// Make newform 1728.3.q.i in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1728_q();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1728_q_Hecke();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_1728_3_q_i();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_1728_3_q_i();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function ConvertToHeckeField(input: pass_field := false, Kf := []) if not pass_field then poly := [144, 144, 276, -84, 133, -2, 15, -2, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-256788, 140724, -29928, -75285, -7826, -5547, 198, -473], [78912, 156924, -132153, 82215, -6901, 10353, -1827, 677], [-177138, -98226, -29928, -75285, -7826, -5547, 198, -473], [-401472, 247356, -232782, 143085, -19444, 16707, -2688, 1013], [194364, 129078, -107691, 68730, -7447, 9966, -2019, 644], [209808, -1245834, 475248, -389565, 47066, -45123, 7182, -2407], [2015568, 544536, 364008, 657585, 95186, 67467, -4878, 5753]]; Rf_basisdens := [1, 159300, 159300, 79650, 159300, 79650, 159300, 318600]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf); return Eltseq(inp_vec); end function; // To make the character of type GrpDrchElt, type "MakeCharacter_1728_q();" function MakeCharacter_1728_q() N := 1728; order := 6; char_gens := [703, 325, 1217]; v := [6, 6, 5]; // chi(gens[i]) = zeta^v[i] assert UnitGenerators(DirichletGroup(N)) eq char_gens; F := CyclotomicField(order); chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]); return MinimalBaseRingCharacter(chi); end function; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1728_q_Hecke();" function MakeCharacter_1728_q_Hecke(Kf) N := 1728; order := 6; char_gens := [703, 325, 1217]; char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; end function; function ExtendMultiplicatively(weight, aps, character) prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec primes := PrimesUpTo(prec); prime_powers := primes; assert #primes eq #aps; log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1) F := Universe(aps); FXY := PolynomialRing(F, 2); // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ... R := PowerSeriesRing(FXY : Precision := log_prec + 1); recursion := Coefficients(1/(1 - X*T + Y*T^2)); coeffs := [F!0: i in [1..(prec+1)]]; coeffs[1] := 1; //a_1 for i := 1 to #primes do p := primes[i]; coeffs[p] := aps[i]; b := p^(weight - 1) * F!character(p); r := 2; p_power := p * p; //deals with powers of p while p_power le prec do Append(~prime_powers, p_power); coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]); p_power *:= p; r +:= 1; end while; end for; Sort(~prime_powers); for pp in prime_powers do for k := 1 to Floor(prec/pp) do if GCD(k, pp) eq 1 then coeffs[pp*k] := coeffs[pp]*coeffs[k]; end if; end for; end for; return coeffs; end function; function qexpCoeffs() // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" weight := 3; raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 1, 0, -1, 0, 0, 0], [-1, 1, 2, 0, 0, 0, -1, 0], [-2, -1, -4, 3, 0, -2, -1, 2], [-1, 1, -3, -2, 3, 6, 2, -2], [0, -3, -4, 1, 1, 1, 4, -2], [-2, -5, 0, -1, 0, 0, 0, 4], [-14, 2, 8, 0, -4, 0, -3, -3], [-10, -2, -9, 