/* 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([25, 5, -11, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([19, 19, -1/5*w^3 + 1/5*w^2 + 1/5*w + 1]) primes_array = [ [5, 5, 1/5*w^3 + 4/5*w^2 - 11/5*w - 7],\ [5, 5, 1/5*w^3 - 6/5*w^2 - 1/5*w + 4],\ [11, 11, w + 1],\ [11, 11, 1/5*w^3 - 1/5*w^2 - 11/5*w + 2],\ [16, 2, 2],\ [19, 19, -1/5*w^3 + 1/5*w^2 + 1/5*w + 1],\ [19, 19, 1/5*w^3 - 1/5*w^2 - 11/5*w],\ [19, 19, -2/5*w^3 + 2/5*w^2 + 17/5*w],\ [19, 19, w - 1],\ [29, 29, w^2 - w - 8],\ [29, 29, 2/5*w^3 - 2/5*w^2 - 7/5*w + 2],\ [31, 31, 1/5*w^3 - 1/5*w^2 - 1/5*w + 2],\ [31, 31, 2/5*w^3 - 2/5*w^2 - 17/5*w + 3],\ [61, 61, -2/5*w^3 - 3/5*w^2 + 12/5*w + 5],\ [61, 61, -4/5*w^3 - 6/5*w^2 + 39/5*w + 15],\ [61, 61, 3/5*w^3 + 2/5*w^2 - 18/5*w - 3],\ [61, 61, 7/5*w^3 + 3/5*w^2 - 62/5*w - 13],\ [71, 71, 1/5*w^3 - 6/5*w^2 - 6/5*w + 4],\ [71, 71, w^2 - 8],\ [79, 79, 3/5*w^3 + 12/5*w^2 - 33/5*w - 22],\ [79, 79, 1/5*w^3 + 4/5*w^2 - 11/5*w - 9],\ [81, 3, -3],\ [101, 101, 3/5*w^3 + 7/5*w^2 - 28/5*w - 15],\ [101, 101, 1/5*w^3 + 9/5*w^2 - 16/5*w - 16],\ [121, 11, 3/5*w^3 - 3/5*w^2 - 18/5*w + 1],\ [131, 131, 1/5*w^3 + 4/5*w^2 - 16/5*w - 8],\ [131, 131, w^2 - 2*w - 2],\ [149, 149, 4/5*w^3 + 1/5*w^2 - 29/5*w - 5],\ [149, 149, 4/5*w^3 - 9/5*w^2 - 19/5*w + 6],\ [151, 151, 2/5*w^3 + 3/5*w^2 - 17/5*w - 3],\ [151, 151, -2/5*w^3 + 7/5*w^2 + 7/5*w - 8],\ [151, 151, 4/5*w^3 + 1/5*w^2 - 29/5*w - 2],\ [151, 151, -4/5*w^3 + 9/5*w^2 + 19/5*w - 9],\ [179, 179, -3/5*w^3 + 8/5*w^2 + 18/5*w - 6],\ [179, 179, 2/5*w^3 + 3/5*w^2 - 12/5*w - 6],\ [211, 211, 1/5*w^3 + 4/5*w^2 - 6/5*w - 10],\ [211, 211, 2/5*w^3 - 7/5*w^2 - 12/5*w + 2],\ [229, 229, 6/5*w^3 - 11/5*w^2 - 41/5*w + 11],\ [229, 229, 2*w^2 - 2*w - 9],\ [229, 229, -4/5*w^3 - 1/5*w^2 + 19/5*w + 2],\ [229, 229, -3/5*w^3 + 3/5*w^2 + 28/5*w - 3],\ [239, 239, 11/5*w^3 + 4/5*w^2 - 96/5*w - 19],\ [239, 239, -6/5*w^3 + 1/5*w^2 + 51/5*w + 3],\ [241, 241, 1/5*w^3 + 4/5*w^2 - 1/5*w - 6],\ [241, 241, 4/5*w^3 - 4/5*w^2 - 29/5*w - 1],\ [251, 251, -1/5*w^3 + 6/5*w^2 + 11/5*w - 12],\ [251, 251, -1/5*w^3 + 11/5*w^2 + 1/5*w - 15],\ [269, 269, 2/5*w^3 + 3/5*w^2 - 17/5*w - 2],\ [269, 269, -2/5*w^3 + 7/5*w^2 + 7/5*w - 9],\ [271, 271, 1/5*w^3 - 1/5*w^2 + 4/5*w],\ [271, 271, 3/5*w^3 - 3/5*w^2 - 28/5*w + 2],\ [281, 281, w^3 - 3*w^2 - 4*w + 13],\ [281, 281, -3/5*w^3 + 3/5*w^2 + 28/5*w - 6],\ [281, 281, -6/5*w^3 + 26/5*w^2 + 11/5*w - 16],\ [281, 281, w^3 + w^2 - 8*w - 9],\ [289, 17, 2/5*w^3 + 3/5*w^2 - 7/5*w - 5],\ [289, 17, 4/5*w^3 - 9/5*w^2 - 29/5*w + 8],\ [311, 311, -w^2 + 2*w + 11],\ [311, 