/* 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([3, -4, -8, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 2, 2]) primes_array = [ [3, 3, w],\ [3, 3, w^3 - 2*w^2 - 6*w + 1],\ [9, 3, w^3 - 2*w^2 - 5*w + 1],\ [16, 2, 2],\ [17, 17, w^3 - w^2 - 8*w - 5],\ [17, 17, -2*w^3 + 4*w^2 + 11*w - 2],\ [17, 17, -w^3 + 3*w^2 + 2*w - 2],\ [17, 17, w^3 - 2*w^2 - 4*w + 1],\ [25, 5, w^3 - 3*w^2 - 3*w + 1],\ [25, 5, w^2 - 2*w - 7],\ [29, 29, w^3 - 2*w^2 - 6*w - 1],\ [29, 29, w - 2],\ [53, 53, w^3 - 3*w^2 - 4*w + 4],\ [53, 53, -w^3 + 3*w^2 + 4*w - 5],\ [61, 61, -w^3 + 2*w^2 + 7*w - 4],\ [61, 61, -4*w^3 + 11*w^2 + 14*w - 8],\ [101, 101, -w^3 + 2*w^2 + 7*w - 1],\ [103, 103, -w^3 + 3*w^2 + 2*w - 5],\ [103, 103, -w^3 + w^2 + 8*w + 2],\ [113, 113, w^3 - 3*w^2 - 3*w + 7],\ [113, 113, w^2 - 2*w - 1],\ [127, 127, w^3 - 2*w^2 - 4*w - 2],\ [127, 127, -2*w^3 + 4*w^2 + 11*w + 1],\ [131, 131, -3*w^3 + 7*w^2 + 14*w - 8],\ [131, 131, -w^2 + w + 8],\ [131, 131, 2*w^3 - 5*w^2 - 9*w + 1],\ [131, 131, -3*w^3 + 6*w^2 + 16*w + 1],\ [139, 139, -2*w^3 + 5*w^2 + 9*w - 2],\ [139, 139, -w^2 + w + 7],\ [157, 157, -2*w^3 + 5*w^2 + 7*w - 5],\ [157, 157, -2*w^3 + 3*w^2 + 13*w + 2],\ [157, 157, -2*w^3 + 5*w^2 + 6*w - 2],\ [157, 157, -3*w^3 + 5*w^2 + 19*w + 4],\ [169, 13, -2*w^3 + 4*w^2 + 10*w - 1],\ [173, 173, 3*w^3 - 8*w^2 - 10*w + 7],\ [173, 173, 2*w^3 - 2*w^2 - 15*w - 8],\ [179, 179, -3*w^2 + 6*w + 17],\ [179, 179, -2*w^3 + 3*w^2 + 12*w + 1],\ [179, 179, -3*w^3 + 7*w^2 + 13*w - 7],\ [179, 179, -2*w^3 + 3*w^2 + 14*w + 4],\ [191, 191, w^3 - 3*w^2 - w + 4],\ [191, 191, -2*w^3 + 3*w^2 + 14*w + 2],\ [199, 199, -w^3 + 8*w + 10],\ [199, 199, 5*w^3 - 15*w^2 - 14*w + 14],\ [211, 211, -w - 4],\ [211, 211, -w^3 + 2*w^2 + 6*w - 5],\ [211, 211, 2*w^3 - 3*w^2 - 12*w - 4],\ [211, 211, -3*w^3 + 7*w^2 + 13*w - 4],\ [257, 257, -3*w^3 + 8*w^2 + 10*w - 2],\ [257, 257, 2*w^3 - 2*w^2 - 16*w - 7],\ [263, 263, 3*w^3 - 6*w^2 - 17*w + 1],\ [263, 263, w^3 - 2*w^2 - 3*w - 1],\ [269, 269, 5*w^3 - 13*w^2 - 19*w + 7],\ [269, 269, -3*w^3 + 6*w^2 + 18*w - 1],\ [277, 277, w^3 - 4*w^2 - w + 5],\ [277, 277, -w^3 + 4*w^2 + w - 11],\ [283, 283, -w^3 + 5*w^2 - w - 14],\ [283, 283, 2*w^3 - 7*w^2 - 4*w + 10],\ [283, 283, -w^3 + w^2 + 9*w + 1],\ [283, 283, w^2 - 4*w - 5],\ [311, 311, -w^3 + w^2 + 9*w + 5],\ [311, 311, w^2 - 4*w - 1],\ [313, 313, -4*w^3 + 9*w^2 + 19*w - 10],\ [313, 313, -2*w^2 + 5*w + 11],\ [337, 337, -3*w^3 + 6*w^2 + 14*w + 2],\ [337, 337, -4*w^3 + 8*w^2 + 21*w + 1],\ [347, 347, -2*w^3 + 6*w^2 + 7*w - 10],\ [347, 347, -w^3 + 4*w^2 + 2*w - 7],\ [361, 19, -4*w^3 + 11*w^2 + 13*w - 10],\ [361, 19, -2*w^3 + 5*w^2 + 8*w - 10],\ [373, 373, 2*w^3 - 5*w^2 - 10*w + 2],\ [373, 373, -w^3 + 2*w^2 + 8*w - 2],\ [373, 373, -2*w^3 + 4*w^2 + 13*w - 1],\ [373, 373, -w^3 + 3*w^2 + 5*w - 8],\ [389, 389, -w^3 + 4*w^2 + 2*w - 10],\ [389, 389, -2*w^3 + 6*w^2 + 7*w - 7],\ [419, 419, -2*w^3 + 5*w^2 + 6*w - 1],\ [419, 419, -3*w^3 + 5*w^2 + 19*w + 5],\ [433, 433, w^2 - 4*w - 2],\ [433, 433, w^3 - w^2 - 9*w - 4],\ [439, 439, -w^3 + 4*w^2 + w - 2],\ [439, 439, w^3 - 4*w^2 - w + 14],\ [443, 443, -2*w^3 + 5*w^2 + 9*w + 1],\ [443, 443, -w^2 + w + 10],\ [467, 467, w^3 - w^2 - 8*w + 4],\ [467, 467, w^3 - 4*w^2 + 2],\ [491, 491, -4*w^3 + 12*w^2 + 10*w - 11],\ [491, 491, 2*w^3 - w^2 - 18*w - 11],\ [491, 491, 3*w^3 - 9*w^2 - 7*w + 11],\ [491, 491, 2*w^3 - 20*w - 19],\ [503, 503, w^3 - 3*w^2 - 4*w + 10],\ [503, 503, -w^3 + w^2 + 10*w - 2],\ [523, 523, -4*w^3 + 9*w^2 + 22*w - 7],\ [523, 523, w^3 - 3*w^2 - 7*w + 5],\ [529, 23, -2*w^3 + 4*w^2 + 10*w + 5],\ [529, 23, -2*w^3 + 4*w^2 + 10*w - 7],\ [547, 547, 2*w^3 - w^2 - 18*w - 14],\ [547, 547, 3*w^3 - 8*w^2 - 14*w + 8],\ [571, 571, 5*w^3 - 11*w^2 - 25*w + 5],\ [571, 571, -4*w^3 + 9*w^2 + 19*w - 11],\ [599, 599, -5*w^3 + 12*w^2 + 20*w - 4],\ [599, 599, 4*w^3 - 6*w^2 - 25*w - 11],\ [601, 601, -4*w^3 + 9*w^2 + 20*w - 5],\ [601, 601, w^3 - 9*w - 14],\ [601, 601, 3*w^3 - 8*w^2 - 11*w + 2],\ [601, 601, -2*w^3 + 3*w^2 + 14*w - 1],\ [641, 641, -2*w^3 + 3*w^2 + 15*w - 5],\ [641, 641, -4*w^3 + 8*w^2 + 23*w - 2],\ [647, 647, -3*w^3 + 8*w^2 + 13*w - 11],\ [647, 647, -w^3 + 4*w^2 + 3*w - 7],\ [659, 659, 3*w^3 - 7*w^2 - 14*w - 2],\ [659, 659, w^3 - w^2 - 6*w - 11],\ [673, 673, -3*w^3 + 7*w^2 + 11*w - 1],\ [673, 673, -4*w^3 + 7*w^2 + 24*w + 5],\ [677, 677, 5*w^3 - 12*w^2 - 21*w + 11],\ [677, 677, -2*w^3 + 6*w^2 + 5*w - 1],\ [677, 677, -2*w^3 + 6*w^2 + 8*w - 7],\ [677, 677, 2*w^3 - 6*w^2 - 8*w + 11],\ [719, 719, -2*w^3 + 6*w^2 + 10*w - 7],\ [719, 719, -4*w^3 + 10*w^2 + 20*w - 13],\ [727, 727, 3*w^3 - 10*w^2 - 8*w + 10],\ [727, 727, 3*w^3 - 8*w^2 - 13*w + 10],\ [727, 727, 2*w^3 - 8*w^2 - 3*w + 23],\ [727, 727, -w^3 + 4*w^2 + 3*w - 8],\ [751, 751, -7*w^3 + 18*w^2 + 25*w - 7],\ [751, 751, w^3 - 7*w^2 + 5*w + 26],\ [757, 757, -4*w^3 + 9*w^2 + 18*w - 7],\ [757, 757, -3*w^3 + 5*w^2 + 17*w + 1],\ [797, 797, 2*w^2 - 4*w - 5],\ [797, 797, 2*w^3 - 6*w^2 - 6*w + 11],\ [823, 823, -2*w^3 + 2*w^2 + 15*w + 14],\ [823, 823, -3*w^3 + 8*w^2 + 10*w - 1],\ [829, 829, -6*w^3 + 16*w^2 + 22*w - 7],\ [829, 829, 5*w^3 - 13*w^2 - 20*w + 8],\ [829, 829, -w^3 + 7*w + 4],\ [829, 829, -5*w^3 + 12*w^2 + 23*w - 14],\ [841, 29, -3*w^3 + 6*w^2 + 15*w - 2],\ [859, 859, -4*w^3 + 7*w^2 + 24*w - 1],\ [859, 859, -5*w^3 + 15*w^2 + 14*w - 13],\ [881, 881, 3*w^3 - 7*w^2 - 16*w + 4],\ [881, 881, -w^3 + 3*w^2 + 6*w - 7],\ [883, 883, -2*w^3 + 3*w^2 + 12*w + 10],\ [883, 883, -3*w^3 + 7*w^2 + 13*w + 2],\ [887, 887, 4*w^3 - 11*w^2 - 11*w + 11],\ [887, 887, 2*w^3 - 7*w^2 - 3*w + 7],\ [887, 887, 4*w^3 - 5*w^2 - 29*w - 10],\ [887, 887, 3*w^2 - 7*w - 16],\ [907, 907, -w^3 + 3*w^2 + 8*w - 5],\ [907, 907, -5*w^3 + 11*w^2 + 28*w - 8],\ [911, 911, -4*w^3 + 10*w^2 + 18*w - 17],\ [911, 911, 2*w^2 - 2*w - 1],\ [919, 919, 4*w^3 - 7*w^2 - 25*w - 4],\ [919, 919, 2*w^3 - 5*w^2 - 5*w + 1],\ [937, 937, -w^3 + 3*w^2 + w - 8],\ [937, 937, -2*w^3 + 3*w^2 + 14*w - 2],\ [953, 953, -2*w^3 + w^2 + 17*w + 19],\ [953, 953, 4*w^3 - 11*w^2 - 13*w + 4],\ [961, 31, 2*w^3 - 3*w^2 - 11*w - 8],\ [961, 31, w^3 - 4*w^2 + 2*w + 11],\ [971, 971, 2*w^2 - 5*w - 4],\ [971, 971, -w^3 + 4*w^2 - 11],\ [991, 991, -2*w^3 + 6*w^2 + 9*w - 8],\ [991, 991, -2*w^3 + 3*w^2 + 13*w - 1],\ [991, 991, 3*w^3 - 8*w^2 - 14*w + 11],\ [991, 991, -2*w^3 + 5*w^2 + 7*w - 8],\ [997, 997, -2*w^3 + 5*w^2 + 6*w + 1],\ [997, 997, -3*w^3 + 5*w^2 + 19*w + 7],\ [997, 997, -6*w^3 + 12*w^2 + 33*w - 4],\ [997, 997, 4*w^3 - 11*w^2 - 15*w + 11]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^12 - 6*x^11 - 12*x^10 + 125*x^9 - 34*x^8 - 904*x^7 + 915*x^6 + 2606*x^5 - 3596*x^4 - 2445*x^3 + 4023*x^2 + 609*x - 1273 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -372653/7579081*e^11 + 976665/7579081*e^10 + 8279049/7579081*e^9 - 19911902/7579081*e^8 - 65845004/7579081*e^7 + 140239013/7579081*e^6 + 222620351/7579081*e^5 - 390648998/7579081*e^4 - 293247421/7579081*e^3 + 340781367/7579081*e^2 + 112394219/7579081*e - 3311209/398899, -92118/7579081*e^11 + 128733/7579081*e^10 + 2061489/7579081*e^9 - 2374484/7579081*e^8 - 15859956/7579081*e^7 + 16115870/7579081*e^6 + 47760254/7579081*e^5 - 54070135/7579081*e^4 - 50664009/7579081*e^3 + 91470644/7579081*e^2 + 27555514/7579081*e - 3425832/398899, -1, -1133527/7579081*e^11 + 2867336/7579081*e^10 + 25003875/7579081*e^9 - 58059038/7579081*e^8 - 196214283/7579081*e^7 + 409658614/7579081*e^6 + 647136534/7579081*e^5 - 1180357892/7579081*e^4 - 816596227/7579081*e^3 + 1210614268/7579081*e^2 + 313208734/7579081*e - 20331751/398899, -3054321/7579081*e^11 + 9584765/7579081*e^10 + 63194538/7579081*e^9 - 198710163/7579081*e^8 - 444390656/7579081*e^7 + 1444032400/7579081*e^6 + 1166434029/7579081*e^5 - 4308602349/7579081*e^4 - 721311950/7579081*e^3 + 4605809652/7579081*e^2 - 38517633/7579081*e - 79810432/398899, 647709/7579081*e^11 - 2011687/7579081*e^10 - 13662886/7579081*e^9 + 41215532/7579081*e^8 + 101047616/7579081*e^7 - 293714180/7579081*e^6 - 303219513/7579081*e^5 + 848040238/7579081*e^4 + 305134744/7579081*e^3 - 861950946/7579081*e^2 - 55570129/7579081*e + 14631176/398899, 765028/7579081*e^11 - 2838863/7579081*e^10 - 15211435/7579081*e^9 + 60137103/7579081*e^8 + 99783308/7579081*e^7 - 449150640/7579081*e^6 - 216955664/7579081*e^5 + 1387294655/7579081*e^4 - 24904600/7579081*e^3 - 1559442876/7579081*e^2 + 190675499/7579081*e + 28096307/398899, -2338053/7579081*e^11 + 7410378/7579081*e^10 + 49471191/7579081*e^9 - 154804776/7579081*e^8 - 364956700/7579081*e^7 + 1132847347/7579081*e^6 + 1080482768/7579081*e^5 - 3393025398/7579081*e^4 - 1065795026/7579081*e^3 + 3617958243/7579081*e^2 + 284848170/7579081*e - 61020961/398899, 1155350/7579081*e^11 - 3794557/7579081*e^10 - 24696229/7579081*e^9 + 80340722/7579081*e^8 + 185498086/7579081*e^7 - 596423796/7579081*e^6 - 568482821/7579081*e^5 + 1812956107/7579081*e^4 + 601670772/7579081*e^3 - 1970214130/7579081*e^2 - 157186739/7579081*e + 34266605/398899, -1823224/7579081*e^11 + 6289327/7579081*e^10 + 36841935/7579081*e^9 - 130466852/7579081*e^8 - 250525132/7579081*e^7 + 943654434/7579081*e^6 + 616416160/7579081*e^5 - 2764190315/7579081*e^4 - 269779497/7579081*e^3 + 2800239764/7579081*e^2 - 104545256/7579081*e - 44423959/398899, -1823224/7579081*e^11 + 6289327/7579081*e^10 + 36841935/7579081*e^9 - 130466852/7579081*e^8 - 250525132/7579081*e^7 + 943654434/7579081*e^6 + 616416160/7579081*e^5 - 2764190315/7579081*e^4 - 269779497/7579081*e^3 + 2800239764/7579081*e^2 - 104545256/7579081*e - 44423959/398899, -2691029/7579081*e^11 + 6690745/7579081*e^10 + 59148016/7579081*e^9 - 135110044/7579081*e^8 - 459587368/7579081*e^7 + 951365452/7579081*e^6 + 1480262441/7579081*e^5 - 2742636121/7579081*e^4 - 1766950320/7579081*e^3 + 2819374764/7579081*e^2 + 625005603/7579081*e - 46835408/398899, 1627275/7579081*e^11 - 4850714/7579081*e^10 - 34404549/7579081*e^9 + 99048548/7579081*e^8 + 253394078/7579081*e^7 - 703360714/7579081*e^6 - 744668145/7579081*e^5 + 2023930678/7579081*e^4 + 693363345/7579081*e^3 - 2030577476/7579081*e^2 - 82172879/7579081*e + 31609527/398899, 4765705/7579081*e^11 - 14859826/7579081*e^10 - 99321647/7579081*e^9 + 306489232/7579081*e^8 + 710962532/7579081*e^7 - 2202898350/7579081*e^6 - 1963374973/7579081*e^5 + 6412725870/7579081*e^4 + 1544247799/7579081*e^3 - 6436698630/7579081*e^2 - 191791967/7579081*e + 101329165/398899, 1675603/7579081*e^11 - 5719453/7579081*e^10 - 33886136/7579081*e^9 + 118598336/7579081*e^8 + 230946198/7579081*e^7 - 859953820/7579081*e^6 - 573707549/7579081*e^5 + 2544480383/7579081*e^4 + 288013658/7579081*e^3 - 2661347306/7579081*e^2 - 9849475/7579081*e + 44939458/398899, 1942186/7579081*e^11 - 7830896/7579081*e^10 - 37056620/7579081*e^9 + 164137900/7579081*e^8 + 227604696/7579081*e^7 - 1207564906/7579081*e^6 - 425716158/7579081*e^5 + 