
// Make newform 1521.4.a.bm in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1521_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1521_4_a_bm();".
// 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_1521_4_a_bm();".
// 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 := [-6889792, 0, 27824160, 0, -37096332, 0, 22251817, 0, -6594119, 0, 1016064, 0, -86197, 0, 4066, 0, -100, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-11, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -19, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-391044378816, 0, -4388820185568, 0, 5263282300108, 0, -1907438218261, 0, 297386572690, 0, -23574535710, 0, 997040927, 0, -21373147, 0, 181489, 0], [-6066047938496, 0, 20720594303136, 0, -19094667050564, 0, 7013938227079, 0, -1205712814214, 0, 109302912522, 0, -5408740661, 0, 138372809, 0, -1433467, 0], [0, 4651421005504, 0, -5346556479264, 0, 1511229363924, 0, 6322010877, 0, -46241455778, 0, 6387634062, 0, -378988503, 0, 10652275, 0, -116153], [-12024332034624, 0, 43016754240096, 0, -37484856803996, 0, 13069033542529, 0, -2180860865738, 0, 193723590182, 0, -9418555363, 0, 236768687, 0, -2408429, 0], [-7630928212544, 0, 31493414863456, 0, -32351261871516, 0, 12457145153985, 0, -2184793422666, 0, 199518257062, 0, -9869708963, 0, 251135855, 0, -2578733, 0], [0, 1081481943360, 0, -11048203120864, 0, 9195094876716, 0, -3027547657725, 0, 487574025762, 0, -42210338830, 0, 2004907351, 0, -49197939, 0, 487481], [1369985559488, 0, -3950268285600, 0, 3400342544004, 0, -1212878963191, 0, 204598542054, 0, -18217576618, 0, 883985541, 0, -22133081, 0, 224043, 0], [-9201566989888, 0, 26703850400864, 0, -23466350016732, 0, 8458301785969, 0, -1437824961002, 0, 128809709446, 0, -6278374259, 0, 157633343, 0, -1597405, 0], [14970699039296, 0, -44215276411488, 0, 37534795227804, 0, -13069033542529, 0, 2180860865738, 0, -193723590182, 0, 9418555363, 0, -236768687, 0, 2408429, 0], [0, -49189596316224, 0, 141921004199520, 0, -104610840892572, 0, 33015383487233, 0, -5202991062346, 0, 445951059110, 0, -21167805411, 0, 523191087, 0, -5256429], [0, 1371275587136, 0, -5254045774432, 0, 5000497301212, 0, -1841699084785, 0, 314739137482, 0, -28222338150, 0, 1375479251, 0, -34535167, 0, 350141], [0, 71133236166720, 0, -212735514144608, 0, 190460319288764, 0, -68983031416761, 0, 11732373656570, 0, -1050840575606, 0, 51228961675, 0, -1287411959, 0, 13068997], [0, 81381209244608, 0, -363787940189856, 0, 362313964310788, 0, -135986926865111, 0, 23552852125222, 0, -2135981850218, 0, 105137553317, 0, -2663050105, 0, 27213579], [0, 292688312489664, 0, -732619140183840, 0, 594279696796884, 0, -205451868798659, 0, 34220825562462, 0, -3033885246066, 0, 147131501289, 0, -3687858637, 0, 37394823]];