311, -1/5*w^3 + 6/5*w^2 + 1/5*w - 1],\ [311, 311, 1/5*w^3 + 4/5*w^2 - 16/5*w + 1],\ [311, 311, -1/5*w^3 - 4/5*w^2 + 11/5*w + 10],\ [349, 349, -1/5*w^3 - 9/5*w^2 + 11/5*w + 16],\ [349, 349, 2/5*w^3 - 12/5*w^2 - 7/5*w + 7],\ [359, 359, -4/5*w^3 + 4/5*w^2 + 19/5*w - 1],\ [359, 359, -w^3 + 3*w^2 + 5*w - 12],\ [359, 359, w^3 - w^2 - 7*w + 2],\ [359, 359, 4/5*w^3 + 6/5*w^2 - 29/5*w - 11],\ [379, 379, 4/5*w^3 + 6/5*w^2 - 34/5*w - 9],\ [379, 379, 3/5*w^3 - 3/5*w^2 - 28/5*w + 7],\ [389, 389, w^2 - 3*w - 8],\ [389, 389, -4/5*w^3 + 14/5*w^2 + 19/5*w - 16],\ [401, 401, -3*w^2 + 2*w + 23],\ [401, 401, 1/5*w^3 - 16/5*w^2 + 4/5*w + 11],\ [409, 409, -w^3 + 2*w^2 + 3*w - 7],\ [409, 409, 7/5*w^3 - 2/5*w^2 - 57/5*w - 2],\ [419, 419, -1/5*w^3 + 1/5*w^2 + 16/5*w - 2],\ [419, 419, 1/5*w^3 - 1/5*w^2 - 16/5*w],\ [421, 421, 2/5*w^3 + 3/5*w^2 - 27/5*w - 3],\ [421, 421, 1/5*w^3 + 4/5*w^2 + 4/5*w - 3],\ [439, 439, -w^3 + 3*w^2 + 5*w - 16],\ [439, 439, -4/5*w^3 - 6/5*w^2 + 29/5*w + 7],\ [439, 439, -3/5*w^3 + 13/5*w^2 + 3/5*w - 11],\ [439, 439, 4/5*w^3 + 6/5*w^2 - 39/5*w - 10],\ [449, 449, 2/5*w^3 - 12/5*w^2 - 2/5*w + 5],\ [449, 449, 4/5*w^3 - 24/5*w^2 - 4/5*w + 15],\ [461, 461, -1/5*w^3 - 4/5*w^2 - 4/5*w + 4],\ [461, 461, 2*w^2 - 2*w - 11],\ [461, 461, 3/5*w^3 + 2/5*w^2 - 33/5*w - 4],\ [479, 479, -w^3 + 3*w^2 + 6*w - 14],\ [479, 479, -3/5*w^3 + 3/5*w^2 + 28/5*w + 3],\ [499, 499, 2/5*w^3 + 8/5*w^2 - 17/5*w - 6],\ [499, 499, -3/5*w^3 + 13/5*w^2 + 13/5*w - 17],\ [509, 509, -w^3 + 2*w^2 + 4*w - 6],\ [509, 509, 6/5*w^3 - 1/5*w^2 - 46/5*w - 4],\ [521, 521, -2/5*w^3 + 2/5*w^2 + 7/5*w - 7],\ [521, 521, -3/5*w^3 + 3/5*w^2 + 23/5*w - 8],\ [529, 23, -7/5*w^3 + 17/5*w^2 + 42/5*w - 15],\ [529, 23, 2/5*w^3 + 13/5*w^2 - 27/5*w - 21],\ [541, 541, 1/5*w^3 - 11/5*w^2 - 1/5*w + 10],\ [541, 541, 3/5*w^3 - 8/5*w^2 - 13/5*w + 13],\ [541, 541, -2*w^2 + w + 13],\ [541, 541, 3/5*w^3 + 2/5*w^2 - 23/5*w + 2],\ [569, 569, w^3 - 2*w^2 - 5*w + 7],\ [569, 569, -1/5*w^3 + 1/5*w^2 + 1/5*w - 6],\ [571, 571, w^2 + w - 9],\ [571, 571, -2/5*w^3 + 12/5*w^2 + 7/5*w - 12],\ [571, 571, 1/5*w^3 + 9/5*w^2 - 11/5*w - 11],\ [571, 571, -2/5*w^3 + 7/5*w^2 + 17/5*w - 4],\ [601, 601, -3/5*w^3 + 13/5*w^2 + 18/5*w - 11],\ [601, 601, -4/5*w^3 + 4/5*w^2 + 34/5*w + 1],\ [631, 631, 2/5*w^3 + 18/5*w^2 - 27/5*w - 30],\ [631, 631, 3/5*w^3 - 23/5*w^2 - 3/5*w + 15],\ [661, 661, -2*w^3 + 17*w + 12],\ [661, 661, -9/5*w^3 + 4/5*w^2 + 74/5*w + 1],\ [691, 691, -7/5*w^3 - 18/5*w^2 + 72/5*w + 35],\ [691, 691, 6/5*w^3 - 31/5*w^2 - 6/5*w + 19],\ [701, 701, -1/5*w^3 + 6/5*w^2 + 16/5*w - 6],\ [701, 701, 2/5*w^3 - 7/5*w^2 - 22/5*w + 8],\ [709, 709, -3/5*w^3 + 13/5*w^2 + 13/5*w - 18],\ [709, 709, 1/5*w^3 + 14/5*w^2 - 11/5*w - 20],\ [709, 709, -2/5*w^3 - 8/5*w^2 + 17/5*w + 5],\ [709, 709, 1/5*w^3 + 4/5*w^2 - 21/5*w - 10],\ [719, 719, -3/5*w^3 + 8/5*w^2 + 13/5*w - 12],\ [719, 719, 3/5*w^3 + 2/5*w^2 - 23/5*w + 1],\ [719, 719, -7/5*w^3 + 17/5*w^2 + 32/5*w - 14],\ [719, 719, 7/5*w^3 + 3/5*w^2 - 52/5*w - 8],\ [739, 739, 1/5*w^3 - 1/5*w^2 - 11/5*w - 5],\ [739, 739, w - 6],\ [761, 761, -4/5*w^3 + 9/5*w^2 + 19/5*w - 11],\ [761, 761, 4/5*w^3 + 1/5*w^2 - 29/5*w],\ [769, 769, 3/5*w^3 - 13/5*w^2 - 13/5*w + 8],\ [769, 769, -2/5*w^3 - 8/5*w^2 + 17/5*w + 15],\ [769, 769, 7/5*w^3 - 17/5*w^2 - 37/5*w + 18],\ [769, 769, -6/5*w^3 - 4/5*w^2 + 41/5*w + 5],\ [809, 809, -1/5*w^3 + 6/5*w^2 - 4/5*w - 14],\ [809, 809, -2/5*w^3 - 3/5*w^2 + 22/5*w - 4],\ [809, 809, -6/5*w^3 + 6/5*w^2 + 41/5*w - 5],\ [809, 809, w^3 - w^2 - 5*w + 4],\ [821, 821, -3/5*w^3 - 17/5*w^2 + 33/5*w + 30],\ [821, 821, -4/5*w^3 + 24/5*w^2 + 9/5*w - 15],\ [829, 829, 7/5*w^3 - 12/5*w^2 - 52/5*w + 10],\ [829, 829, -1/5*w^3 + 11/5*w^2 - 4/5*w - 19],\ [829, 829, -13/5*w^3 + 3/5*w^2 + 108/5*w + 10],\ [829, 829, -w^3 + 4*w^2 + 5*w - 23],\ [839, 839, 4/5*w^3 - 9/5*w^2 - 24/5*w + 4],\ [839, 839, 3/5*w^3 + 2/5*w^2 - 18/5*w - 8],\ [841, 29, -w^3 + w^2 + 6*w - 4],\ [859, 859, 6/5*w^3 - 11/5*w^2 - 36/5*w + 9],\ [859, 859, w^3 - 6*w - 3],\ [881, 881, 4/5*w^3 - 4/5*w^2 - 9/5*w - 1],\ [881, 881, 9/5*w^3 - 24/5*w^2 - 19/5*w + 14],\ [881, 881, -w^3 + 6*w + 2],\ [881, 881, 1/5*w^3 - 1/5*w^2 - 1/5*w - 6],\ [911, 911, w^3 - w^2 - 5*w + 1],\ [911, 911, 6/5*w^3 - 6/5*w^2 - 41/5*w + 2],\ [919, 919, -7/5*w^3 + 12/5*w^2 + 42/5*w - 7],\ [919, 919, -6/5*w^3 + 1/5*w^2 + 36/5*w + 5],\ [929, 929, -2/5*w^3 - 8/5*w^2 + 17/5*w + 16],\ [929, 929, -3/5*w^3 + 13/5*w^2 + 13/5*w - 7],\ [961, 31, w^3 - w^2 - 6*w + 2],\ [971, 971, 8/5*w^3 - 13/5*w^2 - 48/5*w + 6],\ [971, 971, -2/5*w^3 - 3/5*w^2 + 22/5*w],\ [971, 971, 7/5*w^3 - 7/5*w^2 - 57/5*w + 7],\ [971, 971, -1/5*w^3 + 6/5*w^2 - 4/5*w - 10],\ [991, 991, 2/5*w^3 + 8/5*w^2 - 22/5*w - 7],\ [991, 991, -2/5*w^3 + 12/5*w^2 + 2/5*w - 15]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^10 - 5*x^9 - 18*x^8 + 98*x^7 + 105*x^6 - 621*x^5 - 177*x^4 + 1436*x^3 - 88*x^2 - 1088*x + 304 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, 36441/845392*e^9 - 229385/845392*e^8 - 175861/422696*e^7 + 1973431/422696*e^6 - 1253695/845392*e^5 - 19641161/845392*e^4 + 17755471/845392*e^3 + 5445983/211348*e^2 - 6891285/211348*e + 469192/52837, 