3651962366/7579081*e^4 - 206971564/7579081*e^3 - 3997464386/7579081*e^2 + 374527160/7579081*e + 72179764/398899, 129721/398899*e^11 - 364883/398899*e^10 - 2790744/398899*e^9 + 7434044/398899*e^8 + 21102318/398899*e^7 - 52629190/398899*e^6 - 65140479/398899*e^5 + 150489411/398899*e^4 + 70825282/398899*e^3 - 148356264/398899*e^2 - 23190735/398899*e + 45247868/398899, 1247023/7579081*e^11 - 4361774/7579081*e^10 - 24568027/7579081*e^9 + 90764788/7579081*e^8 + 154746570/7579081*e^7 - 655675526/7579081*e^6 - 281383729/7579081*e^5 + 1889450040/7579081*e^4 - 228042285/7579081*e^3 - 1767427178/7579081*e^2 + 324621545/7579081*e + 25089303/398899, -578784/7579081*e^11 + 864128/7579081*e^10 + 13691978/7579081*e^9 - 16073218/7579081*e^8 - 116718834/7579081*e^7 + 102232135/7579081*e^6 + 427501284/7579081*e^5 - 273128564/7579081*e^4 - 615003206/7579081*e^3 + 322284791/7579081*e^2 + 187128382/7579081*e - 9000364/398899, 5328484/7579081*e^11 - 16080961/7579081*e^10 - 112518957/7579081*e^9 + 330543014/7579081*e^8 + 826131290/7579081*e^7 - 2366979957/7579081*e^6 - 2417637532/7579081*e^5 + 6870615187/7579081*e^4 + 2298198035/7579081*e^3 - 6916634227/7579081*e^2 - 522741560/7579081*e + 110692779/398899, -163680/7579081*e^11 - 760051/7579081*e^10 + 6543435/7579081*e^9 + 19264050/7579081*e^8 - 87111312/7579081*e^7 - 186059870/7579081*e^6 + 494227044/7579081*e^5 + 826643589/7579081*e^4 - 1150824715/7579081*e^3 - 1554287850/7579081*e^2 + 726164518/7579081*e + 42218921/398899, -6213414/7579081*e^11 + 18683575/7579081*e^10 + 130697011/7579081*e^9 - 385972842/7579081*e^8 - 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5001517100/7579081*e^6 - 2625634158/7579081*e^5 + 15195381641/7579081*e^4 + 98875729/7579081*e^3 - 16595551454/7579081*e^2 + 1309037998/7579081*e + 301913889/398899, -3164766/7579081*e^11 + 12995503/7579081*e^10 + 62324861/7579081*e^9 - 281376580/7579081*e^8 - 403830732/7579081*e^7 + 2159348962/7579081*e^6 + 865513216/7579081*e^5 - 6904439795/7579081*e^4 + 71931747/7579081*e^3 + 8178773456/7579081*e^2 - 557282228/7579081*e - 150153493/398899, 15252669/7579081*e^11 - 48035092/7579081*e^10 - 320233215/7579081*e^9 + 994990432/7579081*e^8 + 2331727166/7579081*e^7 - 7191600718/7579081*e^6 - 6724707853/7579081*e^5 + 21120488564/7579081*e^4 + 6137252085/7579081*e^3 - 21685700592/7579081*e^2 - 1285162135/7579081*e + 354341295/398899, -107765/398899*e^11 + 238327/398899*e^10 + 2512966/398899*e^9 - 4899988/398899*e^8 - 21301842/398899*e^7 + 35626692/398899*e^6 + 78633777/398899*e^5 - 109889127/398899*e^4 - 116880812/398899*e^3 + 135899876/398899*e^2 + 44178765/398899*e - 60478624/398899] 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]