
// Make newform 1764.3.z.l in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1764_z();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1764_z_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_1764_3_z_l();".
// 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_1764_3_z_l();".
// 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 := [4, 0, -8, 0, 14, 0, -4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 76, 0, 0, 0, 0, 0, 1], [20, 0, 0, 0, 0, 0, 1, 0], [0, -27, 0, 0, 0, 0, 0, -1], [2, 0, -28, 0, 7, 0, -2, 0], [2, 0, -21, 0, 7, 0, -2, 0], [0, 64, 0, -112, 0, 35, 0, -8], [0, -88, 0, 154, 0, -42, 0, 11]];
    Rf_basisdens := [1, 14, 14, 7, 14, 7, 14, 14];
    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_1764_z();"
function MakeCharacter_1764_z()
    N := 1764;
    order := 6;
    char_gens := [883, 785, 1081];
    v := [6, 6, 1];
    // 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_1764_z_Hecke();"
function MakeCharacter_1764_z_Hecke(Kf)
    N := 1764;
    order := 6;
    char_gens := [883, 785, 1081];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 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<X, Y> := 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<T> := 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], [0, 0, -2, 0, 0, -1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1, -1, 0, 0, -1, 0, -2], [0, 2, 4, -1, 0, 8, -1, 2], [-4, 0, -3, 3, 4, 3, 0, 0], [-16, 0, 4, 0, -8, 2, -2, -1], [0, -2, 0, 4, 2, -9, 2, -1], [-10, -1, -5, -6, 0, 0, 6, 1], [4, 3, 14, 4, -4, -14, 0, 0], [0, 12, 0, -8, 16, -9, -4, 6], [14, 5, -13, -7, 28, -26, -7, 5], [-14, 2, -8, 8, 0, 0, -8, -2], [44, 0, -4, 0, 22, -2, -8, 7], [-18, -6, 26, 4, -18, 26, 8, -12], [-14, 5, -22, 12, 14, 22, 0, 0], [-24, 0, 28, 0, -12, 14, 5, 8], [8, 6, 12, 4, 8, 12, 8, 12], [-28, 1, 43, -8, 0, 0, 8, -1], [28, -8, 6, 15, -28, -6, 0, 0], [0, -8, 0, 0, -54, -28, 0, -4], [-16, 14, -6, -6, -32, -12, -6, 14], [-16, 0, -46, 0, -8, -23, 11, 16], [-28, 4, 10, 3, -56, 20, 3, 4], [-2, 3, 11, 37, 2, -11, 0, 0], [-16, 0, -36, 0, -8, -18, -4, 9], [0, 6, 0, 4, -82, -73, 2, 3], [-2, 0, 43, -20, -2, 43, -40, 0], [-64, -8, 12, 14, 0, 0, -14, 8], [-56, -6, 92, 0, 0, 0, 0, 6], [-16, 0, 92, 0, -8, 46, -10, 28], [62, 21, 59, 10, 62, 59, 20, 42], [16, 6, -36, -18, 32, -72, -18, 6], [0, -8, 0, -32, -34, -108, -16, -4], [-62, -10, 40, 24, -62, 40, 48, -20], [-8, 1, -54, 25, 8, 54, 0, 0], [0, -36, 0, 24, -12, -20, 12, -18], [-30, 23, 40, -30, -60, 80, -30, 23], [-32, 0, -34, 0, -16, -17, 3, 46], [-140, 7, 95, -12, -140, 95, -24, 14], [24, 13, -44, 1, 48, -88, 1, 13], [0, 62, 0, -40, -56, 27, -20, 31], [34, -24, 56, -12, 34, 56, -24, -48], [-32, -6, 22, 42, 0, 0, -42, 6], [-32, -10, -20, 