26281/845392*e^9 - 141565/845392*e^8 - 192619/422696*e^7 + 1271011/422696*e^6 + 1354665/845392*e^5 - 13712141/845392*e^4 + 1277283/845392*e^3 + 1328833/52837*e^2 - 708833/211348*e - 294332/52837, 10689/422696*e^9 - 11163/105674*e^8 - 236059/422696*e^7 + 119092/52837*e^6 + 1485911/422696*e^5 - 1406563/105674*e^4 - 945333/211348*e^3 + 8139475/422696*e^2 - 274975/105674*e - 20511/105674, -10201/845392*e^9 + 10063/845392*e^8 + 91057/211348*e^7 - 96901/422696*e^6 - 4512653/845392*e^5 + 1100783/845392*e^4 + 21020571/845392*e^3 - 1823513/422696*e^2 - 6966673/211348*e + 839171/105674, -1, 946/52837*e^9 - 8801/52837*e^8 + 2247/52837*e^7 + 173241/52837*e^6 - 306103/52837*e^5 - 969964/52837*e^4 + 2145325/52837*e^3 + 1122499/52837*e^2 - 2878896/52837*e + 440022/52837, -81/105674*e^9 - 12121/211348*e^8 + 65075/211348*e^7 + 105083/105674*e^6 - 301320/52837*e^5 - 946601/211348*e^4 + 6607773/211348*e^3 - 145085/211348*e^2 - 2205822/52837*e + 617049/52837, -3567/52837*e^9 + 49391/105674*e^8 + 43823/105674*e^7 - 419368/52837*e^6 + 355553/52837*e^5 + 4063511/105674*e^4 - 6027645/105674*e^3 - 4025147/105674*e^2 + 4438356/52837*e - 991932/52837, 28145/422696*e^9 - 222579/422696*e^8 - 5907/52837*e^7 + 1915769/211348*e^6 - 5277931/422696*e^5 - 18529283/422696*e^4 + 37414185/422696*e^3 + 7340803/211348*e^2 - 13233919/105674*e + 2195969/52837, 5203/211348*e^9 + 8863/422696*e^8 - 503653/422696*e^7 + 81015/211348*e^6 + 1549091/105674*e^5 - 2691217/422696*e^4 - 25751183/422696*e^3 + 10128335/422696*e^2 + 3650046/52837*e - 2673459/105674, -14523/422696*e^9 + 110035/422696*e^8 + 7467/211348*e^7 - 807377/211348*e^6 + 1902677/422696*e^5 + 6964787/422696*e^4 - 10803773/422696*e^3 - 2083121/105674*e^2 + 3549371/105674*e + 20496/52837, 107263/845392*e^9 - 734003/845392*e^8 - 321921/422696*e^7 + 5987141/422696*e^6 - 9154129/845392*e^5 - 56282259/845392*e^4 + 74039421/845392*e^3 + 7095089/105674*e^2 - 26359447/211348*e + 1781137/52837, 4567/105674*e^9 - 94137/211348*e^8 + 93411/211348*e^7 + 804351/105674*e^6 - 922503/52837*e^5 - 7767649/211348*e^4 + 22609605/211348*e^3 + 5873723/211348*e^2 - 7788800/52837*e + 1995395/52837, -45765/422696*e^9 + 247989/422696*e^8 + 328437/211348*e^7 - 2254019/211348*e^6 - 1542245/422696*e^5 + 23690669/422696*e^4 - 9942275/422696*e^3 - 7571755/105674*e^2 + 5932607/105674*e - 36088/52837, 17715/422696*e^9 - 66899/422696*e^8 - 206311/211348*e^7 + 631157/211348*e^6 + 3669875/422696*e^5 - 7185835/422696*e^4 - 13226683/422696*e^3 + 3245935/105674*e^2 + 3326805/105674*e - 534074/52837, 36579/422696*e^9 - 