - 4/3*w - 31/3],\ [631, 631, -2/3*w^3 + 2*w^2 + 16/3*w - 23/3],\ [659, 659, -2/3*w^3 + 1/3*w + 16/3],\ [659, 659, 5/3*w^3 - 34/3*w - 13/3],\ [661, 661, w^3 - 8*w + 2],\ [661, 661, 4/3*w^3 + w^2 - 23/3*w + 1/3],\ [661, 661, 5/3*w^3 + w^2 - 31/3*w - 10/3],\ [661, 661, -3*w - 1],\ [691, 691, -4/3*w^3 + 29/3*w + 2/3],\ [691, 691, 1/3*w^3 + 4/3*w - 5/3],\ [691, 691, -5/3*w^3 + w^2 + 25/3*w - 5/3],\ [691, 691, 2*w^3 + w^2 - 10*w - 7],\ [701, 701, -2/3*w^3 - 3*w^2 + 19/3*w + 46/3],\ [701, 701, w^3 - 2*w^2 - 6*w + 8],\ [701, 701, 4/3*w^3 + 2*w^2 - 17/3*w - 29/3],\ [701, 701, 5/3*w^3 + 3*w^2 - 28/3*w - 52/3],\ [709, 709, 1/3*w^3 - 14/3*w + 10/3],\ [709, 709, -2/3*w^3 + 19/3*w + 7/3],\ [719, 719, 3*w^2 - w - 10],\ [719, 719, -4/3*w^3 - 3*w^2 + 23/3*w + 47/3],\ [739, 739, 5/3*w^3 + w^2 - 34/3*w - 10/3],\ [739, 739, 4/3*w^3 + 2*w^2 - 23/3*w - 14/3],\ [761, 761, 7/3*w^3 + 2*w^2 - 41/3*w - 26/3],\ [761, 761, 2*w^3 + 3*w^2 - 12*w - 15],\ [769, 769, -2/3*w^3 + 19/3*w - 8/3],\ [769, 769, -4/3*w^3 + 2*w^2 + 17/3*w - 31/3],\ [769, 769, 1/3*w^3 - 14/3*w - 5/3],\ [769, 769, -1/3*w^3 + 3*w^2 + 2/3*w - 34/3],\ [809, 809, -4/3*w^3 + w^2 + 26/3*w - 7/3],\ [809, 809, w^3 + w^2 - 3*w - 7],\ [811, 811, 4/3*w^3 + w^2 - 17/3*w - 23/3],\ [811, 811, -1/3*w^3 - 2*w^2 + 2/3*w + 38/3],\ [821, 821, 8/3*w^3 + 3*w^2 - 43/3*w - 34/3],\ [821, 821, 2/3*w^3 + 2*w^2 - 7/3*w - 34/3],\ [821, 821, -w^3 - w^2 + 4*w + 8],\ [821, 821, 2/3*w^3 + 4*w^2 - 7/3*w - 58/3],\ [829, 829, 4/3*w^3 + 3*w^2 - 26/3*w - 50/3],\ [829, 829, 7/3*w^3 - 41/3*w - 5/3],\ [839, 839, 2*w^3 + w^2 - 10*w - 3],\ [839, 839, 5/3*w^3 - w^2 - 25/3*w + 17/3],\ [841, 29, 5/3*w^3 - 25/3*w - 7/3],\ [859, 859, -7/3*w^3 - w^2 + 32/3*w + 29/3],\ [859, 859, 2*w^3 + w^2 - 14*w - 6],\ [859, 859, 1/3*w^3 + 2*w^2 - 2/3*w - 47/3],\ [859, 859, -5/3*w^3 - 2*w^2 + 31/3*w + 49/3],\ [881, 881, w^3 + 4*w^2 - 4*w - 20],\ [881, 881, 8/3*w^3 + 2*w^2 - 43/3*w - 34/3],\ [919, 919, -w^3 + 3*w - 7],\ [919, 919, 1/3*w^3 + 3*w^2 - 5/3*w - 35/3],\ [919, 919, -5/3*w^3 + 31/3*w - 23/3],\ [919, 919, 2/3*w^3 + 3*w^2 - 10/3*w - 43/3],\ [929, 929, w^2 - 2*w - 9],\ [929, 929, w^3 + w^2 - 7*w + 1],\ [961, 31, 2/3*w^3 - 10/3*w - 19/3],\ [971, 971, -w^3 - 3*w^2 + 7*w + 16],\ [971, 971, -5/3*w^3 + w^2 + 25/3*w - 8/3],\ [971, 971, 2*w^3 + w^2 - 10*w - 6],\ [971, 971, 1/3*w^3 + 2*w^2 - 8/3*w - 47/3]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^7 - 2*x^6 - 41*x^5 + 46*x^4 + 456*x^3 - 