34, 32, 20, 0, 0], [-32, -12, -56, -20, 0, 0, 20, 12], [68, 20, -24, -24, 136, -48, -24, 20], [6, 61, 20, 30, -6, -20, 0, 0], [80, 0, 96, 0, 40, 48, 43, -13], [0, -6, 0, 44, 22, -159, 22, -3], [58, 3, 117, 6, 0, 0, -6, -3], [8, 3, -26, 43, -8, 26, 0, 0], [-66, 49, 64, -58, -132, 128, -58, 49], [-164, 0, -58, 0, -82, -29, 25, 5], [32, 7, -5, -16, 32, -5, -32, 14], [30, -37, 15, 31, -30, -15, 0, 0], [-232, 0, -132, 0, -116, -66, -34, 13], [40, 4, -6, -40, 40, -6, -80, 8], [-150, -7, -113, -26, 0, 0, 26, 7], [-44, 39, 62, -8, 44, -62, 0, 0], [90, -33, 15, 7, 180, 30, 7, -33], [-8, 35, 10, 96, -16, 20, 96, 35], [10, -7, -88, -50, -10, 88, 0, 0], [48, 0, 8, 0, 24, 4, 21, -10], [0, -68, 0, -60, 60, 26, -30, -34], [0, -8, 0, -72, -16, 224, -36, -4], [28, 34, -49, 20, 0, 0, -20, -34], [46, -13, 143, 2, 46, 143, 4, -26], [-88, -31, 6, -27, -176, 12, -27, -31], [-182, -31, -23, 7, 182, 23, 0, 0], [0, -38, 0, 60, 162, 45, 30, -19], [188, -34, 64, -26, -188, -64, 0, 0], [0, 32, 0, 24, -208, 86, 12, 16], [184, 12, 104, 52, 0, 0, -52, -12], [304, 0, -176, 0, 152, -88, -20, -32], [302, -29, -199, 34, 302, -199, 68, -58], [-152, 0, 316, 0, -76, 158, -37, 50], [0, 26, 0, 124, 78, 53, 62, 13], [196, 32, -10, -57, -196, 10, 0, 0], [-10, 11, -94, 56, -20, -188, 56, 11], [-68, -40, 16, 72, 0, 0, -72, 40], [-320, -47, -79, -36, -320, -79, -72, -94], [-48, -65, 96, 137, -96, 192, 137, -65], [72, 0, 8, 0, 36, 4, 38, -44], [0, -54, 0, -8, -88, 345, -4, -27], [112, -38, -26, 6, 0, 0, -6, 38], [0, 24, 0, -120, 56, -6, -60, 12], [64, -42, 89, 37, 128, 178, 37, -42], [300, -2, -106, 40, 0, 0, -40, 2], [-60, 0, -212, 0, -30, -106, -60, -95], [10, -27, 6, -148, -10, -6, 0, 0], [-402, -18, -208, -76, -402, -208, -152, -36], [390, 5, 127, -42, 0, 0, 42, -5], [0, 60, 0, -72, -138, 362, -36, 30], [80, 28, 50, 94, 160, 100, 94, 28], [52, 0, -186, 0, 26, -93, 41, -73], [6, -29, -77, -111, -6, 77, 0, 0], [320, 0, 48, 0, 160, 24, -10, 20], [0, -96, 0, 112, 530, -122, 56, -48], [310, 16, -168, -108, 0, 0, 108, -16], [360, 8, -40, 34, 360, -40, 68, 16], [-14, -1, 204, -94, 14, -204, 0, 0], [0, 66, 0, 20, 414, 43, 10, 33], [298, -4, 302, 32, 298, 302, 64, -8], [-56, -141, -152, -21, 56, 152, 0, 0], [246, 17, -58, -20, 492, -116, -20, 17], [48, 0, 426, 0, 24, 213, 125, -18], [-400, 53, -227, 60, -400, -227, 120, 106], [400, 24, -48, -3, 800, -96, -3, 24], [312, 0, 256, 0, 156, 128, -30, -100], [-258, -2, -405, 8, -258, -405, 16, -4], [150, -35, -467, 42, 0, 0, -42, 35], [-44, 104, -152, -74, 44, 152, 0, 0], [212, -38, 330, -68, 0, 0, 68, 38], [114, 33, -151, -158, 114, -151, -316, 66], [176, 41, -54, -222, 352, -108, -222, 41], [-90, 1, 298, -104, 90, -298, 0, 0], [0, 118, 0, -164, 542, -175, -82, 