/* 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([29, 29, w^2 - w - 3]) 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^11 - 2*x^10 - 33*x^9 + 64*x^8 + 374*x^7 - 662*x^6 - 1768*x^5 + 2510*x^4 + 3519*x^3 - 2646*x^2 - 2932*x - 200 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -29158751/9726238420*e^10 + 19652909/2431559605*e^9 + 817060649/9726238420*e^8 - 126906273/486311921*e^7 - 1660953576/2431559605*e^6 + 6585009772/2431559605*e^5 + 6293270923/4863119210*e^4 - 49953091447/4863119210*e^3 + 12915425617/9726238420*e^2 + 28088519492/2431559605*e + 1041787423/486311921, -6104306/486311921*e^10 + 29160617/972623842*e^9 + 375618711/972623842*e^8 - 454134549/486311921*e^7 - 1858525710/486311921*e^6 + 9012551375/972623842*e^5 + 6573924206/486311921*e^4 - 15820762146/486311921*e^3 - 6508586523/486311921*e^2 + 13624649402/486311921*e + 3307044654/486311921, -9143953/1945247684*e^10 - 2209169/972623842*e^9 + 302664613/1945247684*e^8 + 38112704/486311921*e^7 - 1676074107/972623842*e^6 - 496133939/486311921*e^5 + 6860976933/972623842*e^4 + 5467738241/972623842*e^3 - 12570473035/1945247684*e^2 - 5496535094/486311921*e - 1996011231/486311921, -886497/1945247684*e^10 - 2623837/972623842*e^9 + 31380507/1945247684*e^8 + 28618549/486311921*e^7 - 99305443/486311921*e^6 - 196144161/972623842*e^5 + 1099954973/972623842*e^4 - 907457343/486311921*e^3 - 6559826795/1945247684*e^2 + 6995430947/972623842*e + 2697866543/486311921, -55746989/9726238420*e^10 + 68525987/4863119210*e^9 + 1724544981/9726238420*e^8 - 424336917/972623842*e^7 - 4257541729/2431559605*e^6 + 10412489378/2431559605*e^5 + 28689112087/4863119210*e^4 - 36022327709/2431559605*e^3 - 38019517407/9726238420*e^2 + 33279806793/2431559605*e + 2048878741/486311921, -66461967/9726238420*e^10 + 42069561/4863119210*e^9 + 2216085973/9726238420*e^8 - 273120671/972623842*e^7 - 6333482192/2431559605*e^6 + 14315818683/4863119210*e^5 + 28989297868/2431559605*e^4 - 56617563549/4863119210*e^3 - 179991860211/9726238420*e^2 + 35112972564/2431559605*e + 3399903767/486311921, -47650727/9726238420*e^10 - 21283539/4863119210*e^9 + 1462794843/9726238420*e^8 + 63928248/486311921*e^7 - 3753742907/2431559605*e^6 - 7441890657/4863119210*e^5 + 29493421721/4863119210*e^4 + 39593831151/4863119210*e^3 - 74991832831/9726238420*e^2 - 39373390221/2431559605*e + 394994687/486311921, 16182637/4863119210*e^10 + 38362244/2431559605*e^9 - 299658189/2431559605*e^8 - 488707003/972623842*e^7 + 8280380153/4863119210*e^6 + 26442321379/4863119210*e^5 - 50450377907/4863119210*e^4 - 54600033886/2431559605*e^3 + 58375122833/2431559605*e^2 + 63353329972/2431559605*e - 2061421541/486311921, -3949917/4863119210*e^10 - 34670234/2431559605*e^9 + 169318523/4863119210*e^8 + 220632153/486311921*e^7 - 1315244239/2431559605*e^6 - 12012354782/2431559605*e^5 + 7419566456/2431559605*e^4 + 51832616611/2431559605*e^3 - 12024162191/4863119210*e^2 - 69025543137/2431559605*e - 4719199118/486311921, -1, 10590994/2431559605*e^10 + 77303667/4863119210*e^9 - 350823086/2431559605*e^8 - 485478507/972623842*e^7 + 4129710461/2431559605*e^6 + 26087259231/4863119210*e^5 - 40683573203/4863119210*e^4 - 54611350204/2431559605*e^3 + 36449921122/2431559605*e^2 + 70520106258/2431559605*e - 293071334/486311921, -105554669/9726238420*e^10 + 86546146/2431559605*e^9 + 3192047081/9726238420*e^8 - 553628988/486311921*e^7 - 15483385753/4863119210*e^6 + 28390481888/2431559605*e^5 + 26591335496/2431559605*e^4 - 102855863044/2431559605*e^3 - 91724329157/9726238420*e^2 + 88275417583/2431559605*e + 3111352341/486311921, 