    Rf_basisdens := [1, 1, 1, 1, 49938423808, 99876847616, 26889920512, 99876847616, 99876847616, 99876847616, 7682834432, 49938423808, 49938423808, 349568966656, 13444960256, 349568966656, 699137933312, 699137933312];
    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_1521_a();"
function MakeCharacter_1521_a()
    N := 1521;
    order := 1;
    char_gens := [677, 847];
    v := [1, 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;

function MakeCharacter_1521_a_Hecke(Kf)
    return MakeCharacter_1521_a();
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 := 4;
    raw_aps := [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-5, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, -3], [-24, 0, -4, 0, 3, 1, 0, 0, -2, 0, 1, 0, 1, 0, 0, 0, 0, 0], [0, -10, 0, 0, 0, 0, 2, 0, 0, -2, 0, 0, 0, -1, 4, 2, 3, -1], [0, -9, 0, -2, 0, 0, -3, 0, 0, -3, 0, 0, 0, -1, 0, 4, 1, -1], [-42, 0, 1, 0, 1, -4, 0, 6, -1, 0, -4, -11, -1, 0, 0, 0, 0, 0], [-32, 0, -5, 0, -2, 11, 0, 2, 9, 0, 2, -2, 3, 0, 0, 0, 0, 0], [0, -6, 0, -6, 0, 0, -9, 0, 0, -7, 0, 0, 0, -5, -2, -12, -5, 2], [46, 0, 11, 0, 3, -12, 0, 6, -3, 0, 3, 2, 4, 0, 0, 0, 0, 0], [0, 2, 0, -8, 0, 0, 3, 0, 0, -1, 0, 0, 0, -10, -4, -8, 2, 1], [0, -35, 0, 8, 0, 0, 3, 0, 0, -1, 0, 0, 0, -4, 27, -4, -4, -5], [0, 22, 0, -10, 0, 0, -1, 0, 0, 23, 0, 0, 0, 9, 11, 17, -6, 10], [-98, 0, 22, 0, -7, -1, 0, 0, 16, 0, -1, -24, -1, 0, 0, 0, 0, 0], [-153, 0, -26, 0, -6, -11, 0, 3, 1, 0, 5, 16, 15, 0, 0, 0, 0, 0], [0, -96, 0, 16, 0, 0, -8, 0, 0, 5, 0, 0, 0, 7, -6, -14, 7, 12], [-183, 0, -13, 0, -21, 2, 0, -12, 11, 0, 1, 33, -9, 0, 0, 0, 0, 0], [175, 0, 16, 0, -15, -20, 0, 1, -33, 0, 1, -14, -12, 0, 0, 0, 0, 0], [0, 2, 0, -2, 0, 0, 3, 0, 0, -2, 0, 0, 0, 15, -19, -17, 26, 13], [0, -50, 0, 2, 0, 0, 3, 0, 0, -4, 0, 0, 0, -11, 6, 36, -9, -11], [-276, 0, 43, 0, -7, -1, 0, 7, -18, 0, 14, -25, -14, 0, 0, 0, 0, 0], [0, -68, 0, 28, 0, 0, 29, 0, 0, -25, 0, 0, 0, -20, 1, -31, 33, -29], [-326, 0, 48, 0, 24, 39, 0, 3, -23, 0, -17, -7, 26, 0, 0, 0, 0, 0], [0, 20, 0, 14, 0, 0, -26, 0, 0, 18, 0, 0, 0, 0, -9, 17, -13, -2], [-405, 0, 69, 0, -10, 30, 0, -5, -88, 0, -23, -12, 20, 0, 0, 0, 0, 0], [0, -116, 0, 12, 0, 0, -14, 0, 0, 25, 0, 0, 0, 46, -18, -6, 18, 1], [-72, 0, -1, 0, 35, -13, 0, 37, -78, 0, -70, -35, 12, 0, 0, 0, 0, 0], [0, -69, 0, -18, 0, 0, 32, 0, 0, -25, 0, 0, 0, -12, -11, 68, 6, -11], [0, 74, 0, -56, 0, 0, -81, 0, 0, 2, 0, 0, 0, 0, -28, -12, -34, 30], [-855, 0, -23, 0, 6, -42, 0, -5, 18, 0, 49, 4, -62, 0, 0, 0, 0, 0], [0, -133, 0, 34, 0, 0, 56, 0, 0, 24, 0, 0, 0, -15, -10, 36, 39, -62], [-1078, 0, 114, 0, -5, -119, 0, 2, 18, 0, -22, 1, -18, 0, 0, 0, 0, 0], [558, 0, 39, 0, -25, 70, 0, 0, -83, 0, -13, -136, 0, 0, 0, 0, 0, 0], [-197, 0, 92, 0, -14, 101, 0, -23, 21, 0, -79, 70, 1, 0, 0, 0, 0, 0], [0, -40, 0, 0, 0, 0, -15, 0, 0, 0, 0, 0, 0, -20, 18, 8, -20, 42], [0, -203, 0, -46, 