1, 0, 3, -2, 4], [0, 0, 12, 0, 6, 12, 1, -1], [-16, -2, 0, 0, 6, -6, 0, 2], [-42, 0, 21, 0, 0, 0, -2, -2], [-14, 7, 14, -7, 12, 6, 7, 0], [15, 3, 10, 2, 0, -6, 3, -6], [-10, -2, 8, -4, 4, 4, 12, -6], [21, 9, -6, 0, -14, 0, -3, -3], [-2, -12, -5, 5, -18, -9, 2, 0], [-5, 5, 0, -10, 0, 0, 5, -5], [42, -12, -84, 12, -4, -4, 0, 0], [-4, 11, 0, -7, -9, 9, 0, -18], [-13, 5, 12, -6, -24, -12, 7, 0], [-19, 1, -22, 2, 0, 2, 1, -2], [-2, 14, -16, -24, 0, 0, 20, -10], [-63, 10, 67, -6, 0, 0, 2, 0], [-36, 12, -21, -27, 0, -15, 12, -24], [6, -6, -42, 12, 6, 12, -7, 7], [-30, -3, 64, 5, -18, -18, -4, 2], [48, 20, 0, -18, -6, 6, 0, -38], [-93, -15, 39, 0, -3, 0, 8, 8], [-78, 2, 0, 4, -6, 6, 0, 2], [-12, -24, -6, 0, 38, 0, 7, 7], [3, -6, 19, -10, 0, -42, -6, 12], [29, -29, -14, 58, 0, 0, -23, 23], [-19, -17, 1, 0, -35, 0, -10, -10], [9, 17, 0, -8, 24, 12, -1, 0], [11, -11, 13, 22, -3, -6, 24, -24], [6, 0, 0, -6, 30, -30, 0, -6], [-102, 6, 54, 0, 42, 0, -3, -3], [-52, 16, -21, -47, 0, 15, 16, -32], [8, 4, -48, -20, 42, 42, 32, -16], [36, 12, 0, -14, -12, 12, 0, -26], [49, -1, 70, -20, 0, 12, -1, 2], [0, 0, -7, 0, 18, 36, 10, -10], [0, -24, -40, 4, 24, 24, 40, -20], [64, 10, 0, 4, -30, 30, 0, -6], [-14, 14, -8, -28, 24, 48, 35, -35], [-47, -27, 42, 22, 12, 6, -17, 0], [34, -25, 68, -9, 0, -12, -25, 50], [-11, 11, 5, -22, -33, -66, -8, 8], [-122, 23, 228, -31, 77, 77, 16, -8], [-162, -42, 60, 0, -32, 0, 29, 29], [-3, 36, 35, -4, -12, -6, -28, 0], [86, 25, -128, -3, 16, 16, -44, 22], [-10, -32, -11, 0, -28, 0, -16, -16], [133, 11, 128, -6, 0, -82, 11, -22], [38, 10, -48, 4, 44, 44, -28, 14], [-30, -2, 0, 0, 12, -12, 0, 2], [112, -66, -169, 9, 66, 33, 48, 0], [-40, 22, -27, -35, 0, -1, 22, -44], [-12, 12, -142, -24, -6, -12, -17, 17], [-49, -47, 1, 0, -51, 0, 20, 20], [-298, 1, 0, 9, -30, 30, 0, 8], [-32, 32, 32, 0, -54, 0, -55, -55], [169, -52, -177, 44, 0, 0, -36, 0], [-12, -18, -29, 35, 0, -105, -18, 36], [205, -33, -228, 10, 96, 48, 13, 0], [10, -10, 185, 20, 6, 12, -28, 28], [-157, -53, 52, 0, 66, 0, -7, -7], [-80, 86, 147, -19, -90, -45, -48, 0], [15, -6, -13, 34, 0, -130, -6, 12], [-8, 26, 80, 6, -108, -108, -64, 32], [-173, -39, 108, -26, 96, 48, 91, 0], [-45, 45, 185, -90, 27, 54, -6, 6], [352, -35, 0, 25, 24, -24, 0, 60], [-186, 42, 114, 0, 54, 0, -3, -3], [240, 12, 295, -67, 0, 83, 12, -24], [108, 22, 0, 16, 42, -42, 0, -6], [244, 38, -103, 0, 84, 0, -10, -10], [-34, 34, -155, -68, -78, -156, 16, -16], [-458, 14, 236, 0, 108, 0, -5, -5], [12, 56, 19, -25, -114, -57, -6, 0], [184, 2, -336, 14, 104, 104, -32, 16], [-56, -33, 0, 33, -21, 21, 0, 66], [-203, 29, 216, -16, -12, -6, 3, 0], [212, 43, 138, 31, 0, 14, 43, -86], [40, 35, -100, -45, 179, 179, 20, -10], [185, 110, -129, -54, 36, 18, -2, 0], [96, -30, 87, 39, 0, 35, -30, 60], [-50, 50, -172, -100, -12, -24, 21, -21], [2, 73, -40, -91, -126, -126, 36, -18], [-159, -21, -88, -50, 0, -146, -21, 42], [-82, -36, 0, 98, 60, -60, 0, 134], [-381, 87, 234, 0, -168, 0, -59, -59], [25, -25, -466, 50, 48, 96, 77, -77], [106, -70, -132, 110, -150, -150, -80, 40], [205, 71, -67, 0, -105, 0, -70, -70], [66, -39, -36, 87, 1, 1, -96, 48], [-168, 34, 0, -4, 0, 