59], [120, -31, -5, 116, 0, 0, -116, 31], [-160, 81, 182, -23, 160, -182, 0, 0], [182, 28, -215, -28, 0, 0, 28, -28], [-72, 0, -286, 0, -36, -143, 177, -88], [324, 37, -147, 72, 324, -147, 144, 74], [16, 0, -72, 0, 8, -36, -84, -58], [78, 123, -91, 2, 0, 0, -2, -123], [0, -32, 0, 248, -238, 49, 124, -16], [-744, 0, -192, 0, -372, -96, -100, 10], [-312, 9, -10, 76, -624, -20, 76, 9], [368, 0, -128, 0, 184, -64, 223, -105], [-28, -50, -330, -72, -28, -330, -144, -100], [28, -25, 209, 116, 0, 0, -116, 25], [0, -72, 0, 40, -344, 158, 20, -36], [34, -68, 397, -28, 0, 0, 28, 68], [224, 0, 890, 0, 112, 445, 3, -20], [296, -85, 84, -185, 592, 168, -185, -85], [440, -122, -81, 41, -440, 81, 0, 0], [4, 146, -252, 54, -4, 252, 0, 0], [274, 63, 127, 3, 548, 254, 3, 63], [294, 66, -18, -68, 294, -18, -136, 132], [56, 36, -144, 280, 112, -288, 280, 36], [0, 116, 0, -92, 4, -158, -46, 58], [210, -16, 370, -84, 210, 370, -168, -32], [390, -63, -85, -22, 0, 0, 22, 63], [-224, -224, 188, 141, 224, -188, 0, 0], [-158, 81, -4, 2, -316, -8, 2, 81], [-368, -77, -34, -91, -736, -68, -91, -77], [-288, -88, 187, 39, 288, -187, 0, 0], [8, 0, 316, 0, 4, 158, 114, 3], [0, -110, 0, -4, 166, -147, -2, -55], [0, 148, 0, 96, -226, 251, 48, 74], [-312, 62, -259, 113, -624, -518, 113, 62], [-462, -2, 446, -20, 0, 0, 20, 2], [116, 0, -348, 0, 58, -174, -208, 153], [-18, 66, 216, -136, -18, 216, -272, 132], [128, 31, 737, -76, 0, 0, 76, -31], [0, 44, 0, -96, 510, -264, -48, 22], [-608, 0, -230, 0, -304, -115, 63, 40], [336, 112, 364, 149, 672, 728, 149, 112], [-66, 19, -355, 101, 66, 355, 0, 0], [0, -202, 0, 148, -266, 3, 74, -101], [-1034, -50, -262, -76, 0, 0, 76, 50], [84, 86, 90, -4, 0, 0, 4, -86], [1472, 0, 180, 0, 736, 90, -122, -30], [-126, -45, 65, 70, -126, 65, 140, -90], [92, -128, -100, 204, -92, 100, 0, 0], [-642, 94, -638, 104, -642, -638, 208, 188], [440, 73, 66, 33, -440, -66, 0, 0], [-372, 16, 506, 92, 0, 0, -92, -16], [-204, 0, -274, 0, -102, -137, 291, 151], [-214, -105, 551, 66, -214, 551, 132, -210], [-4, 62, -198, -281, -8, -396, -281, 62], [0, 6, 0, -40, -868, -425, -20, 3], [-90, -22, 279, 72, -90, 279, 144, -44], [456, 166, 72, -18, -456, -72, 0, 0], [690, 119, -197, -137, 1380, -394, -137, 119], [402, -28, 138, 236, 0, 0, -236, 28], [166, -79, 105, -102, 166, 105, -204, -158], [196, 69, 82, 332, 392, 164, 332, 69], [-928, 0, 292, 0, -464, 146, 115, 45], [0, 344, 0, -8, 222, 148, -4, 172], [-480, 64, 114, -246, -960, 228, -246, 64], [212, -28, -463, 32, 0, 0, -32, 28], [-220, 0, -306, 0, -110, -153, 35, 171], [90, -81, -113, -2, 90, -113, -4, -162], [468, -96, -91, 219, -468, 91, 0, 0], [-760, -4, -374, -140, -760, -374, -280, -8], [-316, 110, 84, 200, 316, -84, 0, 0], [0, 212, 0, -208, 58, 