/* 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, -2, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 2, 2]) primes_array = [ [5, 5, -w^3 + 2*w^2 + 3*w],\ [7, 7, w - 1],\ [11, 11, -w^3 + 2*w^2 + 4*w],\ [13, 13, -2*w^3 + 3*w^2 + 10*w - 2],\ [16, 2, 2],\ [17, 17, -w^3 + 2*w^2 + 5*w - 3],\ [17, 17, -w^3 + w^2 + 5*w],\ [17, 17, -w^2 + 2*w + 1],\ [19, 19, w^2 - w - 2],\ [29, 29, w^3 - 2*w^2 - 5*w],\ [29, 29, w^2 - w - 3],\ [31, 31, -w^3 + 2*w^2 + 3*w - 2],\ [43, 43, 2*w^3 - 3*w^2 - 11*w],\ [47, 47, w^3 - 7*w - 4],\ [53, 53, -2*w^3 + 3*w^2 + 9*w - 1],\ [61, 61, -w - 3],\ [73, 73, -w^3 + 2*w^2 + 3*w - 3],\ [73, 73, w^3 - w^2 - 7*w - 1],\ [81, 3, -3],\ [83, 83, -2*w^3 + 3*w^2 + 9*w + 1],\ [83, 83, -w^3 + 3*w^2 + 2*w - 3],\ [89, 89, w - 4],\ [103, 103, w^3 - 2*w^2 - 6*w + 3],\ [113, 113, 2*w^3 - w^2 - 14*w - 6],\ [121, 11, -w^2 + 2*w + 7],\ [125, 5, -3*w^3 + 4*w^2 + 15*w + 1],\ [139, 139, w^3 - 2*w^2 - 5*w - 3],\ [139, 139, w^3 - 5*w - 5],\ [151, 151, w^3 - 9*w - 7],\ [157, 157, -3*w^3 + 6*w^2 + 13*w - 6],\ [163, 163, w^3 - 3*w^2 - 2*w + 9],\ [167, 167, -w^3 + 7*w + 3],\ [167, 167, -3*w^3 + 4*w^2 + 15*w - 1],\ [173, 173, 2*w^3 - 3*w^2 - 7*w - 2],\ [181, 181, -w^3 + 3*w^2 + 4*w - 5],\ [191, 191, w^3 - w^2 - 4*w - 4],\ [193, 193, -w^3 + w^2 + 7*w + 6],\ [193, 193, 2*w^3 - 2*w^2 - 13*w - 5],\ [199, 199, -2*w^3 + 5*w^2 + 5*w - 5],\ [211, 211, w^3 - w^2 - 8*w - 3],\ [211, 211, -2*w^2 + 3*w + 8],\ [227, 227, -w^3 + 3*w^2 + 4*w - 4],\ [227, 227, w^2 - 3*w - 6],\ [229, 229, w^2 - 5],\ [229, 229, 3*w^3 - 4*w^2 - 17*w - 2],\ [233, 233, 3*w^3 - 6*w^2 - 13*w + 3],\ [233, 233, 3*w^3 - 4*w^2 - 16*w + 1],\ [241, 241, -2*w^3 + 4*w^2 + 8*w + 1],\ [251, 251, 2*w^3 - 2*w^2 - 9*w - 3],\ [277, 277, 3*w^3 - 4*w^2 - 17*w - 1],\ [281, 281, -2*w^2 + 5*w + 1],\ [283, 283, -4*w^3 + 6*w^2 + 19*w - 2],\ [293, 293, -3*w^3 + 6*w^2 + 12*w - 2],\ [307, 307, w^3 - w^2 - 4*w - 5],\ [307, 307, w^3 - w^2 - 8*w - 2],\ [311, 311, 2*w^2 - 3*w - 7],\ [313, 313, -2*w^3 + w^2 + 15*w + 3],\ [317, 317, -w^3 + 3*w^2 + w - 5],\ [331, 331, 2*w^3 - 3*w^2 - 12*w],\ [337, 337, w^3 - w^2 - 8*w - 1],\ [343, 7, -w^3 + 6*w + 8],\ [347, 347, 2*w^3 - 4*w^2 - 7*w + 5],\ [349, 349, 2*w^3 - 4*w^2 - 9*w],\ [353, 353, -3*w^3 + 4*w^2 + 16*w + 5],\ [353, 353, -3*w^3 + 4*w^2 + 14*w],\ [353, 353, -w^3 + 3*w^2 + 4*w - 8],\ [353, 353, -w^3 + 2*w^2 + 8*w - 3],\ [359, 359, -4*w^3 + 7*w^2 + 19*w - 3],\ [367, 367, -w^3 + 2*w^2 + 6*w - 5],\ [373, 373, 5*w^3 - 8*w^2 - 26*w + 3],\ [379, 379, -3*w^3 + 5*w^2 + 17*w - 4],\ [379, 379, -w^3 + 2*w^2 + 4*w - 6],\ [383, 383, w^2 - 6],\ [389, 389, w^3 - 2*w^2 - 3*w - 4],\ [389, 389, w^2 + w - 4],\ [397, 397, -4*w^3 + 5*w^2 + 23*w + 5],\ [397, 397, -3*w^3 + 5*w^2 + 12*w - 2],\ [419, 419, -3*w^3 + 4*w^2 + 17*w + 5],\ [419, 419, -3*w^3 + 5*w^2 + 15*w],\ [419, 419, 3*w^3 - 5*w^2 - 14*w - 1],\ [419, 419, 2*w^3 - 3*w^2 - 10*w - 6],\ [431, 431, 2*w^3 - 2*w^2 - 11*w],\ [433, 433, w^3 - 8*w - 1],\ [433, 433, 2*w^2 - 2*w - 5],\ [439, 439, 2*w^3 - 2*w^2 - 9*w - 6],\ [443, 443, 2*w^3 - 5*w^2 - 7*w],\ [443, 443, 2*w^3 - 5*w^2 - 9*w + 4],\ [457, 457, -2*w^3 + 5*w^2 + 7*w - 4],\ [457, 457, -w^3 + 2*w^2 - 3],\ [461, 461, -4*w^3 + 7*w^2 + 17*w - 7],\ [479, 479, 3*w^3 - 5*w^2 - 12*w + 6],\ [487, 487, -2*w^3 + 5*w^2 + 9*w - 9],\ [487, 487, w^3 - 10*w - 8],\ [499, 499, 2*w^3 - 2*w^2 - 12*w + 1],\ [503, 503, w^2 - 7],\ [503, 503, w^3 + w^2 - 7*w - 8],\ [529, 23, -3*w^3 + 5*w^2 + 12*w + 1],\ [529, 23, 3*w^3 - 4*w^2 - 14*w - 1],\ [541, 541, w^3 - 4*w^2 - 2*w + 11],\ [563, 563, -2*w^3 + w^2 + 15*w + 6],\ [569, 569, w^3 - 2*w^2 - 4*w - 4],\ [601, 601, 3*w^3 - 5*w^2 - 17*w + 1],\ [607, 607, -2*w^3 + 5*w^2 + 7*w - 5],\ [607, 607, -w^3 + 3*w^2 + 6*w - 5],\ [613, 613, -4*w^3 + 5*w^2 + 21*w + 1],\ [613, 613, -5*w^3 + 8*w^2 + 23*w - 4],\ [619, 619, w^3 + w^2 - 9*w - 5],\ [619, 619, -3*w^3 + 4*w^2 + 14*w + 2],\ [641, 641, -6*w^3 + 8*w^2 + 31*w + 3],\ [643, 643, 3*w^3 - 4*w^2 - 17*w - 6],\ [643, 643, -2*w^3 + 5*w^2 + 7*w - 6],\ [647, 647, 3*w^3 - 4*w^2 - 13*w - 10],\ [653, 653, -3*w^3 + 5*w^2 + 14*w + 2],\ [653, 653, w^3 - 2*w^2 - w - 3],\ [661, 661, 5*w^3 - 9*w^2 - 24*w + 5],\ [683, 683, 2*w^3 - 3*w^2 - 13*w - 2],\ [683, 683, 4*w^3 - 7*w^2 - 18*w + 1],\ [719, 719, -w^3 + 2*w^2 + 3*w - 7],\ [719, 719, 3*w^3 - 5*w^2 - 16*w - 1],\ [727, 727, -2*w^3 + 5*w^2 + 8*w - 7],\ [727, 727, 2*w^3 - 5*w^2 - 7*w + 13],\ [733, 733, -2*w^2 + 7],\ [733, 733, -3*w^3 + 5*w^2 + 11*w + 2],\ [739, 739, -6*w^3 + 9*w^2 + 29*w - 3],\ [739, 739, -2*w^3 + 3*w^2 + 7*w + 5],\ [751, 751, 4*w^3 - 5*w^2 - 23*w - 3],\ [757, 757, -w^3 + 10*w + 4],\ [757, 757, 2*w^3 - w^2 - 14*w - 4],\ [769, 769, -w^3 + 4*w^2 + 4*w - 8],\ [769, 769, 4*w^3 - 6*w^2 - 21*w - 2],\ [787, 787, 3*w^3 - 3*w^2 - 19*w - 6],\ [787, 787, -2*w^2 + 6*w + 7],\ [839, 839, -3*w^3 + 3*w^2 + 19*w + 5],\ [841, 29, 3*w^3 - 4*w^2 - 13*w - 4],\ [853, 853, -3*w^3 + 5*w^2 + 14*w + 3],\ [853, 853, 2*w^2 - w - 9],\ [853, 853, 3*w^3 - 5*w^2 - 11*w + 1],\ [853, 853, w - 6],\ [857, 857, 2*w^3 - 2*w^2 - 9*w - 7],\ [857, 857, -4*w^3 + 8*w^2 + 13*w - 1],\ [859, 859, 2*w^3 - 4*w^2 - 13*w + 2],\ [859, 859, w^3 - w^2 - 9*w - 2],\ [863, 863, -2*w^3 - w^2 + 17*w + 11],\ [877, 877, 5*w^3 - 6*w^2 - 28*w - 9],\ [877, 877, -4*w^3 + 5*w^2 + 20*w + 6],\ [881, 881, 4*w^3 - 5*w^2 - 18*w - 8],\ [911, 911, -w^3 + 4*w^2 - w - 8],\ [911, 911, 2*w^3 - 3*w^2 - 13*w - 1],\ [919, 919, w^3 - w^2 - 7*w + 5],\ [929, 929, 2*w^3 - 3*w^2 - 13*w + 1],\ [929, 929, -2*w^3 + 5*w^2 + 6*w - 11],\ [941, 941, -3*w^3 + 7*w^2 + 12*w - 5],\ [953, 953, -2*w^3 + 5*w^2 + 5*w - 7],\ [971, 971, w^3 - w^2 - 3*w - 5],\ [971, 971, 3*w^3 - 7*w^2 - 12*w + 11],\ [977, 977, 3*w^3 - 5*w^2 - 11*w],\ [997, 997, -2*w^3 + 3*w^2 + 6*w - 5],\ [997, 997, -4*w^3 + 7*w^2 + 16*w - 2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^8 - 30*x^6 + 2*x^5 + 229*x^4 - 35*x^3 - 334*x^2 + 184*x - 8 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -641/58998*e^7 - 127/29499*e^6 + 10301/29499*e^5 + 518/9833*e^4 - 185411/58998*e^3 + 26555/58998*e^2 + 216038/29499*e - 78476/29499, -3877/117996*e^7 - 223/29499*e^6 + 59635/58998*e^5 + 4529/19666*e^4 - 958429/117996*e^3 - 227981/117996*e^2 + 796081/58998*e - 13529/29499, -641/58998*e^7 - 127/29499*e^6 + 10301/29499*e^5 + 518/9833*e^4 - 185411/58998*e^3 + 26555/58998*e^2 + 245537/29499*e - 78476/29499, -1, -2413/29499*e^7 - 3431/58998*e^6 + 71204/29499*e^5 + 15359/9833*e^4 - 521341/29499*e^3 - 548683/58998*e^2 + 1235281/58998*e - 107714/29499, 2819/39332*e^7 + 129/19666*e^6 - 41743/19666*e^5 - 2585/19666*e^4 + 617731/39332*e^3 + 27689/39332*e^2 - 193649/9833*e + 51160/9833, -10847/117996*e^7 - 6475/58998*e^6 + 159587/58998*e^5 + 58851/19666*e^4 - 2317535/117996*e^3 - 2229661/117996*e^2 + 624835/29499*e + 115046/29499, 1395/19666*e^7 + 1059/19666*e^6 - 20301/9833*e^5 - 14841/9833*e^4 + 285757/19666*e^3 + 105706/9833*e^2 - 267735/19666*e - 49252/9833, -9011/58998*e^7 - 3304/29499*e^6 + 132107/29499*e^5 + 30200/9833*e^4 - 1899953/58998*e^3 - 1182919/58998*e^2 + 1019243/29499*e - 18956/29499, -1195/58998*e^7 - 3044/29499*e^6 + 17179/29499*e^5 + 28133/9833*e^4 - 232171/58998*e^3 - 1073297/58998*e^2 + 14389/29499*e + 268526/29499, 17155/58998*e^7 + 5930/29499*e^6 - 254515/29499*e^5 - 52060/9833*e^4 + 3764959/58998*e^3 + 1883015/58998*e^2 - 2271370/29499*e + 391510/29499, -2204/9833*e^7 - 2437/19666*e^6 + 65269/9833*e^5 + 30782/9833*e^4 - 483928/9833*e^3 - 343599/19666*e^2 + 1233509/19666*e - 101946/9833, -127/29499*e^7 + 686/29499*e^6 + 2195/29499*e^5 - 6437/9833*e^4 + 2060/29499*e^3 + 108991/29499*e^2 - 19504/29499*e - 2564/29499, -4873/19666*e^7 - 1395/9833*e^6 + 72036/9833*e^5 + 35729/9833*e^4 - 1056553/19666*e^3 - 420625/19666*e^2 + 622045/9833*e - 52752/9833, -288/9833*e^7 + 162/9833*e^6 + 9778/9833*e^5 - 4847/9833*e^4 - 92110/9833*e^3 + 33622/9833*e^2 + 208796/9833*e - 35236/9833, 15355/39332*e^7 + 1989/9833*e^6 - 226417/19666*e^5 - 103725/19666*e^4 + 3302351/39332*e^3 + 1252431/39332*e^2 - 1918789/19666*e + 182489/9833, 24745/117996*e^7 + 5639/58998*e^6 - 370045/58998*e^5 - 46305/19666*e^4 + 5583205/117996*e^3 + 1483259/117996*e^2 - 1862573/29499*e + 526034/29499, 2678/9833*e^7 + 2181/9833*e^6 - 78494/9833*e^5 - 57425/9833*e^4 + 566285/9833*e^3 + 343032/9833*e^2 - 620340/9833*e + 82094/9833, 39593/117996*e^7 + 5648/29499*e^6 - 580583/58998*e^5 - 95703/19666*e^4 + 8361017/117996*e^3 + 3248221/117996*e^2 - 4596281/58998*e + 421891/29499, 2855/39332*e^7 + 674/9833*e^6 - 41125/19666*e^5 - 36083/19666*e^4 + 592371/39332*e^3 + 443847/39332*e^2 - 432305/19666*e + 1867/9833, -38803/117996*e^7 - 14591/58998*e^6 + 571201/58998*e^5 + 129273/19666*e^4 - 8286031/117996*e^3 - 4806521/117996*e^2 + 2245433/29499*e - 23732/29499, 10183/29499*e^7 + 8407/29499*e^6 - 296084/29499*e^5 - 76177/9833*e^4 + 2075125/29499*e^3 + 1488308/29499*e^2 - 1846418/29499*e - 324004/29499, 811/117996*e^7 + 1576/29499*e^6 - 4105/58998*e^5 - 27025/19666*e^4 - 257741/117996*e^3 + 875579/117996*e^2 + 1180217/58998*e - 139453/29499, 2519/19666*e^7 + 2885/19666*e^6 - 37060/9833*e^5 - 38091/9833*e^4 + 527519/19666*e^3 + 225148/9833*e^2 - 440743/19666*e - 20098/9833, -3208/29499*e^7 - 3112/29499*e^6 + 93074/29499*e^5 + 26707/9833*e^4 - 656173/29499*e^3 - 480413/29499*e^2 + 625739/29499*e + 146140/29499, -25579/58998*e^7 - 21871/58998*e^6 + 375394/29499*e^5 + 98491/9833*e^4 - 5441461/58998*e^3 - 1858363/29499*e^2 + 6166175/58998*e + 22382/29499, 13531/39332*e^7 + 4491/19666*e^6 - 198731/19666*e^5 - 117435/19666*e^4 + 2882871/39332*e^3 + 1399817/39332*e^2 - 794323/9833*e + 74256/9833, -8116/29499*e^7 - 7726/29499*e^6 + 239222/29499*e^5 + 72238/9833*e^4 - 1748161/29499*e^3 - 1456955/29499*e^2 + 2028905/29499*e + 242164/29499, 1989/19666*e^7 + 977/9833*e^6 - 29770/9833*e^5 - 26743/9833*e^4 + 457297/19666*e^3 + 332581/19666*e^2 - 345501/9833*e - 14144/9833, 16601/117996*e^7 + 3013/58998*e^6 - 247637/58998*e^5 - 24445/19666*e^4 + 3659201/117996*e^3 + 665167/117996*e^2 - 1074265/29499*e + 664246/29499, 15065/58998*e^7 + 3445/29499*e^6 - 224840/29499*e^5 - 29846/9833*e^4 + 3410495/58998*e^3 + 1054255/58998*e^2 - 2502932/29499*e + 343004/29499, 5803/19666*e^7 + 1748/9833*e^6 - 85570/9833*e^5 - 45623/9833*e^4 + 1253613/19666*e^3 + 555011/19666*e^2 - 770288/9833*e + 52694/9833, -12569/29499*e^7 - 8294/29499*e^6 + 368215/29499*e^5 + 74787/9833*e^4 - 2644985/29499*e^3 - 1421032/29499*e^2 + 2868532/29499*e - 256078/29499, 6989/29499*e^7 + 4058/29499*e^6 - 205807/29499*e^5 - 35211/9833*e^4 + 1492124/29499*e^3 + 614716/29499*e^2 - 1561600/29499*e + 315772/29499, -6121/19666*e^7 - 1044/9833*e^6 + 89944/9833*e^5 + 26866/9833*e^4 - 1311479/19666*e^3 - 360149/19666*e^2 + 792557/9833*e - 56988/9833, -7913/39332*e^7 - 270/9833*e^6 + 121457/19666*e^5 + 12879/19666*e^4 - 1965457/39332*e^3 - 142205/39332*e^2 + 1745875/19666*e - 180543/9833, -1465/9833*e^7 + 419/19666*e^6 + 44754/9833*e^5 - 7209/9833*e^4 - 356627/9833*e^3 + 116581/19666*e^2 + 1201273/19666*e - 167084/9833, 9875/29499*e^7 + 6122/29499*e^6 - 293548/29499*e^5 - 55553/9833*e^4 + 2176283/29499*e^3 + 1082053/29499*e^2 - 2576377/29499*e + 84622/29499, 31567/117996*e^7 + 2782/29499*e^6 - 459427/58998*e^5 - 48167/19666*e^4 + 6539623/117996*e^3 + 1750031/117996*e^2 - 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7832773/58998*e - 515941/29499, -62293/117996*e^7 - 28265/58998*e^6 + 905431/58998*e^5 + 246831/19666*e^4 - 12800917/117996*e^3 - 9118175/117996*e^2 + 3175058/29499*e + 649768/29499, -4340/29499*e^7 - 17/29499*e^6 + 129595/29499*e^5 - 5273/9833*e^4 - 1001555/29499*e^3 + 379109/29499*e^2 + 1819531/29499*e - 1697290/29499, 41276/29499*e^7 + 23489/29499*e^6 - 1221379/29499*e^5 - 202962/9833*e^4 + 8994542/29499*e^3 + 3524341/29499*e^2 - 10750723/29499*e + 1938952/29499, 2519/9833*e^7 + 2885/9833*e^6 - 74120/9833*e^5 - 76182/9833*e^4 + 537352/9833*e^3 + 469962/9833*e^2 - 588238/9833*e - 99194/9833, -10389/39332*e^7 - 3531/19666*e^6 + 151061/19666*e^5 + 94539/19666*e^4 - 2157261/39332*e^3 - 1223487/39332*e^2 + 656201/9833*e + 200822/9833, 4849/58998*e^7 + 3860/29499*e^6 - 72448/29499*e^5 - 40520/9833*e^4 + 1099681/58998*e^3 + 2142563/58998*e^2 - 595318/29499*e - 1395806/29499, -22519/58998*e^7 - 734/29499*e^6 + 339427/29499*e^5 + 4852/9833*e^4 - 5276473/58998*e^3 - 251381/58998*e^2 + 4041247/29499*e - 687688/29499, -34769/58998*e^7 - 5278/29499*e^6 + 515900/29499*e^5 + 35077/9833*e^4 - 7708061/58998*e^3 - 561583/58998*e^2 + 5153453/29499*e - 1546730/29499, 9038/29499*e^7 + 16873/58998*e^6 - 263287/29499*e^5 - 77149/9833*e^4 + 1880933/29499*e^3 + 2990075/58998*e^2 - 4211993/58998*e - 434456/29499, 15587/117996*e^7 + 8102/29499*e^6 - 215879/58998*e^5 - 152421/19666*e^4 + 2649455/117996*e^3 + 6272407/117996*e^2 - 141719/58998*e - 941483/29499, 13789/19666*e^7 + 5033/9833*e^6 - 204135/9833*e^5 - 131345/9833*e^4 + 3009225/19666*e^3 + 1527435/19666*e^2 - 1775173/9833*e + 415446/9833, 35217/39332*e^7 + 14985/19666*e^6 - 511487/19666*e^5 - 404099/19666*e^4 + 7217661/39332*e^3 + 5207271/39332*e^2 - 1788797/9833*e - 341542/9833, 54445/58998*e^7 + 46195/58998*e^6 - 804163/29499*e^5 - 202062/9833*e^4 + 11800285/58998*e^3 + 3569278/29499*e^2 - 14188631/58998*e + 1017862/29499, 395/9833*e^7 + 3243/19666*e^6 - 9382/9833*e^5 - 46785/9833*e^4 + 37493/9833*e^3 + 693457/19666*e^2 + 152991/19666*e - 363976/9833, -58061/39332*e^7 - 8582/9833*e^6 + 853529/19666*e^5 + 440827/19666*e^4 - 12421425/39332*e^3 - 5032785/39332*e^2 + 7261731/19666*e - 670883/9833, -29470/29499*e^7 - 21526/29499*e^6 + 872855/29499*e^5 + 191781/9833*e^4 - 6443563/29499*e^3 - 3522029/29499*e^2 + 7804268/29499*e - 856976/29499, -42241/58998*e^7 - 8093/29499*e^6 + 630178/29499*e^5 + 64831/9833*e^4 - 9423697/58998*e^3 - 1962887/58998*e^2 + 5893216/29499*e - 1635628/29499] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([16, 2, 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]