0, 0, -38], [258, -108, 0, -24, 42, -42, 0, 84], [-94, 119, 160, -53, -132, -66, -13, 0], [-18, -84, -48, 42, 22, 22, 84, -42], [-561, -33, 264, 0, 40, 0, 65, 65], [-205, -20, -281, 96, 0, 28, -20, 40], [47, -47, 268, 94, -36, -72, -35, 35], [360, -25, 0, -15, -27, 27, 0, 10], [222, 3, 330, -111, 0, 166, 3, -6], [194, -26, -248, 96, -124, -124, -140, 70], [-142, 94, 118, 0, 54, 0, -19, -19], [-203, 22, 183, -42, 216, 108, 62, 0], [58, -58, 2, 116, -60, -120, 33, -33], [-40, 62, 0, 36, -30, 30, 0, -26], [184, 2, -91, 0, 48, 0, -38, -38], [304, 19, -236, 49, -168, -84, -117, 0], [666, 56, 0, 2, -6, 6, 0, -54], [257, -68, 175, 150, 0, 4, -68, 136], [-31, 31, 0, -62, -18, -36, 93, -93], [120, -30, -256, 22, 260, 260, 16, -8], [331, -91, -211, 0, 57, 0, 32, 32], [45, -15, -84, 144, 0, 24, -15, 30], [-35, 35, -723, -70, 69, 138, 76, -76], [780, -52, -747, 85, -138, -69, -118, 0], [-70, 4, -45, -29, 0, 161, 4, -8], [-50, 71, -32, -137, -10, -10, 132, -66], [-459, -65, 362, -32, -84, -42, 129, 0], [-38, 26, 216, 44, 78, 78, -140, 70], [512, 114, -409, -11, 78, 39, -92, 0], [64, 182, -176, -206, -74, -74, 48, -24], [705, 93, -576, 36, 96, 48, -165, 0], [-127, 127, 393, -254, 3, 6, 166, -166], [-328, -93, 0, -81, 114, -114, 0, 12], [-58, -62, -2, 0, -28, 0, -79, -79], [24, -24, -182, 48, -6, -12, -11, 11], [254, -88, 0, -56, -132, 132, 0, 32], [283, -79, -181, 0, -91, 0, 2, 2], [-29, 29, 655, -58, 9, 18, 120, -120], [10, 158, -24, -160, -18, -18, 4, -2], [0, 0, 142, 0, 102, 204, -13, 13], [1075, 29, -523, 0, -61, 0, -6, -6], [-108, -9, 148, -25, -71, -71, 68, -34], [306, -17, 0, 43, -60, 60, 0, 60], [-34, -86, -57, 109, 0, -201, -86, 172], [118, -118, 224, 236, -36, -72, -231, 231], [-84, -72, 296, 136, 94, 94, -128, 64], [364, 22, 0, 44, -12, 12, 0, 22], [-69, -9, -142, 82, 0, 60, -9, 18], [118, -140, -209, 49, -246, -123, 42, 0], [52, 74, 209, -231, 0, 27, 74, -148], [57, -57, 412, 114, 108, 216, -83, 83], [20, -122, -48, 118, -26, -26, 8, -4], [-5, 5, -43, -10, -195, -390, -70, 70], [134, 130, -192, -92, -68, -68, -76, 38], [-238, -69, 0, -85, 78, -78, 0, -16], [-156, -60, 48, 0, 172, 0, -35, -35], [-542, 211, 706, -47, -192, -96, -117, 0], [-33, -117, 0, 84, 0, -286, -117, 234], [394, -158, 0, -152, -102, 102, 0, 6], [76, -46, 209, -87, 0, -145, -46, 92], [-336, 114, 0, -96, 24, -24, 0, -210], [465, -141, -303, 0, -9, 0, 68, 68], [-478, 115, -420, -173, 0, 246, 115, -230], [-168, -15, 348, 21, 39, 39, -12, 6], [-50, 50, 208, -100, -138, -276, -55, 55], [-266, -118, 460, 82, -132, -132, 72, -36], [-808, 82, 445, 0, 0, 0, -16, -16], [21, 75, -216, 162, 0, 48, 75, -150], [-490, -78, 0, 44, 168, -168, 0, 122], [-740, 78, 851, 33, -18, -9, -144, 0], [284, 42, 0, -116, -36, 36, 0, -158], [-1279, -29, 625, 0, 23, 0, 74, 74], [120, 123, 106, -109, 0, -24, 123, -246], [-43, 43, 525, -86, 117, 234, -18, 18], [442, 98, -172, 0, -62, 0, 37, 37], [-457, 174, 535, -96, 36, 18, 18, 0], [-228, 228, -46, -456, 96, 192, 357, -357], [-85, -71, 7, 0, -91, 0, 80, 80], [-220, 47, 226, -41, 444, 222, 35, 0], [-94, 76, -160, -250, 150, 150, 348, -174], [526, -140, 0, 34, 174, -174, 0, 174], [178, -134, -165, 147, 246, 