/* This code can be loaded, or copied and pasted, into Magma. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. At the *bottom* of the file, there is code to recreate the Hilbert modular form in Magma, by creating the HMF space and cutting out the corresponding Hecke irreducible subspace. From there, you can ask for more eigenvalues or modify as desired. It is commented out, as this computation may be lengthy. */ P := PolynomialRing(Rationals()); g := P![4, 8, -7, -2, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [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 := [ideal : I in primesArray]; heckePol := x^8 - 14*x^6 + 57*x^4 - 56*x^2 + 4; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -1/4*e^7 + 7/2*e^5 - 55/4*e^3 + 21/2*e, -1, -1/2*e^6 + 11/2*e^4 - 15*e^2 + 6, 1/2*e^6 - 5*e^4 + 23/2*e^2 - 1, 1/2*e^5 - 7/2*e^3 + e, -e^3 + 7*e, 1/2*e^6 - 4*e^4 + 9/2*e^2 + 5, -1/2*e^6 + 13/2*e^4 - 22*e^2 + 12, 1/2*e^7 - 7*e^5 + 59/2*e^3 - 39*e, e^7 - 15*e^5 + 62*e^3 - 46*e, -e^4 + 5*e^2 + 8, 1/2*e^6 - 6*e^4 + 37/2*e^2 - 3, -1/2*e^6 + 9/2*e^4 - 8*e^2 + 4, -e^4 + 9*e^2 - 6, -e^5 + 11*e^3 - 26*e, -e^7 + 12*e^5 - 39*e^3 + 24*e, e^7 - 14*e^5 + 56*e^3 - 47*e, -1/2*e^7 + 13/2*e^5 - 24*e^3 + 16*e, -3/2*e^4 + 21/2*e^2 - 1, e^7 - 14*e^5 + 56*e^3 - 47*e, -1/2*e^7 + 13/2*e^5 - 24*e^3 + 16*e, e^4 - 5*e^2 - 2, -1/2*e^6 + 5*e^4 - 23/2*e^2 + 1, 1/2*e^6 - 11/2*e^4 + 15*e^2 - 6, e^4 - 9*e^2 + 12, -4*e^4 + 28*e^2 - 10, -e^4 + 7*e^2 - 4, -1/2*e^7 + 13/2*e^5 - 28*e^3 + 40*e, 2*e^7 - 26*e^5 + 97*e^3 - 85*e, -1/2*e^6 + 3*e^4 - 3/2*e^2 + 9, 1/2*e^6 - 15/2*e^4 + 33*e^2 - 26, -3/2*e^7 + 41/2*e^5 - 83*e^3 + 82*e, e^7 - 16*e^5 + 75*e^3 - 88*e, -3*e^5 + 25*e^3 - 38*e, 2*e^7 - 26*e^5 + 97*e^3 - 81*e, -3*e^4 + 21*e^2 - 24, -2*e^7 + 28*e^5 - 108*e^3 + 78*e, -2*e^7 + 27*e^5 - 103*e^3 + 90*e, 5*e^4 - 33*e^2 + 12, 5*e^4 - 37*e^2 + 26, e^7 - 13*e^5 + 46*e^3 - 30*e, -1/2*e^7 + 15/2*e^5 - 35*e^3 + 50*e, 2*e^5 - 15*e^3 + 13*e, e^7 - 13*e^5 + 46*e^3 - 30*e, e^6 - 23/2*e^4 + 67/2*e^2 - 23, 2*e^4 - 18*e^2 + 26, 2*e^4 - 10*e^2 - 2, -e^6 + 19/2*e^4 - 39/2*e^2 - 9, -e^7 + 17*e^5 - 82*e^3 + 92*e, -2*e^5 + 16*e^3 - 22*e, e^7 - 15*e^5 + 66*e^3 - 72*e, -1/2*e^7 + 10*e^5 - 109/2*e^3 + 73*e, -e^7 + 14*e^5 - 52*e^3 + 31*e, -5/2*e^7 + 67/2*e^5 - 127*e^3 + 106*e, e^7 - 15*e^5 + 68*e^3 - 90*e, e^7 - 17*e^5 + 78*e^3 - 66*e, -e^7 + 14*e^5 - 56*e^3 + 51*e, -1/2*e^7 + 15/2*e^5 - 31*e^3 + 26*e, e^6 - 13*e^4 + 42*e^2 - 2, -e^6 + 8*e^4 - 7*e^2 - 2, e^5 - 11*e^3 + 26*e, e^7 - 12*e^5 + 39*e^3 - 24*e, e^6 - 9*e^4 + 16*e^2 + 6, -e^6 + 12*e^4 - 37*e^2 + 20, 1/2*e^7 - 13/2*e^5 + 22*e^3 - 14*e, 2*e^7 - 27*e^5 + 102*e^3 - 77*e, 1/2*e^6 - 7*e^4 + 59/2*e^2 - 13, -1/2*e^6 + 7/2*e^4 - 5*e^2 + 22, -1/2*e^5 + 7/2*e^3 + 3*e, -e^7 + 14*e^5 - 54*e^3 + 35*e, 3/2*e^6 - 17*e^4 + 97/2*e^2 - 7, -3/2*e^6 + 29/2*e^4 - 31*e^2 + 14, -e^7 + 14*e^5 - 64*e^3 + 93*e, 3*e^7 - 75/2*e^5 + 267/2*e^3 - 113*e, -2*e^6 + 51/2*e^4 - 177/2*e^2 + 37, 2*e^6 - 33/2*e^4 + 51/2*e^2 - 19, 3*e^3 - 19*e, 2*e^3 - 14*e, -e^5 + 7*e^3 - 2*e, -1/2*e^7 + 11/2*e^5 - 17*e^3 + 18*e, 5*e^4 - 39*e^2 + 28, 5*e^4 - 31*e^2, 7/2*e^7 - 45*e^5 + 333/2*e^3 - 145*e, -2*e^7 + 27*e^5 - 111*e^3 + 124*e, -e^6 + 10*e^4 - 25*e^2 + 34, e^6 - 11*e^4 + 32*e^2 + 6, e^3 + 3*e, -3/2*e^7 + 39/2*e^5 - 68*e^3 + 24*e, 2*e^7 - 30*e^5 + 127*e^3 - 103*e, -5/2*e^7 + 69/2*e^5 - 134*e^3 + 104*e, 3*e^7 - 75/2*e^5 + 259/2*e^3 - 89*e, e^7 - 12*e^5 + 42*e^3 - 49*e, e^7 - 29/2*e^5 + 109/2*e^3 - 19*e, 2*e^5 - 23*e^3 + 55*e, 3/2*e^4 - 21/2*e^2 - 23, e^6 - 14*e^4 + 53*e^2 - 30, -e^6 + 7*e^4 - 4*e^2 - 2, e^7 - 9*e^5 + 20*e^3 - 26*e, -3/2*e^7 + 21*e^5 - 185/2*e^3 + 129*e, -4*e^6 + 42*e^4 - 110*e^2 + 42, 4*e^6 - 42*e^4 + 110*e^2 - 42, -5/2*e^7 + 61/2*e^5 - 104*e^3 + 74*e, 2*e^7 - 29*e^5 + 126*e^3 - 139*e, -5/2*e^5 + 51/2*e^3 - 69*e, 2*e^7 - 32*e^5 + 143*e^3 - 127*e, e^6 - 12*e^4 + 37*e^2 - 12, -e^6 + 9*e^4 - 16*e^2 + 2, e^4 - 7*e^2 - 6, 5*e^5 - 48*e^3 + 95*e, 3/2*e^7 - 29/2*e^5 + 27*e^3 + 20*e, -e^4 - e^2 + 20, -e^6 + 10*e^4 - 25*e^2 + 22, e^6 - 11*e^4 + 32*e^2 - 6, -e^4 + 15*e^2 - 36, -1/2*e^7 + 5/2*e^5 + 12*e^3 - 52*e, 2*e^7 - 27*e^5 + 107*e^3 - 94*e, -4*e^7 + 54*e^5 - 208*e^3 + 170*e, 2*e^7 - 32*e^5 + 147*e^3 - 153*e, 2*e^6 - 30*e^4 + 116*e^2 - 54, -2*e^6 + 12*e^4 + 10*e^2 - 26, 3*e^7 - 42*e^5 + 174*e^3 - 175*e, -7/2*e^7 + 89/2*e^5 - 161*e^3 + 126*e, 1/2*e^6 - 17/2*e^4 + 32*e^2 - 4, 4*e^2 + 10, -4*e^2 + 38, -1/2*e^6 + 2*e^4 + 27/2*e^2 - 25, -e^6 + 9*e^4 - 26*e^2 + 42, e^6 - 12*e^4 + 47*e^2 - 42, e^6 - 19/2*e^4 + 23/2*e^2 + 17, -e^6 + 23/2*e^4 - 51/2*e^2 - 25, -9/2*e^7 + 121/2*e^5 - 236*e^3 + 214*e, 3*e^7 - 39*e^5 + 149*e^3 - 137*e, 5/2*e^6 - 49/2*e^4 + 54*e^2 - 4, 3/2*e^6 - 20*e^4 + 139/2*e^2 - 13, -3/2*e^6 + 23/2*e^4 - 10*e^2 + 8, -5/2*e^6 + 28*e^4 - 157/2*e^2 + 31, -e^7 + 14*e^5 - 62*e^3 + 91*e, 7/2*e^5 - 49/2*e^3 + 11*e, 1/2*e^6 + 1/2*e^4 - 31*e^2 + 46, -1/2*e^6 + 11*e^4 - 99/2*e^2 + 25, 3*e^7 - 37*e^5 + 132*e^3 - 118*e, -3*e^7 + 41*e^5 - 168*e^3 + 182*e, 4*e^7 - 107/2*e^5 + 405/2*e^3 - 167*e, e^7 - 14*e^5 + 50*e^3 - 23*e, 4*e^7 - 103/2*e^5 + 377/2*e^3 - 151*e, -2*e^7 + 28*e^5 - 119*e^3 + 131*e, 3/2*e^7 - 33/2*e^5 + 49*e^3 - 28*e, -3*e^7 + 43*e^5 - 181*e^3 + 185*e, -e^6 + 19/2*e^4 - 47/2*e^2 + 3, e^6 - 23/2*e^4 + 75/2*e^2 - 39, -e^7 + 8*e^5 - 5*e^3 - 30*e, e^7 - 18*e^5 + 95*e^3 - 130*e, -14, -e^6 + 7*e^4 + 2*e^2 - 10, e^6 - 14*e^4 + 47*e^2 + 4, -14]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := 1; // EXAMPLE: // pp := Factorization(2*ZF)[1][1]; // heckeEigenvalues[pp]; print "To reconstruct the Hilbert newform f, type f, iso := Explode(make_newform());"; function make_newform(); M := HilbertCuspForms(F, NN); S := NewSubspace(M); // SetVerbose("ModFrmHil", 1); NFD := NewformDecomposition(S); newforms := [* Eigenform(U) : U in NFD *]; if #newforms eq 0 then; print "No Hilbert newforms at this level"; return 0; end if; print "Testing ", #newforms, " possible newforms"; newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *]; print #newforms, " newforms have the correct Hecke field"; if #newforms eq 0 then; print "No Hilbert newform found with the correct Hecke field"; return 0; end if; autos := Automorphisms(K); xnewforms := [* *]; for f in newforms do; if K eq RationalField() then; Append(~xnewforms, [* f, autos[1] *]); else; flag, iso := IsIsomorphic(K,BaseField(f)); for a in autos do; Append(~xnewforms, [* f, a*iso *]); end for; end if; end for; newforms := xnewforms; for P in primes do; xnewforms := [* *]; for f_iso in newforms do; f, iso := Explode(f_iso); if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then; Append(~xnewforms, f_iso); end if; end for; newforms := xnewforms; if #newforms eq 0 then; print "No Hilbert newform found which matches the Hecke eigenvalues"; return 0; else if #newforms eq 1 then; print "success: unique match"; return newforms[1]; end if; end if; end for; print #newforms, "Hilbert newforms found which match the Hecke eigenvalues"; return newforms[1]; end function;