81161009/9726238420*e^10 - 7461366/2431559605*e^9 - 2665022171/9726238420*e^8 + 41803995/486311921*e^7 + 14861104183/4863119210*e^6 - 1340120443/2431559605*e^5 - 32193932186/2431559605*e^4 - 1887265057/4863119210*e^3 + 159110148387/9726238420*e^2 + 34962254359/4863119210*e + 3273689167/486311921, 401119/1945247684*e^10 - 8090637/486311921*e^9 + 37259783/1945247684*e^8 + 262127618/486311921*e^7 - 658329283/972623842*e^6 - 5577715823/972623842*e^5 + 6015872141/972623842*e^4 + 21652688853/972623842*e^3 - 32453677779/1945247684*e^2 - 11556096996/486311921*e + 3883315673/486311921, 42160539/9726238420*e^10 + 25031613/4863119210*e^9 - 1615822461/9726238420*e^8 - 100954422/486311921*e^7 + 5441980819/2431559605*e^6 + 6940431592/2431559605*e^5 - 30460821026/2431559605*e^4 - 37519903046/2431559605*e^3 + 245634599337/9726238420*e^2 + 64126730997/2431559605*e - 2683207648/486311921, -40843001/2431559605*e^10 + 230228557/4863119210*e^9 + 1194013069/2431559605*e^8 - 739376933/486311921*e^7 - 10595674674/2431559605*e^6 + 38277229123/2431559605*e^5 + 53114813337/4863119210*e^4 - 286856122043/4863119210*e^3 + 27299741079/4863119210*e^2 + 137368769123/2431559605*e - 2078967475/486311921, -6950559/2431559605*e^10 + 77559034/2431559605*e^9 + 153306116/2431559605*e^8 - 959590045/972623842*e^7 - 529206217/4863119210*e^6 + 24117321002/2431559605*e^5 - 9786951071/2431559605*e^4 - 175530011937/4863119210*e^3 + 45739316928/2431559605*e^2 + 79091701177/2431559605*e - 4022554755/486311921, -66047701/9726238420*e^10 + 71871999/2431559605*e^9 + 1973797279/9726238420*e^8 - 884871821/972623842*e^7 - 9633700677/4863119210*e^6 + 21747713207/2431559605*e^5 + 37814146873/4863119210*e^4 - 77481437946/2431559605*e^3 - 131295112873/9726238420*e^2 + 82362652607/2431559605*e + 4467858649/486311921, 38158603/1945247684*e^10 - 17517954/486311921*e^9 - 1173357183/1945247684*e^8 + 1072807097/972623842*e^7 + 5805389503/972623842*e^6 - 10165195907/972623842*e^5 - 10016374344/486311921*e^4 + 32197799965/972623842*e^3 + 27699936117/1945247684*e^2 - 10411116325/486311921*e + 2811725965/486311921, -82266227/9726238420*e^10 + 7848368/2431559605*e^9 + 2718456313/9726238420*e^8 - 57225811/486311921*e^7 - 7944440577/2431559605*e^6 + 3002428119/2431559605*e^5 + 80417359901/4863119210*e^4 - 19846159219/4863119210*e^3 - 327998322331/9726238420*e^2 + 583681899/2431559605*e + 6553876278/486311921, 45331896/2431559605*e^10 - 160305466/2431559605*e^9 - 2666236783/4863119210*e^8 + 1017692857/486311921*e^7 + 24112222903/4863119210*e^6 - 51873661413/2431559605*e^5 - 66489205797/4863119210*e^4 + 377117095493/4863119210*e^3 + 4302547988/2431559605*e^2 - 177582489958/2431559605*e - 5005211242/486311921, 25893074/2431559605*e^10 - 16061304/2431559605*e^9 - 1590053477/4863119210*e^8 + 129849032/486311921*e^7 + 7903392931/2431559605*e^6 - 16293051699/4863119210*e^5 - 53756578573/4863119210*e^4 + 75109669977/4863119210*e^3 + 28694951189/4863119210*e^2 - 44722594872/2431559605*e + 1038439361/486311921, 189227753/9726238420*e^10 - 133967239/4863119210*e^9 - 5459251037/9726238420*e^8 + 850000273/972623842*e^7 + 24178281921/4863119210*e^6 - 20803278611/2431559605*e^5 - 31898376287/2431559605*e^4 + 70486107938/2431559605*e^3 - 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33990662807/4863119210*e^5 - 242431024919/4863119210*e^4 + 142413807091/4863119210*e^3 + 454233156237/4863119210*e^2 - 94192005506/2431559605*e - 29104234413/486311921] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([29, 29, w^2 - w - 3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]