/* 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([11, 9, -9, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([59, 59, -3*w^3 - 4*w^2 + 19*w + 14]) primes_array = [ [5, 5, -2*w^3 - 2*w^2 + 13*w + 8],\ [11, 11, 2*w^3 + 3*w^2 - 12*w - 11],\ [11, 11, -w^3 - 2*w^2 + 6*w + 9],\ [11, 11, w^3 + w^2 - 7*w - 4],\ [11, 11, w - 1],\ [16, 2, 2],\ [19, 19, -w^2 + 4],\ [19, 19, 2*w^3 + 2*w^2 - 13*w - 6],\ [19, 19, 3*w^3 + 4*w^2 - 18*w - 16],\ [19, 19, -w^3 - w^2 + 5*w + 3],\ [49, 7, 2*w^3 + 3*w^2 - 13*w - 15],\ [59, 59, -3*w^3 - 4*w^2 + 19*w + 14],\ [59, 59, 3*w^3 + 4*w^2 - 18*w - 17],\ [59, 59, 2*w^3 + 3*w^2 - 11*w - 13],\ [59, 59, -w^3 - 2*w^2 + 7*w + 9],\ [71, 71, 3*w^3 + 3*w^2 - 17*w - 10],\ [71, 71, -2*w^3 - w^2 + 14*w + 2],\ [71, 71, 5*w^3 + 7*w^2 - 30*w - 25],\ [71, 71, -3*w^3 - 4*w^2 + 16*w + 16],\ [81, 3, -3],\ [89, 89, -2*w^3 - 2*w^2 + 13*w + 4],\ [89, 89, 3*w^3 + 4*w^2 - 17*w - 16],\ [89, 89, 3*w^3 + 4*w^2 - 18*w - 18],\ [89, 89, -4*w^3 - 5*w^2 + 25*w + 18],\ [139, 139, -w^3 + 6*w + 1],\ [139, 139, -3*w^3 - 3*w^2 + 19*w + 8],\ [139, 139, 4*w^3 + 5*w^2 - 24*w - 21],\ [139, 139, -2*w^3 - 2*w^2 + 11*w + 5],\ [151, 151, -6*w^3 - 7*w^2 + 37*w + 27],\ [151, 151, 3*w^3 + 5*w^2 - 19*w - 17],\ [151, 151, -6*w^3 - 8*w^2 + 37*w + 31],\ [151, 151, 4*w^3 + 6*w^2 - 25*w - 26],\ [191, 191, 5*w^3 + 7*w^2 - 31*w - 26],\ [191, 191, -2*w^3 - 3*w^2 + 10*w + 13],\ [191, 191, -5*w^3 - 7*w^2 + 31*w + 29],\ [191, 191, 2*w^3 + 4*w^2 - 13*w - 17],\ [199, 199, 2*w^3 + 4*w^2 - 11*w - 20],\ [199, 199, -2*w^3 - 2*w^2 + 11*w + 3],\ [199, 199, 3*w^3 + 5*w^2 - 17*w - 17],\ [199, 199, 3*w^3 + 3*w^2 - 19*w - 6],\ [211, 211, 3*w^3 + 5*w^2 - 18*w - 19],\ [211, 211, -3*w^3 - 5*w^2 + 18*w + 21],\ [211, 211, 2*w^3 + 4*w^2 - 12*w - 17],\ [211, 211, -4*w^3 - 6*w^2 + 24*w + 23],\ [229, 229, 2*w^3 + 4*w^2 - 13*w - 19],\ [229, 229, -2*w^3 - 3*w^2 + 10*w + 15],\ [229, 229, -5*w^3 - 7*w^2 + 31*w + 24],\ [229, 229, 2*w^3 + 2*w^2 - 14*w - 3],\ [269, 269, 2*w^3 + w^2 - 11*w - 1],\ [269, 269, -6*w^3 - 7*w^2 + 37*w + 24],\ [269, 269, 5*w^3 + 6*w^2 - 29*w - 24],\ [269, 269, 3*w^3 + 2*w^2 - 19*w - 7],\ [281, 281, -4*w^3 - 6*w^2 + 25*w + 25],\ [281, 281, -5*w^3 - 7*w^2 + 31*w + 27],\ [281, 281, w^2 + 2*w - 7],\ [281, 281, 3*w^3 + 5*w^2 - 19*w - 18],\ [331, 331, -2*w^3 - 3*w^2 + 12*w + 7],\ [331, 331, w^3 + w^2 - 7*w - 8],\ [331, 331, w^3 + 2*w^2 - 6*w - 13],\ [331, 331, w - 5],\ [349, 349, 2*w^3 + 5*w^2 - 16*w - 12],\ [349, 349, 2*w^3 + 4*w^2 - 15*w - 10],\ [349, 349, 8*w^3 + 9*w^2 - 50*w - 32],\ [349, 349, 2*w^3 + 4*w^2 - 15*w - 9],\ [401, 401, 4*w^3 + 3*w^2 - 21*w - 12],\ [401, 401, 3*w^3 + 2*w^2 - 16*w - 9],\ [401, 401, -3*w^3 - 6*w^2 + 20*w + 19],\ [401, 401, -4*w^3 - 3*w^2 + 27*w + 9],\ [409, 409, -5*w^3 - 6*w^2 + 31*w + 20],\ [409, 409, 3*w^3 + 4*w^2 - 20*w - 12],\ [409, 409, -3*w^3 - 5*w^2 + 17*w + 23],\ [409, 409, 4*w^3 + 5*w^2 - 23*w - 21],\ [419, 419, -4*w^3 - 4*w^2 + 21*w + 18],\ [419, 419, -4*w^3 - 4*w^2 + 20*w + 19],\ [419, 419, -3*w^3 - 4*w^2 + 16*w + 18],\ [419, 419, -2*w^3 - w^2 + 8*w + 10],\ [421, 421, -6*w^3 - 7*w^2 + 36*w + 26],\ [421, 421, 5*w^3 + 5*w^2 - 31*w - 17],\ [421, 421, 3*w^3 + 2*w^2 - 18*w - 6],\ [421, 421, 4*w^3 + 4*w^2 - 23*w - 14],\ [431, 431, 6*w^3 + 7*w^2 - 36*w - 25],\ [431, 431, -3*w^3 - 2*w^2 + 18*w + 5],\ [431, 431, 4*w^3 + 4*w^2 - 23*w - 15],\ [431, 431, -5*w^3 - 5*w^2 + 31*w + 18],\ [439, 439, w^3 + w^2 - 4*w - 1],\ [439, 439, w^3 + 3*w^2 - 6*w - 10],\ [439, 439, 3*w^3 + 3*w^2 - 20*w - 7],\ [439, 439, 5*w^3 + 7*w^2 - 30*w - 30],\ [479, 479, -4*w^3 - 5*w^2 + 25*w + 15],\ [479, 479, -5*w^3 - 6*w^2 + 32*w + 21],\ [479, 479, 5*w^3 + 7*w^2 - 29*w - 28],\ [479, 479, 2*w^2 + w - 9],\ [491, 491, 7*w^3 + 8*w^2 - 43*w - 31],\ [491, 491, 6*w^3 + 6*w^2 - 37*w - 24],\ [491, 491, 4*w^3 + 3*w^2 - 24*w - 6],\ [491, 491, 5*w^3 + 5*w^2 - 29*w - 21],\ [509, 509, -4*w^3 - 5*w^2 + 25*w + 14],\ [509, 509, -3*w^3 - 4*w^2 + 17*w + 20],\ [509, 509, -6*w^3 - 7*w^2 + 38*w + 25],\ [509, 509, -6*w^3 - 8*w^2 + 35*w + 31],\ [541, 541, 9*w^3 + 12*w^2 - 55*w - 46],\ [541, 541, w^3 + 4*w^2 - 7*w - 17],\ [541, 541, -4*w^3 - 5*w^2 + 21*w + 19],\ [541, 541, -4*w^3 - 3*w^2 + 27*w + 8],\ [571, 571, -3*w^3 - 6*w^2 + 18*w + 23],\ [571, 571, 6*w^3 + 9*w^2 - 36*w - 37],\ [571, 571, 7*w^3 + 10*w^2 - 42*w - 36],\ [571, 571, 2*w^3 + 5*w^2 - 12*w - 24],\ [619, 619, -9*w^3 - 11*w^2 + 55*w + 39],\ [619, 619, -5*w^3 - 4*w^2 + 32*w + 14],\ [619, 619, 7*w^3 + 9*w^2 - 42*w - 36],\ [619, 619, 4*w^3 + 6*w^2 - 26*w - 27],\ [631, 631, 2*w^3 + 5*w^2 - 12*w - 21],\ [631, 631, -7*w^3 - 10*w^2 + 42*w + 39],\ [631, 631, 4*w^3 + 4*w^2 - 27*w - 13],\ [631, 631, w^3 + w^2 - 3*w - 4],\ [641, 641, -9*w^3 - 10*w^2 + 56*w + 36],\ [641, 641, -5*w^3 - 9*w^2 + 30*w + 32],\ [641, 641, -3*w^3 - 6*w^2 + 19*w + 21],\ [641, 641, 4*w^3 + 3*w^2 - 