123, -160, 0], [-589, 81, 718, 48, -72, -36, -177, 0], [-204, 27, -378, 147, 0, 56, 27, -54], [21, -21, -427, 42, 51, 102, 80, -80], [72, -93, -284, 23, 3, 3, 140, -70], [962, 286, -338, 0, -108, 0, -59, -59], [572, 70, 519, -17, 0, -75, 70, -140], [-162, 206, 0, -186, -114, 114, 0, -392], [73, -313, -246, 140, 36, 18, 33, 0], [104, -104, -723, 208, 72, 144, 88, -88], [-688, 82, 1296, -122, 150, 150, 80, -40], [-312, -23, 0, 39, 51, -51, 0, 62], [-36, -93, -100, 157, 0, -126, -93, 186], [-546, -23, 0, 37, -84, 84, 0, 60], [-758, 32, -717, -73, 0, 245, 32, -64], [-182, -205, 144, 95, 186, 186, 220, -110], [748, -298, -523, 0, 68, 0, 48, 48], [58, -58, 1, 116, -294, -588, -106, 106], [-668, -282, 555, 169, 78, 39, -56, 0], [355, 152, 435, -232, 0, 152, 152, -304], [-300, 192, 752, -116, -30, -30, -152, 76], [-1141, 61, 601, 0, 229, 0, -64, -64], [-583, 33, 582, -34, -612, -306, 35, 0], [-41, 41, 245, -82, 45, 90, 204, -204], [-1193, -236, 1071, 114, 132, 66, 8, 0], [314, -83, -480, 157, -132, -132, -148, 74], [-722, 98, 1352, -144, 78, 78, 92, -46], [1002, 160, 0, -22, -30, 30, 0, -182], [-863, 101, 482, 0, 78, 0, 49, 49], [-679, 28, -457, -250, 0, -280, 28, -56], [31, -31, 206, 62, -240, -480, -123, 123], [380, 133, 0, -49, 165, -165, 0, -182], [-719, 59, 389, 0, -375, 0, 6, 6], [-657, -155, 646, 144, 288, 144, -133, 0], [-583, 49, -508, -124, 0, 272, 49, -98], [336, 84, -126, 0, 160, 0, 133, 133], [299, 188, -217, -106, -108, -54, 24, 0], [32, -32, -664, 64, -54, -108, -231, 231], [-179, 2, -149, -32, 0, -74, 2, -4], [-468, -96, 856, 56, 306, 306, 80, -40], [585, -117, -351, 0, 507, 0, 142, 142], [7, -7, 469, 14, -177, -354, -262, 262], [40, -40, 690, 80, -114, -228, 153, -153], [350, 136, -107, 0, 18, 0, -190, -190], [-858, -188, 0, 90, -84, 84, 0, 278], [1795, -121, -958, 0, -24, 0, -3, -3], [154, 106, -121, -73, 162, 81, 40, 0], [-90, -12, 141, -219, 0, 135, -12, 24], [-944, -142, 1848, 122, 150, 150, 40, -20], [-1073, 125, 1148, -50, 12, 6, -25, 0], [-291, -75, -4, -212, 0, -332, -75, 150], [-115, 115, 569, -230, -129, -258, 404, -404], [1034, 80, 0, -22, -90, 90, 0, -102], [202, -202, 392, 404, -126, -252, -147, 147], [1698, 300, -699, 0, -146, 0, -24, -24], [-15, 147, -32, -194, 168, 84, 241, 0], [-159, -81, 166, -244, 0, -116, -81, 162], [-213, 213, 991, -426, 249, 498, 6, -6], [1130, -140, -2232, 154, -74, -74, -28, 14], [279, -69, -174, 0, -250, 0, 25, 25], [-672, 72, 1080, -204, -170, -170, 264, -132], [156, -21, 374, -197, 0, -84, -21, 42], [-958, -84, 0, -142, 6, -6, 0, -58], [44, -44, -470, 88, -6, -12, 201, -201], [196, -32, 0, 2, -300, 300, 0, 34], [-64, 10, 37, 0, -498, 0, -110, -110], [-697, 91, -706, -82, 0, 436, 91, -182], [1306, 310, 0, -100, -12, 12, 0, -410], [205, 389, 92, 0, 60, 0, -71, -71], [-94, 94, -1040, -188, 144, 288, 239, -239], [-56, -148, 128, 156, 482, 482, -16, 8], [-71, 71, 621, -142, -153, -306, -78, 78], [720, -27, -1236, 129, -141, -141, -204, 102], [1486, 14, -736, 0, -158, 0, 1, 1], [-334, 294, 601, -27, -666, -333, -240, 0], [282, -36, 243, 75, 0, 399, -36, 72], [128, 1, 92, 173, -120, -120, -348, 174], [-488, 36, 0, 6, 90, -90, 0, -30], [269, -177, -440, 