168*x^2 - 1208*x - 496 K. = NumberField(heckePol) hecke_eigenvalues_array = [-2733/285556*e^6 + 3871/142778*e^5 + 99441/285556*e^4 - 82167/142778*e^3 - 236521/71389*e^2 + 86802/71389*e + 444954/71389, e, 876/71389*e^6 - 6243/285556*e^5 - 71505/142778*e^4 + 151843/285556*e^3 + 726465/142778*e^2 - 218804/71389*e - 566404/71389, -741/285556*e^6 + 1363/142778*e^5 + 15207/285556*e^4 - 9846/71389*e^3 + 8358/71389*e^2 + 11310/71389*e - 88276/71389, -741/285556*e^6 + 1363/142778*e^5 + 15207/285556*e^4 - 9846/71389*e^3 + 8358/71389*e^2 + 11310/71389*e - 88276/71389, 305/142778*e^6 - 9277/285556*e^5 - 2791/142778*e^4 + 277803/285556*e^3 - 27923/142778*e^2 - 407586/71389*e + 66504/71389, 397/285556*e^6 - 895/71389*e^5 - 4101/285556*e^4 + 55349/142778*e^3 - 82657/142778*e^2 - 178896/71389*e + 99223/71389, -378/71389*e^6 + 1227/285556*e^5 + 14694/71389*e^4 - 26893/285556*e^3 - 236707/142778*e^2 + 534/71389*e + 33174/71389, 5757/285556*e^6 - 2549/71389*e^5 - 216993/285556*e^4 + 54530/71389*e^3 + 473228/71389*e^2 - 230648/71389*e - 511302/71389, -107/285556*e^6 - 3171/285556*e^5 + 15491/285556*e^4 + 54141/285556*e^3 - 137763/142778*e^2 + 106838/71389*e + 467226/71389, 771/285556*e^6 + 1499/285556*e^5 - 43569/285556*e^4 - 12491/285556*e^3 + 253423/142778*e^2 - 60613/71389*e - 121450/71389, -1, -3581/285556*e^6 + 7297/285556*e^5 + 130139/285556*e^4 - 149577/285556*e^3 - 594089/142778*e^2 + 133561/71389*e + 823904/71389, 2283/285556*e^6 + 68/71389*e^5 - 102345/285556*e^4 + 7201/142778*e^3 + 245065/71389*e^2 - 203925/71389*e - 154624/71389, -3053/142778*e^6 + 5134/71389*e^5 + 116413/142778*e^4 - 138649/71389*e^3 - 594150/71389*e^2 + 773271/71389*e + 1318686/71389, -4831/285556*e^6 + 10813/142778*e^5 + 169665/285556*e^4 - 163327/71389*e^3 - 404191/71389*e^2 + 1069425/71389*e + 1186178/71389, -349/285556*e^6 + 2585/71389*e^5 + 15833/285556*e^4 - 84119/71389*e^3 - 120922/71389*e^2 + 471234/71389*e + 807384/71389, 5437/285556*e^6 - 3835/142778*e^5 - 200021/285556*e^4 + 26289/71389*e^3 + 896737/142778*e^2 - 37898/71389*e - 653858/71389, -6267/285556*e^6 + 2440/71389*e^5 + 270813/285556*e^4 - 132151/142778*e^3 - 734823/71389*e^2 + 426298/71389*e + 1078306/71389, 378/71389*e^6 - 1227/285556*e^5 - 14694/71389*e^4 + 26893/285556*e^3 + 379485/142778*e^2 - 214701/71389*e - 747064/71389, -916/71389*e^6 + 3753/142778*e^5 + 37874/71389*e^4 - 139857/142778*e^3 - 357815/71389*e^2 + 672124/71389*e + 