0, 0, 11, 0, 0, 31, 0, 0, 0, -23, -52, -28, 3, -3], [0, -13, 0, 22, 0, 0, -46, 0, 0, -28, 0, 0, 0, -6, 109, -72, -28, 46], [-281, 0, 244, 0, -20, 13, 0, 45, 21, 0, -65, -30, -53, 0, 0, 0, 0, 0], [0, 50, 0, 16, 0, 0, 61, 0, 0, 44, 0, 0, 0, 71, 62, 58, 7, 7], [-1092, 0, 91, 0, 53, -90, 0, 46, 95, 0, 123, 2, -26, 0, 0, 0, 0, 0], [0, -102, 0, 70, 0, 0, 3, 0, 0, -16, 0, 0, 0, 21, 47, -21, -14, 39], [316, 0, -163, 0, 74, 75, 0, 46, 133, 0, 93, 33, 10, 0, 0, 0, 0, 0], [1005, 0, -154, 0, -1, -50, 0, -59, 193, 0, 68, 142, -10, 0, 0, 0, 0, 0], [-1063, 0, 105, 0, 2, 8, 0, -71, 144, 0, -1, 30, 22, 0, 0, 0, 0, 0], [0, 263, 0, 26, 0, 0, 23, 0, 0, 94, 0, 0, 0, 61, -161, 52, -7, 47], [-2473, 0, 12, 0, 15, -2, 0, -65, 101, 0, 6, 46, -8, 0, 0, 0, 0, 0], [0, 408, 0, 120, 0, 0, 105, 0, 0, -37, 0, 0, 0, 34, 228, 50, 88, -53], [0, 182, 0, 20, 0, 0, -142, 0, 0, -55, 0, 0, 0, 18, 14, -64, -26, 89], [-899, 0, 164, 0, 16, -26, 0, 70, -96, 0, -160, 25, 37, 0, 0, 0, 0, 0], [0, 634, 0, 34, 0, 0, -156, 0, 0, -18, 0, 0, 0, 45, 225, -69, -44, 87], [0, 810, 0, -82, 0, 0, 59, 0, 0, 20, 0, 0, 0, -73, 34, 190, -49, -19], [0, 482, 0, 8, 0, 0, -55, 0, 0, -2, 0, 0, 0, 95, 72, -150, -49, 19], [0, 877, 0, 138, 0, 0, 27, 0, 0, -73, 0, 0, 0, -125, 213, -94, 61, -96], [-2450, 0, -189, 0, -18, -14, 0, 7, 126, 0, 183, -17, 41, 0, 0, 0, 0, 0], [1371, 0, 135, 0, -142, -250, 0, -31, 176, 0, 37, -62, -154, 0, 0, 0, 0, 0], [0, -244, 0, 118, 0, 0, -68, 0, 0, 122, 0, 0, 0, 11, -136, 0, 43, 23], [-120, 0, 66, 0, 59, 247, 0, -164, -88, 0, -55, 114, 53, 0, 0, 0, 0, 0], [0, -1143, 0, -82, 0, 0, 71, 0, 0, -50, 0, 0, 0, -123, 204, -36, -7, -78], [-1939, 0, -194, 0, 45, 192, 0, 137, 173, 0, -22, 104, -38, 0, 0, 0, 0, 0], [0, 468, 0, -164, 0, 0, -73, 0, 0, 22, 0, 0, 0, 95, 36, -144, 33, 135], [1646, 0, 362, 0, 127, -16, 0, 119, -231, 0, -98, -241, -103, 0, 0, 0, 0, 0], [0, 192, 0, -40, 0, 0, 60, 0, 0, 3, 0, 0, 0, -78, 235, 113, -63, 33], [-1847, 0, 27, 0, 74, -66, 0, 197, -198, 0, 123, 218, -40, 0, 0, 0, 0, 0], [-528, 0, -141, 0, 187, 170, 0, 16, -117, 0, -8, 199, 157, 0, 0, 0, 0, 0], [0, 614, 0, -182, 0, 0, 71, 0, 0, 110, 0, 0, 0, 99, 145, -77, -82, 93], [-1568, 0, 78, 0, -46, 296, 0, -100, 198, 0, 90, -150, 10, 0, 0, 0, 0, 0], [0, -1342, 0, 26, 0, 0, -57, 0, 0, 30, 0, 0, 0, 41, -38, -244, 67, 179], [0, -984, 0, -136, 0, 0, -27, 0, 0, 162, 0, 0, 0, 185, -142, 192, 59, 171], [363, 0, -430, 0, -65, -172, 0, -23, 125, 0, 155, 326, -144, 0, 0, 0, 0, 0], [-1877, 0, 159, 0, 68, 178, 0, 79, -46, 0, -79, -304, 32, 0, 0, 0, 0, 0], [-1908, 0, -511, 0, 2, -161, 0, -34, -43, 0, -116, 216, -11, 0, 0, 0, 0, 0], [0, -276, 0, -284, 0, 0, -341, 0, 0, -44, 0, 0, 0, -93, -192, -126, -193, 59], [0, 1151, 0, 200, 0, 0, 232, 0, 0, -34, 0, 0, 0, 8, -472, -234, 212, -159], [306, 0, -50, 0, -101, 349, 0, -60, -352, 0, -45, 158, 175, 0, 0, 