/* 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, 3, 4, -5, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([27, 3, -w^3 + w^2 + 3*w - 2]) primes_array = [ [3, 3, w^4 - w^3 - 5*w^2 + 3*w + 3],\ [9, 3, -w^4 + 5*w^2 - 3],\ [9, 3, -w^4 + w^3 + 5*w^2 - 3*w - 2],\ [13, 13, -w^4 + w^3 + 4*w^2 - 3*w - 1],\ [17, 17, w^4 - w^3 - 5*w^2 + 3*w + 1],\ [19, 19, -w^3 + w^2 + 4*w - 2],\ [23, 23, -w^2 + 3],\ [31, 31, w^3 - 4*w + 2],\ [32, 2, 2],\ [37, 37, w^3 - 3*w - 1],\ [53, 53, -2*w^4 + w^3 + 9*w^2 - 3*w - 2],\ [59, 59, -w^4 + 5*w^2 + w - 4],\ [61, 61, -w^4 + w^3 + 5*w^2 - 4*w],\ [67, 67, -w^4 + 6*w^2 + 2*w - 4],\ [71, 71, 2*w^4 - w^3 - 9*w^2 + 4*w + 5],\ [79, 79, 2*w^4 - w^3 - 10*w^2 + 2*w + 7],\ [83, 83, -w^4 + 2*w^3 + 5*w^2 - 7*w - 2],\ [83, 83, -w^4 + w^3 + 4*w^2 - 3*w + 3],\ [83, 83, w^4 - w^3 - 5*w^2 + 4*w - 1],\ [83, 83, -w^4 + w^3 + 4*w^2 - 4*w - 2],\ [83, 83, -2*w^4 + w^3 + 9*w^2 - 3*w - 5],\ [97, 97, -w^4 - w^3 + 6*w^2 + 4*w - 4],\ [101, 101, -w^4 + 4*w^2 + 2*w - 2],\ [107, 107, -2*w^4 + w^3 + 11*w^2 - 3*w - 7],\ [127, 127, w^3 - w^2 - 4*w],\ [131, 131, w^4 - w^3 - 4*w^2 + 2*w - 1],\ [137, 137, w^4 - w^3 - 6*w^2 + 3*w + 3],\ [139, 139, -2*w^4 + w^3 + 9*w^2 - 3*w + 1],\ [157, 157, 2*w^4 - w^3 - 9*w^2 + 4*w + 2],\ [163, 163, -w^2 - 2*w + 3],\ [167, 167, w^4 - 5*w^2 + 2*w + 1],\ [169, 13, -2*w^4 + 2*w^3 + 9*w^2 - 7*w - 5],\ [169, 13, w^4 - 2*w^3 - 5*w^2 + 8*w + 2],\ [191, 191, 2*w^4 - w^3 - 9*w^2 + 3*w + 4],\ [193, 193, 2*w^4 - 2*w^3 - 8*w^2 + 7*w],\ [199, 199, -w^4 + 2*w^3 + 5*w^2 - 9*w - 3],\ [211, 211, -3*w^4 + w^3 + 13*w^2 - 2*w - 1],\ [227, 227, 2*w^3 - w^2 - 9*w + 1],\ [233, 233, -w^3 + w^2 + 4*w + 1],\ [251, 251, -3*w^4 + 14*w^2 + w - 2],\ [257, 257, -2*w^4 + 2*w^3 + 10*w^2 - 6*w - 5],\ [257, 257, -w^4 + w^3 + 5*w^2 - 6*w - 1],\ [257, 257, 2*w^4 - 2*w^3 - 11*w^2 + 9*w + 9],\ [257, 257, -2*w^4 + 2*w^3 + 9*w^2 - 6*w - 3],\ [257, 257, 2*w^4 - w^3 - 10*w^2 + 5*w + 5],\ [269, 269, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2],\ [271, 271, -w^4 + 4*w^2 + w - 3],\ [277, 277, -2*w^4 + 9*w^2 + w - 4],\ [281, 281, w^4 - w^3 - 3*w^2 + 3*w - 4],\ [283, 283, 2*w^4 - 3*w^3 - 8*w^2 + 13*w],\ [289, 17, -w^4 + w^3 + 7*w^2 - 5*w - 5],\ [289, 17, 4*w^4 - 3*w^3 - 19*w^2 + 11*w + 6],\ [293, 293, -2*w^4 + 11*w^2 - 7],\ [317, 317, w^3 - 6*w],\ [337, 337, -w^4 + 2*w^3 + 5*w^2 - 9*w - 1],\ [337, 337, w^4 - 2*w^3 - 5*w^2 + 6*w + 3],\ [337, 337, -w^4 + w^3 + 5*w^2 - 4*w - 6],\ [337, 337, -2*w^3 + 8*w + 1],\ [337, 337, w^2 + w - 5],\ [347, 347, -2*w^4 + w^3 + 9*w^2 - 5*w - 3],\ [349, 349, 3*w^4 - 2*w^3 - 13*w^2 + 8*w + 2],\ [353, 353, w^4 - 3*w^3 - 4*w^2 + 12*w + 2],\ [359, 359, 2*w^3 - w^2 - 7*w + 3],\ [361, 19, w^4 - 2*w^3 - 4*w^2 + 7*w + 3],\ [361, 19, w^4 - w^3 - 3*w^2 + 3*w - 3],\ [367, 367, -2*w^4 + w^3 + 9*w^2 - w - 2],\ [379, 379, -2*w^3 + w^2 + 7*w + 2],\ [379, 379, -2*w^4 + w^3 + 10*w^2 - 4*w],\ [379, 379, 2*w^3 - w^2 - 6*w - 2],\ [379, 379, -2*w^3 + 2*w^2 + 6*w - 3],\ [379, 379, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 6],\ [383, 383, 2*w^4 + w^3 - 10*w^2 - 7*w + 4],\ [383, 383, -2*w^3 + w^2 + 10*w],\ [383, 383, w^3 + w^2 - 5*w - 1],\ [383, 383, 3*w^4 + 3*w^3 - 15*w^2 - 15*w + 4],\ [383, 383, -2*w^4 + 3*w^3 + 10*w^2 - 10*w - 2],\ [389, 389, 3*w^4 - w^3 - 14*w^2 + 3*w + 3],\ [397, 397, w^4 - 6*w^2 - 2*w + 5],\ [397, 397, w^4 - 2*w^3 - 2*w^2 + 6*w - 5],\ [397, 397, 2*w^4 - 2*w^3 - 11*w^2 + 10*w + 6],\ [397, 397, w^4 - w^3 - 3*w^2 + 4*w - 6],\ [397, 397, w^3 + w^2 - 3*w - 4],\ [401, 401, w^3 - w^2 - 6*w + 3],\ [401, 401, -w^4 + 4*w^2 + 2*w - 3],\ [401, 401, -w^4 + 6*w^2 - w - 7],\ [421, 421, -w^4 - w^3 + 5*w^2 + 5*w - 4],\ [421, 421, -w^4 + w^3 + 6*w^2 - 3*w - 2],\ [421, 421, -w^4 - w^3 + 6*w^2 + 6*w - 5],\ [421, 421, 3*w^4 - w^3 - 15*w^2 + w + 10],\ [421, 421, 2*w^4 - 9*w^2 + 2*w + 4],\ [431, 431, 2*w^4 - 11*w^2 - 2*w + 11],\ [439, 439, 2*w^4 - 3*w^3 - 9*w^2 + 10*w + 6],\ [443, 443, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 2],\ [449, 449, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4],\ [461, 461, 3*w^4 - 4*w^3 - 14*w^2 + 14*w + 5],\ [463, 463, w^3 - 6*w - 1],\ [467, 467, 3*w^4 - 3*w^3 - 13*w^2 + 10*w],\ [487, 487, 3*w^4 - w^3 - 14*w^2 + w + 3],\ [487, 487, 2*w^2 - w - 4],\ [487, 487, -2*w^4 + 2*w^3 + 9*w^2 - 8*w - 5],\ [487, 487, 2*w^4 - 10*w^2 - 3*w + 4],\ [487, 487, -2*w^4 + w^3 + 8*w^2 - 3*w - 1],\ [499, 499, -w^4 + 2*w^3 + 4*w^2 - 8*w - 2],\ [499, 499, -2*w^4 + w^3 + 10*w^2 - w - 9],\ [499, 499, -w^3 - 2*w^2 + 2*w + 5],\ [499, 499, -w^4 + 4*w^2 + 4*w],\ [499, 499, -2*w^4 + w^3 + 12*w^2 - 4*w - 9],\ [509, 509, 3*w^4 - 2*w^3 - 16*w^2 + 6*w + 9],\ [521, 521, -2*w^4 + w^3 + 10*w^2 - 5*w - 4],\ [523, 523, w - 4],\ [529, 23, -w^3 + 2*w^2 + 4*w - 4],\ [529, 23, w^4 + 2*w^3 - 6*w^2 - 11*w + 4],\ [569, 569, -2*w^4 + 11*w^2 - 10],\ [571, 571, -2*w^4 + w^3 + 10*w^2 - 3*w - 2],\ [593, 593, -2*w^4 + 3*w^3 + 10*w^2 - 11*w - 8],\ [613, 613, -2*w^3 + 2*w^2 + 9*w - 5],\ [617, 617, -2*w^4 + 10*w^2 - w - 7],\ [631, 631, -2*w^4 + 2*w^3 + 7*w^2 - 8*w + 3],\ [641, 641, -2*w^4 + 2*w^3 + 8*w^2 - 9*w + 3],\ [643, 643, -3*w^4 + w^3 + 13*w^2 - w + 1],\ [643, 643, w^4 + w^3 - 3*w^2 - 6*w - 2],\ [643, 643, -w^4 - w^3 + 6*w^2 + 3*w - 4],\ [643, 643, 2*w^4 - w^3 - 10*w^2 + 4*w + 1],\ [643, 643, 2*w^4 - 2*w^3 - 8*w^2 + 6*w + 5],\ [647, 647, -2*w^3 + w^2 + 6*w - 4],\ [659, 659, 4*w^4 - 3*w^3 - 19*w^2 + 11*w + 10],\ [661, 661, 2*w^4 + 3*w^3 - 11*w^2 - 14*w + 6],\ [673, 673, w^4 - 3*w^3 - 3*w^2 + 10*w - 1],\ [683, 683, w^4 - 2*w^3 - 4*w^2 + 6*w - 4],\ [701, 701, w^3 - w^2 - 2*w + 4],\ [727, 727, -w^4 + 6*w^2 + 2*w - 7],\ [743, 743, w^4 - w^3 - 7*w^2 + 5*w + 9],\ [769, 769, w^4 + 2*w^3 - 5*w^2 - 9*w + 4],\ [787, 787, w^4 + w^3 - 7*w^2 - 3*w + 5],\ [797, 797, -2*w^4 - w^3 + 10*w^2 + 4*w - 4],\ [797, 797, w^4 - 3*w^3 - 5*w^2 + 12*w + 5],\ [797, 797, -3*w^4 + 2*w^3 + 13*w^2 - 8*w],\ [797, 797, -w^4 + w^3 + 3*w^2 - 5*w - 1],\ [797, 797, 2*w^4 - 11*w^2 + 2*w + 7],\ [809, 809, -4*w^4 + 2*w^3 + 18*w^2 - 6*w + 1],\ [809, 809, 3*w^4 - 3*w^3 - 14*w^2 + 11*w + 8],\ [809, 809, w^4 - 2*w^3 - 2*w^2 + 8*w - 4],\ [809, 809, 2*w^4 - 8*w^2 - 2*w + 3],\ [809, 809, 2*w^4 - 3*w^3 - 9*w^2 + 13*w + 4],\ [821, 821, -2*w^4 + 2*w^3 + 8*w^2 - 6*w + 3],\ [823, 823, -2*w^4 + 3*w^3 + 8*w^2 - 12*w - 1],\ [829, 829, 2*w^4 - 3*w^3 - 10*w^2 + 13*w + 4],\ [839, 839, -2*w^4 + 12*w^2 - w - 10],\ [863, 863, 2*w^4 - w^3 - 9*w^2 + 5],\ [877, 877, -w^4 + w^3 + 5*w^2 - 5*w - 7],\ [881, 881, 2*w^4 - w^3 - 9*w^2 + w - 1],\ [887, 887, 3*w^4 - 3*w^3 - 15*w^2 + 11*w + 6],\ [907, 907, -3*w^4 + w^3 + 13*w^2 - w - 5],\ [919, 919, w^4 + w^3 - 3*w^2 - 5*w - 3],\ [929, 929, 2*w^4 - 3*w^3 - 10*w^2 + 9*w + 4],\ [937, 937, -2*w^4 + 10*w^2 + 3*w - 7],\ [941, 941, -w^4 + 2*w^3 + 3*w^2 - 6*w - 2],\ [961, 31, 2*w^4 + 2*w^3 - 10*w^2 - 12*w + 5],\ [961, 31, -w^4 + 5*w^2 - w - 7],\ [967, 967, -2*w^4 + w^3 + 9*w^2 - 3*w + 2],\ [991, 991, -3*w^4 + w^3 + 15*w^2 - 4*w - 5],\ [997, 997, -3*w^4 + 15*w^2 - 5],\ [997, 997, 2*w^4 - w^3 - 9*w^2 + 6*w + 2],\ [997, 997, 2*w^4 - w^3 - 11*w^2 + w + 7],\ [997, 997, 2*w^3 - 5*w - 1],\ [997, 997, 2*w^4 - 3*w^3 - 6*w^2 + 5*w + 1]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^9 - 3*x^8 - 71*x^7 + 194*x^6 + 1730*x^5 - 4389*x^4 - 16430*x^3 + 39516*x^2 + 44872*x - 103808 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, e, 1, 6896623/4877833864*e^8 - 15795193/4877833864*e^7 - 441686219/4877833864*e^6 + 366912919/2438916932*e^5 + 1170160041/609729233*e^4 - 9361496893/4877833864*e^3 - 9390696649/609729233*e^2 + 6371631913/1219458466*e + 23947426614/609729233, -13843703/9755667728*e^8 + 41067663/9755667728*e^7 + 794433119/9755667728*e^6 - 122933209/609729233*e^5 - 7129203363/4877833864*e^4 + 