379, -104, 106], [-144, 26, -208, 412, 144, 208, 0, 0], [-624, 0, -44, 0, -312, -22, 17, 231], [-116, -5, -487, -84, 0, 0, 84, 5], [0, -224, 0, 176, 756, -524, 88, -112], [900, 0, 190, 0, 450, 95, -253, 23], [-574, 47, 227, 110, -574, 227, 220, 94], [-728, 26, 31, 269, 728, -31, 0, 0], [-18, -12, 589, 140, -18, 589, 280, -24], [0, 152, 0, 16, 448, -565, 8, 76], [516, 214, -205, -179, 1032, -410, -179, 214], [-916, 0, 344, 0, -458, 172, -70, 229], [-290, -26, 530, 176, -290, 530, 352, -52], [944, 40, 36, -82, 1888, 72, -82, 40], [464, 0, 1016, 0, 232, 508, 21, -164], [360, -67, -110, 383, -360, 110, 0, 0], [-92, 176, -60, -300, -184, -120, -300, 176], [-200, 182, 425, 59, 200, -425, 0, 0], [-248, 0, 304, 0, -124, 152, -272, 300], [0, 162, 0, -312, -404, -79, -156, 81], [-156, -56, 148, 110, 0, 0, -110, 56], [168, 269, 370, 124, -168, -370, 0, 0], [0, 80, 0, -128, -132, -651, -64, 40], [-462, 127, 233, -37, -924, 466, -37, 127], [-622, -118, -422, -196, 0, 0, 196, 118], [-1376, 0, -404, 0, -688, -202, 10, -20], [394, -105, 428, 158, -394, -428, 0, 0], [1136, 0, -736, 0, 568, -368, -243, 51], [1066, 10, 624, -88, 1066, 624, -176, 20], [578, 245, -117, -159, -578, 117, 0, 0], [0, 218, 0, -332, 758, 1, -166, 109], [62, 5, -367, 94, 0, 0, -94, -5], [0, -344, 0, 216, -682, 295, 108, -172], [492, 59, -326, -42, 984, -652, -42, 59], [0, -108, 0, 72, -310, -94, 36, -54], [0, -64, 0, -80, 780, -146, -40, -32], [-540, 28, 346, -154, -1080, 692, -154, 28], [84, 10, 1061, -68, 0, 0, 68, -10], [-1136, 0, 138, 0, -568, 69, 231, -12], [384, -3, -567, 36, 384, -567, 72, -6], [616, 0, -484, 0, 308, -242, 178, -43], [0, 250, 0, 80, -980, 121, 40, 125], [830, -80, 65, -20, 830, 65, -40, -160], [-632, -189, -590, -228, 632, 590, 0, 0], [570, -150, 140, 172, 0, 0, -172, 150], [-728, 78, 226, -146, -728, 226, -292, 156], [-8, -147, -590, 20, -16, -1180, 20, -147], [-324, -364, 130, 182, 324, -130, 0, 0], [1808, 0, 364, 0, 904, 182, 29, -81], [0, -264, 0, 8, -1050, -416, 4, -132], [498, -53, -571, 62, 0, 0, -62, 53], [56, 154, 488, 224, 112, 976, 224, 154], [1478, 49, -151, -6, 1478, -151, -12, 98], [-392, 0, -664, 0, -196, -332, -242, 70], [116, 74, -176, 524, -116, 176, 0, 0], [0, 144, 0, 400, 202, -642, 200, 72], [-626, -63, -57, -55, -1252, -114, -55, -63], [932, 0, 440, 0, 466, 220, 182, 173], [-472, 111, -302, 606, -944, -604, 606, 111], [774, 155, -266, -136, -774, 266, 0, 0], [-406, -72, 1156, 52, -406, 1156, 104, -144], [-232, -75, -745, -112, 0, 0, 112, 75], [240, 136, -566, -76, 0, 0, 76, -136], [-972, 0, 198, 0, -486, 99, -221, -109], [44, 171, -761, 0, 44, -761, 0, 342], [-744, 121, 6, -117, -1488, 12, -117, 121], [182, 35, 389, 195, -182, -389, 0, 0], [0, 186, 0, 100, -970, -203, 