0, 0, 0], [0, 470, 0, 206, 0, 0, 66, 0, 0, -105, 0, 0, 0, 11, 652, -194, -29, -122], [-3035, 0, -141, 0, -208, -102, 0, -289, -62, 0, -128, 98, -6, 0, 0, 0, 0, 0], [0, 1745, 0, -8, 0, 0, 110, 0, 0, -124, 0, 0, 0, -254, 246, -118, 44, -181], [812, 0, 106, 0, -29, -27, 0, -56, 404, 0, 223, 318, -213, 0, 0, 0, 0, 0], [0, -454, 0, 288, 0, 0, 41, 0, 0, -142, 0, 0, 0, 75, 380, -118, 75, -37], [1991, 0, -211, 0, -39, 123, 0, 241, -338, 0, 179, 119, -58, 0, 0, 0, 0, 0], [-2687, 0, -350, 0, -150, 9, 0, -55, 171, 0, 258, 134, 97, 0, 0, 0, 0, 0], [0, -131, 0, -300, 0, 0, 30, 0, 0, 329, 0, 0, 0, 22, -278, 258, -172, 58], [0, -96, 0, -28, 0, 0, -11, 0, 0, 109, 0, 0, 0, -164, -122, -46, -114, 105], [2918, 0, 766, 0, -231, -7, 0, 138, 232, 0, -336, -217, -174, 0, 0, 0, 0, 0], [0, -52, 0, 234, 0, 0, 182, 0, 0, -43, 0, 0, 0, 93, -189, 205, 268, -148], [-360, 0, 142, 0, 254, 36, 0, 160, 258, 0, 86, -152, -56, 0, 0, 0, 0, 0], [0, -21, 0, 14, 0, 0, -177, 0, 0, -190, 0, 0, 0, 53, -73, -352, -67, 127], [0, 1480, 0, -208, 0, 0, 242, 0, 0, 299, 0, 0, 0, -130, -290, 328, -14, -159], [-556, 0, -251, 0, 47, -63, 0, 73, 468, 0, -54, 81, 394, 0, 0, 0, 0, 0], [0, 2171, 0, 40, 0, 0, 0, 0, 0, -287, 0, 0, 0, -93, 460, -196, -67, -265], [320, 0, -398, 0, -78, -250, 0, 86, 640, 0, 596, 796, -78, 0, 0, 0, 0, 0], [0, 600, 0, 164, 0, 0, 354, 0, 0, 62, 0, 0, 0, -19, -130, 220, 351, -401], [0, 184, 0, -364, 0, 0, 74, 0, 0, -312, 0, 0, 0, -322, -404, -34, -36, -116], [0, 1566, 0, -4, 0, 0, 157, 0, 0, -36, 0, 0, 0, -90, -488, 320, 4, -362], [-707, 0, 209, 0, 25, -469, 0, -9, -116, 0, -182, -570, -271, 0, 0, 0, 0, 0], [1987, 0, -485, 0, -226, -20, 0, 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2646, 4726, 0, 1295, 1733, -4053, 0, 0, 0, 0, 0], [0, 21203, 0, -926, 0, 0, -719, 0, 0, -7401, 0, 0, 0, -4513, 2787, -5314, -1899, -2072], [0, 26784, 0, 1294, 0, 0, -4311, 0, 0, -1216, 0, 0, 0, -1858, -3721, -3947, -819, 1338], [-11334, 0, 3746, 0, -1140, 272, 0, -4330, -2652, 0, 5266, -260, -1352, 0, 0, 0, 0, 0], [0, 22354, 0, -848, 0, 0, -3550, 0, 0, 848, 0, 0, 0, 206, 2132, 3128, -2198, 216], [21101, 0, 3301, 0, 1294, 3434, 0, 3991, -1210, 0, 6562, 1041, 1163, 0, 0, 0, 0, 0], [-73006, 0, -480, 0, -3418, -2258, 0, -1994, 1946, 0, 2719, 5190, -3140, 0, 0, 0, 0, 0], [0, -13846, 0, -2466, 0, 0, 514, 0, 0, 2445, 0, 0, 0, -707, 970, 1422, 349, 2412], [-18002, 0, -6121, 0, -1070, -2771, 0, 5234, 7, 0, 3992, -1530, -551, 0, 0, 0, 0, 0], [0, 9199, 0, -3408, 0, 0, -2617, 0, 0, 1478, 0, 0, 0, 1062, -6398, 5754, -2640, 1603], [41838, 0, 19265, 0, -2396, 5, 0, 1160, -5991, 0, -6254, -7784, -2297, 0, 0, 0, 0, 0], [0, -56, 0, -482, 0, 0, 44, 0, 0, 271, 0, 0, 0, -767, 3668, -1574, 649, 150], [-42165, 0, -4512, 0, 835, -1765, 0, -3536, -596, 0, 1085, 10323, 818, 0, 0, 0, 0, 0], [0, -10252, 0, 4682, 0, 0, 2588, 0, 0, 