/* 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([1, 7, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([17, 17, 2*w^3 + w^2 - 10*w - 2]) primes_array = [ [2, 2, -w + 1],\ [4, 2, -w^2 - w + 3],\ [5, 5, w^3 - 5*w + 1],\ [11, 11, -w^3 + 6*w - 2],\ [13, 13, -w^2 + 4],\ [17, 17, 2*w^3 + w^2 - 10*w - 2],\ [37, 37, w^3 + w^2 - 6*w - 3],\ [37, 37, -w^3 + 6*w - 4],\ [41, 41, -w^3 + 5*w - 5],\ [43, 43, w^3 - 4*w + 4],\ [43, 43, -2*w^3 + 11*w - 2],\ [47, 47, -2*w^3 - w^2 + 12*w],\ [47, 47, 2*w - 1],\ [61, 61, -w^3 + w^2 + 6*w - 5],\ [71, 71, w^3 + w^2 - 4*w - 3],\ [71, 71, 2*w^3 + 2*w^2 - 9*w - 2],\ [79, 79, w^3 + w^2 - 6*w - 5],\ [81, 3, -3],\ [83, 83, -3*w^3 - w^2 + 16*w + 3],\ [97, 97, w^3 + w^2 - 6*w + 1],\ [103, 103, -2*w^3 - 2*w^2 + 7*w - 2],\ [103, 103, -3*w^3 + 14*w - 12],\ [107, 107, -w^3 + 3*w + 3],\ [125, 5, -2*w^3 - w^2 + 8*w - 2],\ [127, 127, w^3 - w^2 - 4*w + 1],\ [131, 131, 2*w^3 - 10*w + 3],\ [137, 137, -5*w^3 - 2*w^2 + 26*w + 2],\ [137, 137, w^3 - 7*w + 1],\ [139, 139, -2*w^3 - w^2 + 8*w],\ [149, 149, w^3 - 2*w^2 - 7*w + 9],\ [149, 149, -2*w^3 + 11*w],\ [151, 151, 2*w^2 + 2*w - 9],\ [151, 151, -5*w^3 - 2*w^2 + 27*w + 1],\ [151, 151, w^3 - 7*w + 3],\ [151, 151, 2*w^3 + 2*w^2 - 11*w - 8],\ [157, 157, w^3 - 7*w - 3],\ [169, 13, 6*w^3 + 4*w^2 - 32*w - 11],\ [173, 173, w^3 + w^2 - 4*w - 7],\ [191, 191, -3*w^3 + 14*w - 10],\ [193, 193, 2*w^3 - 12*w + 3],\ [197, 197, w + 4],\ [197, 197, w^3 + 2*w^2 - 6*w - 6],\ [197, 197, 2*w^3 + w^2 - 10*w + 2],\ [197, 197, 2*w^2 - 3],\ [199, 199, w^3 + 2*w^2 - 4*w - 4],\ [223, 223, -w^3 - 2*w^2 + 5*w + 7],\ [229, 229, 5*w^3 + 3*w^2 - 26*w - 5],\ [233, 233, 2*w + 3],\ [241, 241, 2*w^3 + 2*w^2 - 10*w - 3],\ [251, 251, w^3 - 2*w^2 - 5*w + 7],\ [263, 263, 2*w^3 + w^2 - 12*w + 2],\ [269, 269, 2*w^3 + 3*w^2 - 14*w - 4],\ [271, 271, 3*w^3 + 2*w^2 - 15*w - 7],\ [281, 281, 2*w^2 + w - 4],\ [281, 281, w^2 - 8],\ [283, 283, -3*w^3 + 18*w - 2],\ [283, 283, -w^3 - w^2 + 4*w - 3],\ [283, 283, -w^3 + w^2 + 6*w - 3],\ [283, 283, w^3 + 2*w^2 - 5*w - 3],\ [293, 293, -4*w^3 + 19*w - 16],\ [307, 307, -w^3 + 2*w^2 + 6*w - 8],\ [311, 311, 3*w^3 + 2*w^2 - 17*w - 9],\ [311, 311, w^3 + 2*w^2 - 6*w - 8],\ [313, 313, -4*w^3 - 2*w^2 + 17*w - 6],\ [317, 317, 2*w^3 - 9*w + 2],\ [317, 317, 3*w^2 + 2*w - 10],\ [331, 331, 5*w^3 + 4*w^2 - 26*w - 12],\ [337, 337, 2*w^3 + w^2 - 8*w + 6],\ [347, 347, 2*w^3 - 9*w + 4],\ [353, 353, w^3 - 5*w - 3],\ [359, 359, -w^3 + w + 3],\ [367, 367, -2*w^3 + 12*w - 7],\ [367, 367, 2*w^3 + 4*w^2 - 6*w - 7],\ [379, 379, -w^3 - 2*w^2 + w + 3],\ [379, 379, -3*w^2 - 2*w + 12],\ [397, 397, -2*w^3 - w^2 + 6*w + 2],\ [397, 397, -w^3 + 3*w - 7],\ [397, 397, -2*w^3 - 2*w^2 + 9*w + 6],\ [397, 397, w^3 + 2*w^2 - 5*w - 5],\ [401, 401, 3*w^3 + 3*w^2 - 16*w - 13],\ [401, 401, w^3 + 2*w^2 - 6*w - 4],\ [431, 431, 3*w^3 + 2*w^2 - 14*w - 4],\ [431, 431, 2*w^3 + w^2 - 10*w + 4],\ [433, 433, -2*w^3 + 2*w^2 + 13*w - 12],\ [433, 433, -w^3 + 4*w^2 + 8*w - 18],\ [439, 439, 2*w^3 + 2*w^2 - 8*w + 3],\ [439, 439, -w^3 + 3*w^2 + 6*w - 11],\ [443, 443, -7*w^3 - 2*w^2 + 39*w + 1],\ [457, 457, w^3 + 2*w^2 - 3*w - 7],\ [461, 461, w^2 - 2*w - 4],\ [461, 461, 4*w^3 + 2*w^2 - 20*w - 5],\ [463, 463, 3*w^3 + 2*w^2 - 13*w - 3],\ [467, 467, -3*w^3 + 15*w - 11],\ [467, 467, -4*w^3 + 18*w + 1],\ [479, 479, 2*w^2 + 2*w - 11],\ [487, 487, -3*w^3 - 2*w^2 + 19*w + 1],\ [487, 487, 4*w^3 + 2*w^2 - 19*w],\ [491, 491, 3*w^3 + 2*w^2 - 18*w],\ [499, 499, -w^3 + 3*w - 5],\ [499, 499, -6*w^3 - 2*w^2 + 31*w - 2],\ [523, 523, 2*w^2 + 3*w - 10],\ [541, 541, 2*w^3 + 3*w^2 - 8*w - 6],\ [547, 547, -3*w^3 - w^2 + 16*w - 3],\ [563, 563, -w^3 + 5*w - 7],\ [563, 563, 2*w^2 + w - 2],\ [571, 571, 3*w^3 + 2*w^2 - 12*w + 6],\ [577, 577, -2*w^3 + w^2 + 8*w - 6],\ [577, 577, -2*w^3 - 2*w^2 + 12*w + 11],\ [587, 587, -5*w^3 - w^2 + 26*w - 5],\ [593, 593, -w^3 + 8*w - 4],\ [599, 599, -w^3 + w^2 + 4*w - 9],\ [601, 601, 3*w^3 + w^2 - 14*w + 1],\ [607, 607, 2*w^2 + 2*w - 3],\ [607, 607, 4*w^3 - 24*w + 1],\ [613, 613, -3*w^3 - 3*w^2 + 16*w + 9],\ [617, 617, -2*w^3 + 13*w - 6],\ [619, 619, -4*w^3 + 20*w - 1],\ [631, 631, -5*w^3 - 2*w^2 + 29*w + 5],\ [643, 643, -w^3 - 2*w^2 + 6*w + 10],\ [647, 647, -5*w^3 - 4*w^2 + 20*w - 4],\ [647, 647, -3*w^3 - 3*w^2 + 18*w + 7],\ [653, 653, 4*w^3 - 20*w + 3],\ [653, 653, w^3 - 8*w + 2],\ [659, 659, -9*w^3 - 4*w^2 + 48*w + 4],\ [659, 659, -5*w^3 - 2*w^2 + 25*w - 1],\ [661, 661, 4*w^3 + 2*w^2 - 21*w],\ [673, 673, -3*w^3 + w^2 + 18*w - 9],\ [683, 683, 2*w^3 + 2*w^2 - 7*w - 4],\ [691, 691, -5*w^3 - 3*w^2 + 28*w + 7],\ [691, 691, 3*w^3 - 15*w + 5],\ [701, 701, -2*w^3 + 2*w^2 + 11*w - 10],\ [701, 701, -w^3 + 8*w - 10],\ [709, 709, -4*w^3 - w^2 + 20*w - 4],\ [709, 709, w^3 + 4*w^2 - w - 13],\ [719, 719, -2*w^3 + 12*w - 9],\ [739, 739, -6*w^3 - 4*w^2 + 29*w + 6],\ [739, 739, -w^3 + w^2 + 8*w - 7],\ [769, 769, -w^3 + w^2 + 8*w - 5],\ [769, 769, 3*w^3 - 2*w^2 - 15*w + 19],\ [773, 773, -4*w^3 + 22*w - 9],\ [797, 797, -2*w^3 + 13*w - 2],\ [827, 827, -w^3 + 6*w - 8],\ [827, 827, -4*w^2 - 3*w + 14],\ [839, 839, w^2 + 4*w - 6],\ [857, 857, 7*w^3 + 2*w^2 - 37*w + 1],\ [863, 863, 3*w^3 - 14*w + 2],\ [881, 881, -3*w^3 + 2*w^2 + 15*w - 13],\ [883, 883, 2*w^3 - w^2 - 10*w + 14],\ [887, 887, -5*w^3 - 4*w^2 + 25*w + 11],\ [887, 887, 3*w^3 - 14*w + 8],\ [907, 907, 4*w^3 - 20*w + 13],\ [907, 907, 2*w^2 + 4*w - 5],\ [907, 907, -3*w^3 + 18*w - 4],\ [907, 907, w - 6],\ [947, 947, 3*w^3 + 3*w^2 - 12*w - 5],\ [947, 947, 2*w^3 + 3*w^2 - 10*w - 10],\ [961, 31, 2*w^3 - w^2 - 12*w + 4],\ [961, 31, -w^3 + 2*w^2 + 5*w - 5],\ [971, 971, -4*w^3 - w^2 + 20*w - 2],\ [991, 991, -2*w^3 + 3*w^2 + 14*w - 18],\ [997, 997, 7*w^3 + 2*w^2 - 36*w + 6]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^13 + 3*x^12 - 13*x^11 - 43*x^10 + 52*x^9 + 219*x^8 - 43*x^7 - 473*x^6 - 121*x^5 + 398*x^4 + 168*x^3 - 99*x^2 - 36*x + 2 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -380/641*e^12 - 507/641*e^11 + 5766/641*e^10 + 6563/641*e^9 - 30728/641*e^8 - 29780/641*e^7 + 68410/641*e^6 + 56084/641*e^5 - 60615/641*e^4 - 35483/641*e^3 + 18075/641*e^2 + 2172/641*e - 437/641, 1565/641*e^12 + 2670/641*e^11 - 23595/641*e^10 - 36349/641*e^9 + 125505/641*e^8 + 174770/641*e^7 - 279329/641*e^6 - 352835/641*e^5 + 240740/641*e^4 + 261826/641*e^3 - 61359/641*e^2 - 42817/641*e - 604/641, 2185/641*e^12 + 3396/641*e^11 - 33475/641*e^10 - 45910/641*e^9 + 182455/641*e^8 + 219951/641*e^7 - 425087/641*e^6 - 446196/641*e^5 + 408630/641*e^4 + 337195/641*e^3 - 139667/641*e^2 - 58000/641*e + 7160/641, -104/641*e^12 - 800/641*e^11 + 1099/641*e^10 + 11897/641*e^9 - 3329/641*e^8 - 61522/641*e^7 + 741/641*e^6 + 129589/641*e^5 + 10670/641*e^4 - 100443/641*e^3 - 8278/641*e^2 + 18394/641*e + 906/641, 1, 616/641*e^12 + 498/641*e^11 - 9320/641*e^10 - 5430/641*e^9 + 48415/641*e^8 + 19295/641*e^7 - 99898/641*e^6 - 25830/641*e^5 + 71608/641*e^4 + 5754/641*e^3 - 8856/641*e^2 + 1155/641*e - 830/641, 1140/641*e^12 + 2162/641*e^11 - 17298/641*e^10 - 29945/641*e^9 + 94107/641*e^8 + 146389/641*e^7 - 222537/641*e^6 - 299657/641*e^5 + 226074/641*e^4 + 227598/641*e^3 - 93326/641*e^2 - 40489/641*e + 5798/641, -5595/641*e^12 - 9312/641*e^11 + 85251/641*e^10 + 127020/641*e^9 - 461707/641*e^8 - 612992/641*e^7 + 1065224/641*e^6 + 1245888/641*e^5 - 1001269/641*e^4 - 934730/641*e^3 + 324217/641*e^2 + 151442/641*e - 12684/641, -225/641*e^12 - 646/641*e^11 + 3296/641*e^10 + 9461/641*e^9 - 17452/641*e^8 - 49413/641*e^7 + 41265/641*e^6 + 110465/641*e^5 - 45244/641*e^4 - 98208/641*e^3 + 23497/641*e^2 + 22574/641*e - 2342/641, -339/641*e^12 - 93/641*e^11 + 5154/641*e^10 + 597/641*e^9 - 25773/641*e^8 - 1298/641*e^7 + 44481/641*e^6 + 7295/641*e^5 - 6700/641*e^4 - 17895/641*e^3 - 27168/641*e^2 + 12585/641*e + 3360/641, -5595/641*e^12 - 8671/641*e^11 + 85251/641*e^10 + 116764/641*e^9 - 459784/641*e^8 - 556584/641*e^7 + 1047276/641*e^6 + 1121534/641*e^5 - 953194/641*e^4 - 837939/641*e^3 + 283193/641*e^2 + 143109/641*e - 11402/641, 