50, 93], [-692, -36, -535, 56, -692, -535, 112, -72], [1012, 25, -94, -358, -1012, 94, 0, 0], [994, -89, -7, 129, 1988, -14, 129, -89], [-624, 0, 712, 0, -312, 356, -197, -11], [754, 140, -576, 20, 754, -576, 40, 280], [666, -3, -457, 154, 0, 0, -154, 3], [-184, 148, -52, -499, 184, 52, 0, 0], [360, 35, 394, 401, 720, 788, 401, 35], [-4, 40, 513, -269, 4, -513, 0, 0], [1024, 0, -616, 0, 512, -308, 212, -310], [498, -16, 696, 44, 0, 0, -44, 16], [-320, 158, 45, 93, -640, 90, 93, 158], [62, -76, -896, 136, 0, 0, -136, 76], [838, 35, -757, 162, 838, -757, 324, 70], [-1376, 0, -124, 0, -688, -62, -117, -305], [-144, 20, -326, -120, -144, -326, -240, 40], [-528, 89, -50, -147, 528, 50, 0, 0], [0, 112, 0, -560, 324, -612, -280, 56], [-472, -211, 278, -269, -944, 556, -269, -211], [304, 0, 236, 0, 152, 118, -308, 347], [0, -298, 0, 276, -330, 7, 138, -149], [-376, 36, -336, -84, -376, -336, -168, 72], [0, 32, 0, 328, 1110, 16, 164, 16], [-492, 0, 856, 0, -246, 428, -38, -131], [-428, -340, -552, -100, 428, 552, 0, 0], [504, 82, 64, 92, 504, 64, 184, 164], [-504, 4, -62, 590, -1008, -124, 590, 4], [-92, 184, 306, -41, -184, 612, -41, 184], [-2144, 0, 864, 0, -1072, 432, 394, -330], [0, 30, 0, 400, 896, 351, 200, 15], [110, 56, -1074, -32, 110, -1074, -64, 112], [948, -178, 350, -118, 0, 0, 118, 178], [332, -199, -46, -38, -332, 46, 0, 0], [-12, 102, 283, -43, -24, 566, -43, 102], [-546, -86, -778, 68, -546, -778, 136, -172], [1084, -40, 136, -28, -1084, -136, 0, 0], [0, -58, 0, 588, 386, 145, 294, -29], [590, 11, -168, -438, 1180, -336, -438, 11], [-730, 112, -1172, 80, 0, 0, -80, -112], [358, 195, -397, -379, -358, 397, 0, 0], [-2728, 0, -8, 0, -1364, -4, 156, -68], [384, -108, -855, 427, 768, -1710, 427, -108], [-476, 0, 296, 0, -238, 148, 354, -571], [-24, -326, 28, 250, -48, 56, 250, -326], [224, 0, -264, 0, 112, -132, 339, -574], [0, -246, 0, 716, 762, 253, 358, -123], [-416, 186, -1276, -36, -416, -1276, -72, 372]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1764_z_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1764_3_z_l();".
// 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_1764_3_z_l(:prec:=8)
    chi := MakeCharacter_1764_z();
    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_1764_3_z_l();".
// 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_1764_3_z_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1764_z();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![196, -336, -256, 768, 1010, 0, -32, 0, 1]>,<11,R![10837264, -158016, 410512, 5952, 12084, 96, 124, 0, 1]>,<13,R![2979076, 0, 1723280, 0, 123284, 0, 712, 0, 1]>,<19,R![285745216, 199602432, 62839360, 11430144, 1297976, 92928, 4040, 96, 1]>],Snew);
    return Vf;
end function;