27400742327/9755667728*e^3 + 10865077791/1219458466*e^2 - 23279125491/2438916932*e - 8589679268/609729233, -12421749/9755667728*e^8 + 64196373/9755667728*e^7 + 545162001/9755667728*e^6 - 383133495/1219458466*e^5 - 2170369555/4877833864*e^4 + 41008677985/9755667728*e^3 - 9916896813/2438916932*e^2 - 31080571187/2438916932*e + 13365759444/609729233, 4749455/2438916932*e^8 - 2814211/1219458466*e^7 - 316237917/2438916932*e^6 + 234271557/2438916932*e^5 + 7109990125/2438916932*e^4 - 706910003/609729233*e^3 - 15630733520/609729233*e^2 + 2256521212/609729233*e + 42315035036/609729233, 3924433/9755667728*e^8 + 51102519/9755667728*e^7 - 612171265/9755667728*e^6 - 166724416/609729233*e^5 + 11745117473/4877833864*e^4 + 38693706167/9755667728*e^3 - 18456091081/609729233*e^2 - 28742985191/2438916932*e + 54988837174/609729233, 1110785/1219458466*e^8 + 1747405/1219458466*e^7 - 172941495/2438916932*e^6 - 60097101/609729233*e^5 + 4611359983/2438916932*e^4 + 4643107225/2438916932*e^3 - 47499587669/2438916932*e^2 - 11571776261/1219458466*e + 33910417397/609729233, -10061083/4877833864*e^8 + 27015651/4877833864*e^7 + 662125007/4877833864*e^6 - 174920022/609729233*e^5 - 7229337249/2438916932*e^4 + 21205240407/4877833864*e^3 + 29636518959/1219458466*e^2 - 20414317239/1219458466*e - 34110686844/609729233, 18947363/9755667728*e^8 - 13036411/9755667728*e^7 - 1353890987/9755667728*e^6 + 27362910/609729233*e^5 + 16452057215/4877833864*e^4 - 2632811507/9755667728*e^3 - 39177782547/1219458466*e^2 - 3948693749/2438916932*e + 57672782382/609729233, 2183213/9755667728*e^8 + 39942967/9755667728*e^7 - 285880033/9755667728*e^6 - 510294025/2438916932*e^5 + 4798643937/4877833864*e^4 + 26518184067/9755667728*e^3 - 27156408899/2438916932*e^2 - 10773568437/2438916932*e + 18349204666/609729233, -1328759/1219458466*e^8 + 8234831/1219458466*e^7 + 38476261/1219458466*e^6 - 199512678/609729233*e^5 + 318003377/609729233*e^4 + 5563306443/1219458466*e^3 - 10797784539/609729233*e^2 - 9018802336/609729233*e + 44403312334/609729233, -13550519/9755667728*e^8 + 77790363/9755667728*e^7 + 581387923/9755667728*e^6 - 926063541/2438916932*e^5 - 1980182899/4877833864*e^4 + 46256704791/9755667728*e^3 - 13722137543/2438916932*e^2 - 22059875473/2438916932*e + 16108330818/609729233, 10063295/4877833864*e^8 - 18053839/4877833864*e^7 - 650588795/4877833864*e^6 + 98517027/609729233*e^5 + 7014896797/2438916932*e^4 - 8279293427/4877833864*e^3 - 29358846675/1219458466*e^2 + 2694567/1219458466*e + 39572463662/609729233, 2322205/1219458466*e^8 + 1051253/1219458466*e^7 - 