2824/641*e^12 + 4564/641*e^11 - 42810/641*e^10 - 62113/641*e^9 + 229147/641*e^8 + 300594/641*e^7 - 515614/641*e^6 - 619794/641*e^5 + 459844/641*e^4 + 484752/641*e^3 - 132504/641*e^2 - 95342/641*e + 940/641, 2647/641*e^12 + 4090/641*e^11 - 40465/641*e^10 - 54790/641*e^9 + 219888/641*e^8 + 258620/641*e^7 - 509305/641*e^6 - 512041/641*e^5 + 481566/641*e^4 + 370676/641*e^3 - 155924/641*e^2 - 57294/641*e + 4294/641, -5482/641*e^12 - 8640/641*e^11 + 84174/641*e^10 + 117206/641*e^9 - 460808/641*e^8 - 563416/641*e^7 + 1081806/641*e^6 + 1145597/641*e^5 - 1049461/641*e^4 - 865947/641*e^3 + 351862/641*e^2 + 145324/641*e - 8676/641, -3058/641*e^12 - 5723/641*e^11 + 46084/641*e^10 + 79106/641*e^9 - 246733/641*e^8 - 386777/641*e^7 + 561350/641*e^6 + 797111/641*e^5 - 513718/641*e^4 - 621169/641*e^3 + 156505/641*e^2 + 123588/641*e - 3068/641, -4365/641*e^12 - 7148/641*e^11 + 66891/641*e^10 + 97393/641*e^9 - 365619/641*e^8 - 469401/641*e^7 + 857590/641*e^6 + 953966/641*e^5 - 834402/641*e^4 - 722462/641*e^3 + 293797/641*e^2 + 127307/641*e - 16718/641, -3355/641*e^12 - 5345/641*e^11 + 51127/641*e^10 + 72544/641*e^9 - 276154/641*e^8 - 349596/641*e^7 + 631332/641*e^6 + 716139/641*e^5 - 577363/641*e^4 - 553914/641*e^3 + 168650/641*e^2 + 103721/641*e - 3698/641, 3725/641*e^12 + 5923/641*e^11 - 56775/641*e^10 - 79997/641*e^9 + 307018/641*e^8 + 381966/641*e^7 - 704959/641*e^6 - 769094/641*e^5 + 656878/641*e^4 + 574007/641*e^3 - 213728/641*e^2 - 99662/641*e + 12136/641, -3861/641*e^12 - 5342/641*e^11 + 59149/641*e^10 + 70457/641*e^9 - 320296/641*e^8 - 328153/641*e^7 + 732209/641*e^6 + 645373/641*e^5 - 674186/641*e^4 - 463452/641*e^3 + 206601/641*e^2 + 73126/641*e - 6908/641, 759/641*e^12 + 2239/641*e^11 - 11392/641*e^10 - 32445/641*e^9 + 63008/641*e^8 + 164302/641*e^7 - 157405/641*e^6 - 341269/641*e^5 + 172477/641*e^4 + 259163/641*e^3 - 76477/641*e^2 - 41604/641*e + 8020/641, 1944/641*e^12 + 2479/641*e^11 - 29862/641*e^10 - 32104/641*e^9 + 161631/641*e^8 + 146478/641*e^7 - 365760/641*e^6 - 282906/641*e^5 + 321193/641*e^4 + 200261/641*e^3 - 76814/641*e^2 - 32251/641*e - 5200/641, -7889/641*e^12 - 13793/641*e^11 + 119243/641*e^10 + 188869/641*e^9 - 638449/641*e^8 - 913682/641*e^7 + 1444535/641*e^6 + 1855608/641*e^5 - 1304526/641*e^4 - 1389678/641*e^3 + 395632/641*e^2 + 231272/641*e - 16540/641, 1945/641*e^12 + 3818/641*e^11 - 29361/641*e^10 - 53168/641*e^9 + 158156/641*e^8 + 261599/641*e^7 - 364405/641*e^6 - 540324/641*e^5 + 337892/641*e^4 + 417817/641*e^3 - 94177/641*e^2 - 73834/641*e - 8500/641, 3926/641*e^12 + 7124/641*e^11 - 59275/641*e^10 - 98645/641*e^9 + 316848/641*e^8 + 484388/641*e^7 - 714003/641*e^6 - 1003839/641*e^5 + 635307/641*e^4 + 776299/641*e^3 - 181396/641*e^2 - 138306/641*e + 2656/641, 9953/641*e^12 + 16061/641*e^11 - 151803/641*e^10 - 217985/641*e^9 + 821524/641*e^8 + 1046387/641*e^7 - 1885506/641*e^6 - 2115808/641*e^5 + 1739290/641*e^4 + 1577191/641*e^3 - 524036/641*e^2 - 254324/641*e + 10196/641, 264/641*e^12 + 946/641*e^11 - 4269/641*e^10 - 14323/641*e^9 + 26793/641*e^8 + 75208/641*e^7 - 81365/641*e^6 - 157218/641*e^5 + 117682/641*e^4 + 103744/641*e^3 - 69269/641*e^2 + 6905/641*e + 5688/641, 4979/641*e^12 + 7532/641*e^11 - 75931/641*e^10 - 101078/641*e^9 + 409446/641*e^8 + 480240/641*e^7 - 930712/641*e^6 - 966222/641*e^5 + 843126/641*e^4 + 725138/641*e^3 - 249338/641*e^2 - 128880/641*e + 9668/641, 4090/641*e^12 + 7498/641*e^11 - 61723/641*e^10 - 103920/641*e^9 + 330258/641*e^8 + 511140/641*e^7 - 747542/641*e^6 - 1062462/641*e^5 + 673410/641*e^4 + 825498/641*e^3 - 193785/641*e^2 - 147934/641*e - 1386/641, 7647/641*e^12 + 13460/641*e^11 - 115490/641*e^10 - 184767/641*e^9 + 617254/641*e^8 + 897517/641*e^7 - 1389768/641*e^6 - 1835525/641*e^5 + 1233081/641*e^4 + 1389661/641*e^3 - 343620/641*e^2 - 229322/641*e + 6198/641, -6551/641*e^12 - 11242/641*e^11 + 99224/641*e^10 + 154111/641*e^9 - 532420/641*e^8 - 748706/641*e^7 + 1208248/641*e^6 + 1535149/641*e^5 - 1098199/641*e^4 - 1168243/641*e^3 + 333524/641*e^2 + 199968/641*e - 10322/641, -3184/641*e^12 - 5213/641*e^11 + 48340/641*e^10 + 71097/641*e^9 - 259506/641*e^8 - 345169/641*e^7 + 585484/641*e^6 + 718336/641*e^5 - 514671/641*e^4 - 585605/641*e^3 + 128306/641*e^2 + 135050/641*e + 2492/641, 648/641*e^12 - 242/641*e^11 - 10595/641*e^10 + 5751/641*e^9 + 60928/641*e^8 - 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2261418/641*e^6 - 1995465/641*e^5 + 2266572/641*e^4 + 1470384/641*e^3 - 820143/641*e^2 - 234985/641*e + 33732/641, -10857/641*e^12 - 15027/641*e^11 + 166829/641*e^10 + 199065/641*e^9 - 906277/641*e^8 - 932118/641*e^7 + 2072709/641*e^6 + 1839974/641*e^5 - 1875528/641*e^4 - 1305693/641*e^3 + 524662/641*e^2 + 166735/641*e - 16620/641, -21002/641*e^12 - 36904/641*e^11 + 317949/641*e^10 + 506368/641*e^9 - 1708011/641*e^8 - 2456961/641*e^7 + 3890996/641*e^6 + 5012742/641*e^5 - 3562823/641*e^4 - 3789245/641*e^3 + 1097910/641*e^2 + 655946/641*e - 18832/641, -8122/641*e^12 - 12331/641*e^11 + 124300/641*e^10 + 166850/641*e^9 - 675535/641*e^8 - 805260/641*e^7 + 1566623/641*e^6 + 1666537/641*e^5 - 1503874/641*e^4 - 1309180/641*e^3 + 518291/641*e^2 + 228832/641*e - 32224/641] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([17, 17, 2*w^3 + w^2 - 10*w - 2])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]