495126/71389, -185/285556*e^6 - 3539/71389*e^5 + 32121/285556*e^4 + 118643/71389*e^3 - 148784/71389*e^2 - 635920/71389*e + 698402/71389, 14839/285556*e^6 - 19973/142778*e^5 - 579103/285556*e^4 + 263070/71389*e^3 + 2932671/142778*e^2 - 1511492/71389*e - 2303220/71389, -6213/285556*e^6 + 17365/285556*e^5 + 248317/285556*e^4 - 500455/285556*e^3 - 1326063/142778*e^2 + 665942/71389*e + 1357578/71389, 13025/285556*e^6 - 16829/142778*e^5 - 463261/285556*e^4 + 184630/71389*e^3 + 2051605/142778*e^2 - 762400/71389*e - 1938770/71389, -2709/285556*e^6 + 5561/142778*e^5 + 105307/285556*e^4 - 87153/71389*e^3 - 299799/71389*e^2 + 518527/71389*e + 791174/71389, 339/285556*e^6 + 3012/71389*e^5 - 6379/285556*e^4 - 119776/71389*e^3 - 126370/71389*e^2 + 949184/71389*e + 1213824/71389, 1829/142778*e^6 - 8351/285556*e^5 - 29317/71389*e^4 + 147311/285556*e^3 + 461713/142778*e^2 - 119707/71389*e - 653070/71389, -2881/142778*e^6 + 10695/142778*e^5 + 55430/71389*e^4 - 312955/142778*e^3 - 1265137/142778*e^2 + 1155024/71389*e + 1807462/71389, -6077/285556*e^6 + 6361/142778*e^5 + 233965/285556*e^4 - 82771/71389*e^3 - 996175/142778*e^2 + 566176/71389*e + 368746/71389, 6575/285556*e^6 - 3494/71389*e^5 - 219329/285556*e^4 + 127619/142778*e^3 + 398682/71389*e^2 - 156715/71389*e - 394970/71389, -6551/285556*e^6 + 4339/71389*e^5 + 225195/285556*e^4 - 109879/71389*e^3 - 390571/71389*e^2 + 445662/71389*e + 27300/71389, -2057/71389*e^6 + 7760/71389*e^5 + 74296/71389*e^4 - 501035/142778*e^3 - 1325695/142778*e^2 + 1823892/71389*e + 1428134/71389, -4165/142778*e^6 + 16675/285556*e^5 + 76987/71389*e^4 - 254583/285556*e^3 - 1430333/142778*e^2 - 135870/71389*e + 1170312/71389, 17127/285556*e^6 - 12717/71389*e^5 - 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6425/71389*e^5 - 166390/71389*e^4 + 99163/71389*e^3 + 1282046/71389*e^2 - 386108/71389*e + 21142/71389, -427/285556*e^6 - 645/285556*e^5 + 32463/285556*e^4 + 83955/285556*e^3 - 116093/142778*e^2 - 485691/71389*e - 389220/71389, 24205/285556*e^6 - 19323/71389*e^5 - 895595/285556*e^4 + 551108/71389*e^3 + 4158993/142778*e^2 - 3525590/71389*e - 5328580/71389, 24057/285556*e^6 - 74339/285556*e^5 - 955565/285556*e^4 + 2198589/285556*e^3 + 4891829/142778*e^2 - 3249047/71389*e - 5083970/71389, 25591/285556*e^6 - 24066/71389*e^5 - 949473/285556*e^4 + 1438767/142778*e^3 + 2280198/71389*e^2 - 4989554/71389*e - 5965796/71389, -18917/285556*e^6 + 15134/71389*e^5 + 665105/285556*e^4 - 386044/71389*e^3 - 1446495/71389*e^2 + 1678940/71389*e + 3322450/71389, 1547/285556*e^6 - 2049/71389*e^5 - 91865/285556*e^4 + 120015/142778*e^3 + 305680/71389*e^2 - 188934/71389*e + 2173168/71389, -4234/71389*e^6 + 65869/285556*e^5 + 309913/142778*e^4 - 1816641/285556*e^3 - 3047219/142778*e^2 + 2437740/71389*e + 2785212/71389, 4150/71389*e^6 - 10450/71389*e^5 - 350975/142778*e^4 + 663403/142778*e^3 + 1874084/71389*e^2 - 2342436/71389*e - 3920064/71389, 25813/285556*e^6 - 64999/285556*e^5 - 930907/285556*e^4 + 1636991/285556*e^3 + 4103643/142778*e^2 - 2441725/71389*e - 2834650/71389, -30347/285556*e^6 + 46379/142778*e^5 + 1191011/285556*e^4 - 638500/71389*e^3 - 6014127/142778*e^2 + 3544190/71389*e + 4985712/71389, 31921/285556*e^6 - 66423/142778*e^5 - 1079957/285556*e^4 + 1855547/142778*e^3 + 2331346/71389*e^2 - 5469416/71389*e - 5100134/71389, -1097/71389*e^6 + 182/71389*e^5 + 118149/142778*e^4 - 108807/142778*e^3 - 828562/71389*e^2 + 938672/71389*e + 4709364/71389, 4979/142778*e^6 - 24579/142778*e^5 - 90542/71389*e^4 + 659151/142778*e^3 + 1793039/142778*e^2 - 1264156/71389*e - 1291018/71389, -2298/71389*e^6 + 61851/285556*e^5 + 161663/142778*e^4 - 2024139/285556*e^3 - 1858297/142778*e^2 + 3270143/71389*e + 4321842/71389, -803/142778*e^6 + 11785/285556*e^5 - 11845/142778*e^4 - 126111/285556*e^3 + 853879/142778*e^2 - 54391/71389*e - 1299058/71389, 6305/285556*e^6 - 52001/285556*e^5 - 249627/285556*e^4 + 1864337/285556*e^3 + 1507229/142778*e^2 - 3782161/71389*e - 1934108/71389, 28273/285556*e^6 - 37747/142778*e^5 - 1114921/285556*e^4 + 1011763/142778*e^3 + 2883199/71389*e^2 - 2772343/71389*e - 5611604/71389, 13523/285556*e^6 - 8728/71389*e^5 - 591403/285556*e^4 + 272752/71389*e^3 + 1854454/71389*e^2 - 2137664/71389*e - 5391666/71389, 13119/285556*e^6 - 11054/71389*e^5 - 380795/285556*e^4 + 418869/142778*e^3 + 961019/142778*e^2 - 493308/71389*e - 606538/71389, 14739/285556*e^6 - 30233/285556*e^5 - 627341/285556*e^4 + 963437/285556*e^3 + 3126839/142778*e^2 - 2084835/71389*e - 795058/71389, 2546/71389*e^6 - 20563/142778*e^5 - 230803/142778*e^4 + 404401/71389*e^3 + 1323894/71389*e^2 - 3724890/71389*e - 2052360/71389, -2649/285556*e^6 + 4893/71389*e^5 + 48583/285556*e^4 - 77396/71389*e^3 - 130709/142778*e^2 - 222580/71389*e + 942834/71389] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([31,31,2/3*w^3 - 7/3*w + 2/3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]