6, -360, -180, 165, 0], [-4, 10, 488, 230, -348, -348, -480, 240], [760, -141, 0, 101, 321, -321, 0, 242], [201, 409, 192, -16, 36, 18, -377, 0], [-320, 107, -602, 175, 0, -18, 107, -214], [87, -87, -515, 174, -15, -30, -72, 72], [-204, 42, 0, 252, -12, 12, 0, 210], [-713, -283, 215, 0, -191, 0, 110, 110], [964, -93, -924, 133, -480, -240, -173, 0], [194, -74, -288, 124, 416, 416, -100, 50], [-515, -88, -725, 298, 0, -400, -88, 176], [-55, 55, 912, -110, 150, 300, 5, -5], [-897, -111, 393, 0, -455, 0, -34, -34], [-93, 93, -707, -186, 81, 162, 210, -210], [66, 8, 0, -10, 426, -426, 0, -18], [445, -272, -581, 136, -204, -102, 0, 0], [184, -184, -34, 368, 84, 168, 25, -25], [252, -252, -1275, 504, -12, -24, 94, -94], [710, -214, 0, -40, -24, 24, 0, 174], [964, -184, -574, 0, 726, 0, 155, 155], [416, -300, -505, 211, 66, 33, -122, 0], [-512, -561, 0, -79, 63, -63, 0, 482], [-268, 139, -288, -119, 0, 124, 139, -278], [-362, 194, 278, 0, 260, 0, -127, -127], [-94, 94, -570, -188, 450, 900, 197, -197], [-39, 15, 416, -470, 0, 14, 15, -30], [188, 126, -285, -223, -390, -195, 320, 0], [3, -3, 786, 6, 18, 36, -7, 7], [510, 18, -244, 370, -224, -224, -776, 388], [-516, -322, 0, 92, -120, 120, 0, 414], [527, -311, -419, 0, 231, 0, -56, -56], [-603, -655, 106, 158, 72, 36, 339, 0], [-186, 147, 412, -127, 307, 307, -40, 20], [-42, -78, -205, 241, 0, -127, -78, 156], [-932, -49, 1780, 7, -42, -42, 84, -42], [1678, 248, -715, 0, -350, 0, -72, -72], [1391, 295, -548, 0, -240, 0, -171, -171], [1186, 110, -1073, 3, 198, 99, -116, 0], [973, 239, -367, 0, -619, 0, -20, -20], [201, 29, -140, 32, 0, 0, -93, 0], [702, 126, 227, 349, 0, 279, 126, -252], [-580, 124, 1424, 8, 30, 30, -264, 132], [-1115, 183, 1118, -180, 1044, 522, 177, 0], [-36, 36, -619, -72, 168, 336, 88, -88], [-466, -170, 1304, 356, -346, -346, -372, 186], [690, -380, 0, 194, 390, -390, 0, 574]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1728_q_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1728_3_q_i();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. function MakeNewformModFrm_1728_3_q_i(:prec:=8) chi := MakeCharacter_1728_q(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); S := BaseChange(S, Kf); maxprec := NextPrime(1999) - 1; while true do trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]); B := Basis(S, prec + 1); S_basismat := Matrix([AbsEltseq(g): g in B]); if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then S_basismat := ChangeRing(S_basismat,Kf); f_lincom := Solution(S_basismat,trunc_vec); f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]]; return f; end if; error if prec eq maxprec, "Unable to distinguish newform within newspace"; prec := Min(Ceiling(1.25 * prec), maxprec); end while; end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_1728_3_q_i();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function MakeNewformModSym_1728_3_q_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1728_q(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<5,R![16, 528, 5612, -6468, 2661, -294, -37, 6, 1]>,<7,R![11664, -3888, 4860, -108, 1197, -126, 69, 6, 1]>],Snew); return Vf; end function;