158395/422696*e^8 - 378723/211348*e^7 + 1615817/211348*e^6 + 4528355/422696*e^5 - 18601923/422696*e^4 - 5327195/422696*e^3 + 6823735/105674*e^2 - 902707/105674*e - 530878/52837, -7715/52837*e^9 + 165231/211348*e^8 + 432393/211348*e^7 - 1429593/105674*e^6 - 618443/105674*e^5 + 14207879/211348*e^4 - 3858855/211348*e^3 - 16601387/211348*e^2 + 2426162/52837*e - 241389/52837, -10025/211348*e^9 + 26945/105674*e^8 + 132409/211348*e^7 - 205662/52837*e^6 - 772711/211348*e^5 + 2033599/105674*e^4 + 731995/52837*e^3 - 8177545/211348*e^2 - 697468/52837*e + 1213445/52837, 13809/105674*e^9 - 125351/211348*e^8 - 511027/211348*e^7 + 1145753/105674*e^6 + 751634/52837*e^5 - 11825659/211348*e^4 - 5231341/211348*e^3 + 14080801/211348*e^2 + 603822/52837*e + 196775/52837, 37735/845392*e^9 - 378487/845392*e^8 + 194697/422696*e^7 + 3079901/422696*e^6 - 14980289/845392*e^5 - 27000295/845392*e^4 + 91742569/845392*e^3 + 2529119/211348*e^2 - 32132533/211348*e + 2716902/52837, -5065/422696*e^9 + 10035/105674*e^8 - 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9672719/211348*e^6 - 19375663/422696*e^5 + 100961657/422696*e^4 + 27670709/422696*e^3 - 70679297/211348*e^2 - 1198511/105674*e + 3269671/52837, -54795/105674*e^9 + 755261/211348*e^8 + 680037/211348*e^7 - 6305803/105674*e^6 + 2385411/52837*e^5 + 61426901/211348*e^4 - 78666037/211348*e^3 - 69529231/211348*e^2 + 27488318/52837*e - 4780633/52837, -54163/211348*e^9 + 719373/422696*e^8 + 1114225/422696*e^7 - 7060479/211348*e^6 + 1883253/105674*e^5 + 78211189/422696*e^4 - 94105957/422696*e^3 - 90802707/422696*e^2 + 18784069/52837*e - 6049189/105674, -6816/52837*e^9 + 22304/52837*e^8 + 158742/52837*e^7 - 302175/52837*e^6 - 1836815/52837*e^5 + 1422681/52837*e^4 + 9357398/52837*e^3 - 4599441/52837*e^2 - 12914622/52837*e + 6450858/52837, 52971/422696*e^9 - 368313/422696*e^8 - 25890/52837*e^7 + 2744099/211348*e^6 - 5607665/422696*e^5 - 22854065/422696*e^4 + 41934971/422696*e^3 + 10480961/211348*e^2 - 17597921/105674*e + 1309363/52837, -49651/211348*e^9 + 308829/211348*e^8 + 159984/52837*e^7 - 3158597/105674*e^6 + 1353789/211348*e^5 + 37144661/211348*e^4 - 32627115/211348*e^3 - 25846029/105674*e^2 + 13316913/52837*e - 2043430/52837, 41161/422696*e^9 + 134207/422696*e^8 - 1322169/211348*e^7 + 174623/211348*e^6 + 32172321/422696*e^5 - 15773905/422696*e^4 - 125124001/422696*e^3 + 13388545/105674*e^2 + 34606319/105674*e - 6277750/52837] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([19, 19, -1/5*w^3 + 1/5*w^2 + 1/5*w + 1])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]