22*w - 10],\ [701, 701, -2*w^3 - 5*w^2 + 14*w + 20],\ [701, 701, -6*w^3 - 7*w^2 + 36*w + 34],\ [701, 701, 4*w^3 + 6*w^2 - 23*w - 17],\ [701, 701, -9*w^3 - 12*w^2 + 56*w + 46],\ [719, 719, 3*w^3 + 5*w^2 - 17*w - 25],\ [719, 719, -3*w^3 - 2*w^2 + 17*w + 3],\ [719, 719, 7*w^3 + 8*w^2 - 43*w - 26],\ [719, 719, -6*w^3 - 7*w^2 + 35*w + 29],\ [751, 751, 2*w^3 + 2*w^2 - 15*w - 4],\ [751, 751, 5*w^3 + 8*w^2 - 30*w - 34],\ [751, 751, w^3 + w^2 - 9*w - 5],\ [751, 751, -4*w^3 - 7*w^2 + 24*w + 26],\ [769, 769, 6*w^3 + 8*w^2 - 37*w - 27],\ [769, 769, -w^3 - 3*w^2 + 7*w + 16],\ [769, 769, 2*w^3 + w^2 - 14*w - 5],\ [769, 769, -3*w^3 - 4*w^2 + 16*w + 19],\ [821, 821, 6*w^3 + 6*w^2 - 37*w - 23],\ [821, 821, 5*w^3 + 5*w^2 - 29*w - 20],\ [821, 821, -7*w^3 - 8*w^2 + 42*w + 27],\ [821, 821, 4*w^3 + 3*w^2 - 24*w - 7],\ [829, 829, 7*w^3 + 8*w^2 - 44*w - 28],\ [829, 829, 7*w^3 + 9*w^2 - 41*w - 35],\ [829, 829, -2*w^3 - w^2 + 10*w + 2],\ [829, 829, 2*w^3 - 13*w + 2],\ [839, 839, 4*w^3 + 8*w^2 - 25*w - 26],\ [839, 839, -3*w^3 - 3*w^2 + 15*w + 16],\ [839, 839, -w^3 - 4*w^2 + 8*w + 10],\ [839, 839, -3*w^3 - 7*w^2 + 18*w + 26],\ [841, 29, -5*w^3 - 5*w^2 + 30*w + 16],\ [841, 29, -5*w^3 - 5*w^2 + 30*w + 19],\ [859, 859, 5*w^3 + 6*w^2 - 32*w - 20],\ [859, 859, -4*w^3 - 5*w^2 + 23*w + 23],\ [859, 859, -5*w^3 - 6*w^2 + 31*w + 18],\ [859, 859, -2*w^3 - w^2 + 13*w + 6],\ [911, 911, 9*w^3 + 12*w^2 - 56*w - 48],\ [911, 911, 6*w^3 + 5*w^2 - 38*w - 14],\ [911, 911, 7*w^3 + 8*w^2 - 40*w - 28],\ [911, 911, 10*w^3 + 12*w^2 - 61*w - 46],\ [929, 929, -7*w^3 - 10*w^2 + 39*w + 39],\ [929, 929, -10*w^3 - 13*w^2 + 60*w + 45],\ [929, 929, -6*w^3 - 7*w^2 + 32*w + 29],\ [929, 929, -3*w^3 - w^2 + 12*w + 12],\ [961, 31, -5*w^3 - 5*w^2 + 30*w + 17],\ [961, 31, -5*w^3 - 5*w^2 + 30*w + 18],\ [991, 991, 10*w^3 + 12*w^2 - 61*w - 45],\ [991, 991, -6*w^3 - 9*w^2 + 38*w + 38],\ [991, 991, 6*w^3 + 8*w^2 - 37*w - 38],\ [991, 991, -8*w^3 - 11*w^2 + 49*w + 43]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^8 - 10*x^7 + 29*x^6 - 100*x^4 + 64*x^3 + 78*x^2 - 50*x - 1 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, 105/809*e^7 - 1150/809*e^6 + 3755/809*e^5 - 841/809*e^4 - 12858/809*e^3 + 8911/809*e^2 + 11299/809*e - 4762/809, -134/809*e^7 + 1275/809*e^6 - 3020/809*e^5 - 3095/809*e^4 + 13728/809*e^3 + 1464/809*e^2 - 11384/809*e - 102/809, -53/809*e^7 + 619/809*e^6 - 2088/809*e^5 + 255/809*e^4 + 