/* 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([4, 8, -7, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([9, 3, -1/2*w^2 + 1/2*w + 2]) primes_array = [ [2, 2, 1/2*w^2 + 1/2*w - 1],\ [2, 2, -1/2*w^2 + 3/2*w],\ [9, 3, -1/2*w^2 + 1/2*w + 2],\ [13, 13, 1/2*w^3 - w^2 - 5/2*w + 2],\ [13, 13, -1/2*w^3 + 1/2*w^2 + 3*w - 1],\ [23, 23, 1/2*w^3 - w^2 - 5/2*w],\ [23, 23, -1/2*w^3 + 1/2*w^2 + 3*w - 3],\ [37, 37, 1/2*w^3 - 1/2*w^2 - 3*w - 3],\ [37, 37, 1/2*w^3 - w^2 - 5/2*w + 6],\ [59, 59, -1/2*w^3 + 7/2*w],\ [59, 59, -1/2*w^3 + 3/2*w^2 + 2*w - 3],\ [61, 61, -w^3 + 2*w^2 + 7*w - 7],\ [61, 61, w^3 - w^2 - 6*w + 3],\ [61, 61, w^3 - 2*w^2 - 5*w + 3],\ [61, 61, w^3 - w^2 - 8*w + 1],\ [71, 71, -1/2*w^3 + 1/2*w^2 + 5*w - 5],\ [71, 71, -w^3 + 3/2*w^2 + 15/2*w - 6],\ [83, 83, -1/2*w^3 + 3/2*w^2 + 2*w - 7],\ [83, 83, 1/2*w^3 - 7/2*w - 4],\ [97, 97, 2*w - 1],\ [107, 107, 1/2*w^3 + 1/2*w^2 - 4*w - 1],\ [107, 107, 3/2*w^3 - 7/2*w^2 - 9*w + 13],\ [109, 109, -3/2*w^2 + 7/2*w + 4],\ [109, 109, 1/2*w^3 + 1/2*w^2 - 4*w - 5],\ [109, 109, -1/2*w^3 + 2*w^2 + 3/2*w - 8],\ [109, 109, 3/2*w^2 + 1/2*w - 6],\ [121, 11, w^2 - w - 5],\ [121, 11, w^2 - w - 3],\ [131, 131, -w^3 + 11*w - 1],\ [131, 131, -w^3 + 3*w^2 + 8*w - 9],\ [157, 157, -w^3 + w^2 + 6*w + 1],\ [157, 157, 1/2*w^3 - 2*w^2 - 11/2*w + 4],\ [167, 167, w^2 - 3*w - 3],\ [167, 167, 1/2*w^3 - 3/2*w^2 - 2*w + 1],\ [167, 167, 3/2*w^3 - 5/2*w^2 - 10*w + 9],\ [167, 167, w^2 + w - 5],\ [169, 13, w^2 - w - 9],\ [179, 179, w^3 - 7*w - 5],\ [179, 179, 1/2*w^3 + 1/2*w^2 - 2*w - 3],\ [181, 181, -1/2*w^3 + 1/2*w^2 + 3*w - 5],\ [181, 181, 1/2*w^3 - w^2 - 5/2*w - 2],\ [191, 191, -1/2*w^3 + 3/2*w + 2],\ [191, 191, 2*w - 5],\ [191, 191, -2*w - 3],\ [191, 191, -1/2*w^3 + 2*w^2 + 3/2*w - 10],\ [193, 193, -1/2*w^3 + w^2 + 9/2*w - 4],\ [193, 193, 1/2*w^3 - 5/2*w^2 + w + 5],\ [193, 193, -1/2*w^3 - w^2 + 5/2*w + 4],\ [193, 193, 1/2*w^3 - 1/2*w^2 - 5*w + 1],\ [227, 227, -1/2*w^3 + 3/2*w^2 + 4*w - 9],\ [227, 227, w^3 - 2*w^2 - 7*w + 5],\ [227, 227, w^3 - w^2 - 8*w + 3],\ [227, 227, 1/2*w^3 - 11/2*w - 4],\ [239, 239, 1/2*w^3 - 3/2*w^2 - 4*w + 3],\ [239, 239, 1/2*w^3 - 11/2*w + 2],\ [251, 251, 1/2*w^3 - w^2 - 1/2*w - 2],\ [251, 251, -1/2*w^3 + 1/2*w^2 + w - 3],\ [263, 263, -1/2*w^3 - w^2 + 17/2*w],\ [263, 263, 1/2*w^3 - 5/2*w^2 - 5*w + 7],\ [277, 277, w^3 - w^2 - 6*w + 1],\ [277, 277, -w^3 + 2*w^2 + 5*w - 5],\ [311, 311, -w^3 + 5/2*w^2 + 13/2*w - 8],\ [311, 311, -w^3 + 1/2*w^2 + 17/2*w],\ [337, 337, 2*w^3 - 3*w^2 - 13*w + 9],\ [337, 337, 5/2*w^2 + 3/2*w - 8],\ [347, 347, 2*w^3 - 7/2*w^2 - 25/2*w + 14],\ [347, 347, 2*w^3 - 5/2*w^2 - 27/2*w],\ [349, 349, w^3 - 3/2*w^2 - 11/2*w + 4],\ [349, 349, -w^3 + 3/2*w^2 + 11/2*w - 2],\ [359, 359, -1/2*w^3 + w^2 + 1/2*w + 4],\ [359, 359, 1/2*w^3 - 1/2*w^2 - w + 5],\ [373, 373, 1/2*w^3 - 3/2*w + 4],\ [373, 373, 1/2*w^3 - 3/2*w^2 - 3],\ [383, 383, w^3 - 3/2*w^2 - 15/2*w + 8],\ [383, 383, -2*w^2 + 1],\ [409, 409, 1/2*w^3 + w^2 - 13/2*w - 4],\ [409, 409, -1/2*w^3 + 5/2*w^2 + 3*w - 9],\ [431, 431, 3/2*w^3 - 5/2*w^2 - 10*w + 5],\ [431, 431, 1/2*w^3 - 5/2*w^2 - 3*w + 13],\ [431, 431, 1/2*w^3 + w^2 - 13/2*w - 8],\ [431, 431, 3/2*w^3 - 2*w^2 - 21/2*w + 6],\ [433, 433, 2*w^3 - 4*w^2 - 14*w + 13],\ [433, 433, w^3 - 1/2*w^2 - 9/2*w + 6],\ [443, 443, 2*w^3 - 11/2*w^2 - 21/2*w + 24],\ [443, 443, w^3 - 5*w - 3],\ [457, 457, -2*w^3 + 4*w^2 + 12*w - 13],\ [457, 457, -w^3 + 1/2*w^2 + 9/2*w],\ [467, 467, -3/2*w^3 + 7/2*w^2 + 7*w - 7],\ [467, 467, 3/2*w^2 + 1/2*w - 8],\ [467, 467, w^3 - 7/2*w^2 - 15/2*w + 12],\ [467, 467, 3/2*w^3 - w^2 - 19/2*w + 2],\ [503, 503, -2*w^2 + 11],\ [503, 503, 2*w^3 - 2*w^2 - 12*w + 7],\ [503, 503, 2*w^3 - 4*w^2 - 10*w + 5],\ [503, 503, -2*w^2 + 4*w + 9],\ [529, 23, 3/2*w^2 - 3/2*w - 8],\ [541, 541, -1/2*w^3 + w^2 + 1/2*w + 6],\ [541, 541, 1/2*w^3 - 1/2*w^2 - w + 7],\ [563, 563, 1/2*w^2 + 3/2*w - 6],\ [563, 563, 1/2*w^2 - 5/2*w - 4],\ [577, 577, -1/2*w^3 + 5/2*w^2 + w - 11],\ [577, 577, -1/2*w^3 - w^2 + 9/2*w + 8],\ [587, 587, -w^3 + 1/2*w^2 + 13/2*w - 2],\ [587, 587, -w^3 + 5/2*w^2 + 9/2*w - 4],\ [599, 599, 1/2*w^3 + w^2 - 9/2*w - 12],\ [599, 599, -1/2*w^3 + 5/2*w^2 + w - 15],\ [601, 601, 1/2*w^3 - 1/2*w^2 - 3*w - 5],\ [601, 601, -1/2*w^3 + w^2 + 5/2*w - 8],\ [625, 5, -5],\ [659, 659, 3/2*w^3 - 1/2*w^2 - 8*w + 3],\ [659, 659, 3/2*w^3 - 4*w^2 - 9/2*w + 4],\ [673, 673, 2*w^3 - 2*w^2 - 16*w - 1],\ [673, 673, w^3 - 3/2*w^2 - 7/2*w - 8],\ [673, 673, w^3 - 3/2*w^2 - 7/2*w + 12],\ [673, 673, 2*w^3 - 4*w^2 - 14*w + 17],\ [683, 683, -1/2*w^3 - w^2 + 9/2*w - 2],\ [683, 683, -w^3 + 5/2*w^2 + 9/2*w - 12],\ [683, 683, -w^3 + 1/2*w^2 + 13/2*w + 6],\ [683, 683, 3/2*w^3 - w^2 - 11/2*w + 10],\ [709, 709, -1/2*w^3 + w^2 + 9/2*w - 10],\ [709, 709, 1/2*w^3 - 1/2*w^2 - 5*w - 5],\ [719, 719, -3/2*w^3 + 3/2*w^2 + 11*w - 3],\ [719, 719, -3/2*w^3 + 3*w^2 + 19/2*w - 8],\ [733, 733, w^3 - 3*w^2 - 8*w + 13],\ [733, 733, -w^3 + 7/2*w^2 + 7/2*w - 12],\ [733, 733, w^3 + 1/2*w^2 - 15/2*w - 6],\ [733, 733, -3/2*w^3 + 9/2*w^2 + 8*w - 19],\ [757, 757, w^3 - w^2 - 4*w + 1],\ [757, 757, -2*w^3 + 7/2*w^2 + 25/2*w - 10],\ [769, 769, -2*w^2 + 4*w + 7],\ [769, 769, -2*w^2 + 9],\ [827, 827, -1/2*w^3 - 1/2*w^2 + 6*w - 1],\ [827, 827, -1/2*w^3 + 2*w^2 + 7/2*w - 4],\ [829, 829, 1/2*w^3 + 2*w^2 - 7/2*w - 12],\ [829, 829, 7/2*w^3 - 8*w^2 - 41/2*w + 32],\ [829, 829, 3/2*w^3 - w^2 - 15/2*w - 2],\ [829, 829, -w^3 + w^2 + 8*w + 5],\ [839, 839, 3/2*w^3 - w^2 - 23/2*w],\ [839, 839, -3/2*w^3 + 7/2*w^2 + 9*w - 11],\ [853, 853, 1/2*w^3 - 3/2*w^2 - 2*w + 11],\ [853, 853, -1/2*w^3 + 7/2*w + 8],\ [863, 863, 2*w^3 - 11/2*w^2 - 25/2*w + 26],\ [863, 863, 3/2*w^3 - 5/2*w^2 - 10*w + 13],\ [887, 887, 2*w^3 - 7/2*w^2 - 29/2*w + 8],\ [887, 887, 2*w^3 - 5/2*w^2 - 31/2*w + 8],\ [911, 911, 2*w^3 - w^2 - 15*w - 3],\ [911, 911, 2*w^3 - 5*w^2 - 11*w + 17],\ [947, 947, -w^3 + 3*w^2 + 6*w - 9],\ [947, 947, w^3 - 9*w - 1],\ [961, 31, -3/2*w^3 + 3*w^2 + 23/2*w - 8],\ [961, 31, 3/2*w^3 - 3/2*w^2 - 13*w + 5],\ [983, 983, -1/2*w^3 + 2*w^2 - 1/2*w - 6],\ [983, 983, 1/2*w^3 + 1/2*w^2 - 2*w - 5],\ [997, 997, w^3 - 13*w + 7],\ [997, 997, 1/2*w^3 - 5/2*w^2 - w + 9],\ [997, 997, 1/2*w^3 + w^2 - 9/2*w - 6],\ [997, 997, -w^3 + 3*w^2 + 10*w - 5]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^6 - 13*x^4 + 48*x^2 - 43 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, e, -1, -e^4 + 8*e^2 - 12, -e^4 + 8*e^2 - 12, -e^3 + 7*e, -e^3 + 7*e, e^2 - 6, e^2 - 6, -e^5 + 9*e^3 - 15*e, -e^5 + 9*e^3 - 15*e, 2*e^2 - 11, -e^4 + 8*e^2 - 19, -e^4 + 8*e^2 - 19, 2*e^2 - 11, e^5 - 10*e^3 + 23*e, e^5 - 10*e^3 + 23*e, 2*e^5 - 18*e^3 + 33*e, 2*e^5 - 18*e^3 + 33*e, 2*e^4 - 22*e^2 + 44, 3*e^3 - 16*e, 3*e^3 - 16*e, e^4 - 13*e^2 + 26, e^4 - 10*e^2 + 19, e^4 - 10*e^2 + 19, e^4 - 13*e^2 + 26, -e^4 + 8*e^2 + 6, 2*e^4 - 13*e^2 + 12, -e^3 + 8*e, -e^3 + 8*e, -e^4 + 4*e^2 + 13, -e^4 + 4*e^2 + 13, -5*e^3 + 26*e, e^3 - 8*e, e^3 - 8*e, -5*e^3 + 26*e, -e^4 + 12*e^2 - 17, e^5 - 11*e^3 + 18*e, e^5 - 11*e^3 + 18*e, -3*e^4 + 23*e^2 - 17, -3*e^4 + 23*e^2 - 17, e^5 - 7*e^3 + e, -e^5 + 11*e^3 - 24*e, -e^5 + 11*e^3 - 24*e, e^5 - 7*e^3 + e, -e^4 + 13*e^2 - 37, 3*e^2 - 23, 3*e^2 - 23, -e^4 + 13*e^2 - 37, e^5 - 8*e^3 + 12*e, -2*e^5 + 18*e^3 - 29*e, -2*e^5 + 18*e^3 - 29*e, e^5 - 8*e^3 + 12*e, -2*e^5 + 18*e^3 - 35*e, -2*e^5 + 18*e^3 - 35*e, -e^5 + 4*e^3 + 14*e, -e^5 + 4*e^3 + 14*e, e^5 - 6*e^3 + e, e^5 - 6*e^3 + e, e^4 - 17*e^2 + 50, e^4 - 17*e^2 + 50, e^5 - 9*e^3 + 19*e, e^5 - 9*e^3 + 19*e, -5*e^2 + 38, -5*e^2 + 38, e^3 - 3*e, e^3 - 3*e, 4*e^4 - 27*e^2 + 24, 4*e^4 - 27*e^2 + 24, -e^3 + 4*e, -e^3 + 4*e, -4*e^4 + 39*e^2 - 73, -4*e^4 + 39*e^2 - 73, -e^5 + 11*e^3 - 20*e, -e^5 + 11*e^3 - 20*e, 4*e^4 - 34*e^2 + 67, 4*e^4 - 34*e^2 + 67, 2*e^5 - 20*e^3 + 37*e, 4*e^5 - 38*e^3 + 77*e, 4*e^5 - 38*e^3 + 77*e, 2*e^5 - 20*e^3 + 37*e, -e^2 + 8, -e^2 + 8, -e^5 + 8*e^3 - 18*e, -e^5 + 8*e^3 - 18*e, -e^4 + 13*e^2 - 33, -e^4 + 13*e^2 - 33, e^5 - 4*e^3 - 9*e, -4*e^3 + 22*e, -4*e^3 + 22*e, e^5 - 4*e^3 - 9*e, -2*e^3 + 6*e, 3*e^5 - 27*e^3 + 43*e, 3*e^5 - 27*e^3 + 43*e, -2*e^3 + 6*e, -e^4 + 5*e^2 + 31, 2*e^4 - 13*e^2 + 22, 2*e^4 - 13*e^2 + 22, e^3 - e, e^3 - e, -5*e^4 + 30*e^2 + 8, -5*e^4 + 30*e^2 + 8, e^5 - 16*e^3 + 59*e, e^5 - 16*e^3 + 59*e, 3*e^5 - 29*e^3 + 63*e, 3*e^5 - 29*e^3 + 63*e, -4*e^4 + 40*e^2 - 69, -4*e^4 + 40*e^2 - 69, -3*e^4 + 23*e^2 + 7, 3*e^5 - 29*e^3 + 53*e, 3*e^5 - 29*e^3 + 53*e, 4*e^4 - 26*e^2 + 29, -7*e^4 + 46*e^2 - 18, -7*e^4 + 46*e^2 - 18, 4*e^4 - 26*e^2 + 29, 3*e^5 - 29*e^3 + 51*e, 4*e^3 - 13*e, 4*e^3 - 13*e, 3*e^5 - 29*e^3 + 51*e, -4*e^4 + 43*e^2 - 83, -4*e^4 + 43*e^2 - 83, -e^5 + 12*e^3 - 42*e, -e^5 + 12*e^3 - 42*e, 2*e^4 - 30*e^2 + 67, 9*e^4 - 74*e^2 + 121, 9*e^4 - 74*e^2 + 121, 2*e^4 - 30*e^2 + 67, -5*e^4 + 35*e^2 - 17, -5*e^4 + 35*e^2 - 17, -6*e^4 + 54*e^2 - 80, -6*e^4 + 54*e^2 - 80, 2*e^5 - 13*e^3 - 4*e, 2*e^5 - 13*e^3 - 4*e, 9*e^4 - 74*e^2 + 102, -4*e^4 + 33*e^2 - 73, -4*e^4 + 33*e^2 - 73, 9*e^4 - 74*e^2 + 102, -2*e^5 + 17*e^3 - 41*e, -2*e^5 + 17*e^3 - 41*e, 7*e^4 - 56*e^2 + 71, 7*e^4 - 56*e^2 + 71, -3*e^5 + 27*e^3 - 47*e, -3*e^5 + 27*e^3 - 47*e, 2*e^5 - 18*e^3 + 36*e, 2*e^5 - 18*e^3 + 36*e, e^5 - 12*e^3 + 36*e, e^5 - 12*e^3 + 36*e, -e^5 + 14*e^3 - 44*e, -e^5 + 14*e^3 - 44*e, -2*e^4 + 26*e^2 - 55, -2*e^4 + 26*e^2 - 55, -e^5 + 20*e^3 - 76*e, -e^5 + 20*e^3 - 76*e, 7*e^4 - 54*e^2 + 72, -2*e^4 + 13*e^2 - 7, -2*e^4 + 13*e^2 - 7, 7*e^4 - 54*e^2 + 72] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([9, 3, -1/2*w^2 + 1/2*w + 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]