85363137/609729233*e^6 - 33324698/609729233*e^5 + 4174123613/1219458466*e^4 + 467334412/609729233*e^3 - 38364188073/1219458466*e^2 + 140312377/609729233*e + 46151751272/609729233, -1213625/609729233*e^8 + 6679675/1219458466*e^7 + 145511643/1219458466*e^6 - 300920953/1219458466*e^5 - 1467933256/609729233*e^4 + 1881154418/609729233*e^3 + 24158746007/1219458466*e^2 - 4982459280/609729233*e - 34819943402/609729233, -498674/609729233*e^8 + 21478293/2438916932*e^7 + 12397896/609729233*e^6 - 1109773903/2438916932*e^5 + 1016730309/2438916932*e^4 + 16269730389/2438916932*e^3 - 12763388285/1219458466*e^2 - 13979496101/609729233*e + 13717478946/609729233, -7700417/9755667728*e^8 - 10165071/9755667728*e^7 + 779088013/9755667728*e^6 + 33293451/1219458466*e^5 - 12288304943/4877833864*e^4 + 1401802829/9755667728*e^3 + 69189934731/2438916932*e^2 - 9748063291/2438916932*e - 51134821686/609729233, -6281781/4877833864*e^8 - 479079/4877833864*e^7 + 512512577/4877833864*e^6 + 21261733/1219458466*e^5 - 7007810457/2438916932*e^4 - 1940265051/4877833864*e^3 + 35297186211/1219458466*e^2 + 873340361/1219458466*e - 45210908020/609729233, 11903217/9755667728*e^8 - 34810653/9755667728*e^7 - 810537601/9755667728*e^6 + 442876239/2438916932*e^5 + 9711656855/4877833864*e^4 - 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56227462057/1219458466*e - 102940380982/609729233, -5059263/4877833864*e^8 + 19530925/4877833864*e^7 + 268375851/4877833864*e^6 - 462867413/2438916932*e^5 - 1292249637/1219458466*e^4 + 12775520057/4877833864*e^3 + 6702945579/609729233*e^2 - 8488689361/1219458466*e - 27244279638/609729233, -5678123/4877833864*e^8 + 81271195/4877833864*e^7 + 176594475/4877833864*e^6 - 550096879/609729233*e^5 + 873129389/2438916932*e^4 + 66263394947/4877833864*e^3 - 9567681639/609729233*e^2 - 61348486305/1219458466*e + 35307663636/609729233, -19277813/4877833864*e^8 + 123030115/4877833864*e^7 + 1024166641/4877833864*e^6 - 3107111677/2438916932*e^5 - 2057896701/609729233*e^4 + 88448480871/4877833864*e^3 + 8775145929/609729233*e^2 - 78519844987/1219458466*e + 18041024006/609729233, 64580213/9755667728*e^8 - 359108073/9755667728*e^7 - 3076233025/9755667728*e^6 + 4324830713/2438916932*e^5 + 17352149025/4877833864*e^4 - 225468253973/9755667728*e^3 + 9953455899/2438916932*e^2 + 154261766259/2438916932*e - 37253862162/609729233] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([3, 3, w^4 - w^3 - 5*w^2 + 3*w + 3])] = 1 AL_eigenvalues[ZF.ideal([9, 3, -w^4 + w^3 + 5*w^2 - 3*w - 2])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]