8062/809*e^3 - 5253/809*e^2 - 6181/809*e + 1795/809, 11/809*e^7 - 159/809*e^6 + 586/809*e^5 - 404/809*e^4 - 330/809*e^3 - 253/809*e^2 - 2060/809*e + 2375/809, 91/809*e^7 - 727/809*e^6 + 1097/809*e^5 + 3424/809*e^4 - 6775/809*e^3 - 6947/809*e^2 + 8552/809*e + 4718/809, 21/809*e^7 - 230/809*e^6 + 751/809*e^5 - 330/809*e^4 - 1439/809*e^3 + 326/809*e^2 + 1289/809*e + 342/809, -187/809*e^7 + 1894/809*e^6 - 5108/809*e^5 - 2840/809*e^4 + 21790/809*e^3 - 2980/809*e^2 - 21610/809*e + 4120/809, -48/809*e^7 + 179/809*e^6 + 826/809*e^5 - 3753/809*e^4 + 1440/809*e^3 + 4340/809*e^2 - 1675/809*e + 3610/809, -72/809*e^7 + 673/809*e^6 - 1997/809*e^5 + 1247/809*e^4 + 3778/809*e^3 - 5625/809*e^2 - 1299/809*e + 4606/809, 11/809*e^7 - 159/809*e^6 + 1395/809*e^5 - 5258/809*e^4 + 3715/809*e^3 + 11073/809*e^2 - 6105/809*e - 861/809, -1, 433/809*e^7 - 4126/809*e^6 + 10785/809*e^5 + 4175/809*e^4 - 42114/809*e^3 + 15929/809*e^2 + 37172/809*e - 10284/809, 71/809*e^7 - 585/809*e^6 + 1576/809*e^5 - 1578/809*e^4 - 512/809*e^3 + 4030/809*e^2 - 573/809*e - 924/809, -43/809*e^7 + 548/809*e^6 - 1923/809*e^5 + 329/809*e^4 + 7762/809*e^3 - 8719/809*e^2 - 2832/809*e + 11088/809, 7/809*e^7 + 193/809*e^6 - 1907/809*e^5 + 3935/809*e^4 + 3835/809*e^3 - 11487/809*e^2 - 5503/809*e + 3350/809, 62/809*e^7 - 602/809*e^6 + 1832/809*e^5 - 512/809*e^4 - 5096/809*e^3 + 1810/809*e^2 + 2804/809*e + 4708/809, -234/809*e^7 + 1985/809*e^6 - 4670/809*e^5 - 599/809*e^4 + 11874/809*e^3 - 7562/809*e^2 - 4424/809*e + 1621/809, -189/809*e^7 + 1261/809*e^6 - 287/809*e^5 - 10783/809*e^4 + 11333/809*e^3 + 20527/809*e^2 - 14837/809*e - 7932/809, -214/809*e^7 + 1843/809*e^6 - 4340/809*e^5 - 2069/809*e^4 + 16128/809*e^3 - 3168/809*e^2 - 17142/809*e - 1636/809, 575/809*e^7 - 5296/809*e^6 + 13128/809*e^5 + 5873/809*e^4 - 47183/809*e^3 + 15090/809*e^2 + 31172/809*e - 9705/809, -92/809*e^7 + 815/809*e^6 - 1518/809*e^5 - 3755/809*e^4 + 10850/809*e^3 + 2116/809*e^2 - 11233/809*e + 582/809, 689/809*e^7 - 6429/809*e^6 + 15818/809*e^5 + 8820/809*e^4 - 57884/809*e^3 + 7614/809*e^2 + 43139/809*e - 4728/809, -272/809*e^7 + 2902/809*e^6 - 9342/809*e^5 + 3812/809*e^4 + 23531/809*e^3 - 21250/809*e^2 - 4368/809*e + 12906/809, 42/809*e^7 - 460/809*e^6 + 1502/809*e^5 + 958/809*e^4 - 10968/809*e^3 + 3888/809*e^2 + 15522/809*e - 934/809, 264/809*e^7 - 2198/809*e^6 + 3547/809*e^5 + 9720/809*e^4 - 20864/809*e^3 - 13353/809*e^2 + 21752/809*e + 8460/809, 297/809*e^7 - 3484/809*e^6 + 11777/809*e^5 - 2818/809*e^4 - 36416/809*e^3 + 18248/809*e^2 + 24471/809*e - 6258/809, 17/809*e^7 + 122/809*e^6 - 1742/809*e^5 + 4009/809*e^4 + 3535/809*e^3 - 14144/809*e^2 - 5390/809*e + 12643/809, -152/809*e^7 + 432/809*e^6 + 3964/809*e^5 - 14716/809*e^4 - 294/809*e^3 + 30193/809*e^2 - 9484/809*e - 5018/809, -402/809*e^7 + 3825/809*e^6 - 10678/809*e^5 + 423/809*e^4 + 34712/809*e^3 - 23923/809*e^2 - 26062/809*e + 8593/809, 51/809*e^7 - 443/809*e^6 + 1246/809*e^5 + 701/809*e^4 - 11238/809*e^3 + 11771/809*e^2 + 20235/809*e - 12229/809, 886/809*e^7 - 8394/809*e^6 + 21900/809*e^5 + 6071/809*e^4 - 73502/809*e^3 + 25735/809*e^2 + 47064/809*e - 17353/809, -824/809*e^7 + 7792/809*e^6 - 20877/809*e^5 - 1729/809*e^4 + 62743/809*e^3 - 27970/809*e^2 - 43451/809*e + 10735/809, -276/809*e^7 + 2445/809*e^6 - 6172/809*e^5 - 748/809*e^4 + 21224/809*e^3 - 20349/809*e^2 - 11856/809*e + 17117/809, -13/809*e^7 + 1144/809*e^6 - 7900/809*e^5 + 11877/809*e^4 + 17379/809*e^3 - 30443/809*e^2 - 13819/809*e + 11461/809, 176/809*e^7 - 1735/809*e^6 + 6140/809*e^5 - 6464/809*e^4 - 12561/809*e^3 + 25076/809*e^2 + 9108/809*e - 15394/809, 6/809*e^7 - 528/809*e^6 + 4144/809*e^5 - 7722/809*e^4 - 9888/809*e^3 + 24132/809*e^2 + 6378/809*e + 3796/809, 571/809*e^7 - 4944/809*e^6 + 10635/809*e^5 + 10212/809*e^4 - 42209/809*e^3 - 998/809*e^2 + 30156/809*e + 7450/809, -302/809*e^7 + 3115/809*e^6 - 9028/809*e^5 - 455/809*e^4 + 25240/809*e^3 + 2092/809*e^2 - 16033/809*e - 17400/809, -490/809*e^7 + 4288/809*e^6 - 9703/809*e^5 - 8480/809*e^4 + 47060/809*e^3 - 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21542/809*e^6 + 51669/809*e^5 + 24526/809*e^4 - 176241/809*e^3 + 40354/809*e^2 + 128038/809*e - 46849/809, -722/809*e^7 + 6097/809*e^6 - 11104/809*e^5 - 24597/809*e^4 + 71818/809*e^3 + 13370/809*e^2 - 78218/809*e + 6502/809, -563/809*e^7 + 4240/809*e^6 - 4840/809*e^5 - 26980/809*e^4 + 58149/809*e^3 + 17803/809*e^2 - 87990/809*e + 15679/809, 920/809*e^7 - 8150/809*e^6 + 20034/809*e^5 + 5190/809*e^4 - 71286/809*e^3 + 55695/809*e^2 + 49228/809*e - 45461/809, -1414/809*e^7 + 12790/809*e^6 - 30612/809*e^5 - 20657/809*e^4 + 133028/809*e^3 - 41097/809*e^2 - 124546/809*e + 40074/809] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([59, 59, -3*w^3 - 4*w^2 + 19*w + 14])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]