438, 0, 0, 0, 1239, 1830, -1916, 1559, -2551], [0, -4195, 0, -312, 0, 0, 752, 0, 0, 322, 0, 0, 0, 638, 3732, 5742, 1806, -2851], [-16311, 0, -10677, 0, 3487, -3934, 0, -1110, 2723, 0, 4985, 8565, 4061, 0, 0, 0, 0, 0], [84340, 0, 3099, 0, 894, 4569, 0, -4316, 4327, 0, 3266, 4356, 11, 0, 0, 0, 0, 0], [0, -33168, 0, 406, 0, 0, -1321, 0, 0, -2621, 0, 0, 0, -2167, -6536, 1176, -1915, 1254], [-24960, 0, 9822, 0, -3806, -5472, 0, 1732, 1294, 0, -4150, -652, -1900, 0, 0, 0, 0, 0], [0, 10862, 0, 3084, 0, 0, -780, 0, 0, -831, 0, 0, 0, 1027, -7910, -6930, 871, 350], [-39402, 0, 12525, 0, 3569, 2504, 0, 4790, -10331, 0, -2078, -7422, 3104, 0, 0, 0, 0, 0], [0, 9701, 0, 536, 0, 0, -4953, 0, 0, -3311, 0, 0, 0, 1072, -2550, -1894, -1744, 2192], [0, 4377, 0, 4560, 0, 0, -1273, 0, 0, 62, 0, 0, 0, 2910, 2836, -416, 2106, 1], [17873, 0, 383, 0, 4496, -3156, 0, 1437, 3898, 0, -3921, -30, 770, 0, 0, 0, 0, 0], [33313, 0, 7582, 0, -2715, -2305, 0, -416, 518, 0, -4351, -1345, 796, 0, 0, 0, 0, 0], [0, -1418, 0, 4378, 0, 0, 1439, 0, 0, 899, 0, 0, 0, 3077, 1708, 1408, 2177, 416], [0, 6518, 0, 892, 0, 0, 3768, 0, 0, 2095, 0, 0, 0, 4487, 2334, 2612, 2505, -1606], [15425, 0, -1316, 0, -117, -5752, 0, -1427, 3215, 0, 9417, 1299, -251, 0, 0, 0, 0, 0], [0, -6668, 0, -1420, 0, 0, -577, 0, 0, -3548, 0, 0, 0, -5700, 568, -274, 32, -3604], [0, -22432, 0, 1864, 0, 0, -1113, 0, 0, 1756, 0, 0, 0, -1203, -4888, -1092, -1265, -245], [0, -4482, 0, -1268, 0, 0, -85, 0, 0, -1358, 0, 0, 0, 542, -1901, 131, -1553, -2332], [70159, 0, 3286, 0, -1540, 4863, 0, -3447, -5609, 0, -7261, -6422, 1725, 0, 0, 0, 0, 0], [0, -1604, 0, -856, 0, 0, -5536, 0, 0, -5772, 0, 0, 0, -5485, 4226, -6480, -3139, 2195], [0, -19314, 0, -2956, 0, 0, -3860, 0, 0, -159, 0, 0, 0, 1025, -1698, 5890, -3763, 3368], [30062, 0, -17269, 0, 4554, 827, 0, 1894, 11499, 0, 8091, 9459, 1722, 0, 0, 0, 0, 0], [0, 2316, 0, 1170, 0, 0, 4410, 0, 0, 3682, 0, 0, 0, 4731, 2226, 5276, 3031, -2767], [0, -16261, 0, -3238, 0, 0, 1071, 0, 0, 463, 0, 0, 0, -5030, 5559, 878, -742, -3689], [0, 10718, 0, 5950, 0, 0, 2746, 0, 0, -3000, 0, 0, 0, 259, 1834, 694, 3289, -4665], [-34967, 0, -8204, 0, 3106, -4775, 0, -2063, 7953, 0, 5153, 9906, 1513, 0, 0, 0, 0, 0], [0, 8546, 0, 5436, 0, 0, -1443, 0, 0, 71, 0, 0, 0, 832, -5480, -3920, 3792, 2655]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1521_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1521_4_a_bm();".
// 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_1521_4_a_bm(:prec:=18)
    chi := MakeCharacter_1521_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_1521_4_a_bm();".
// 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_1521_4_a_bm( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1521_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![-6889792, 0, 27824160, 0, -37096332, 0, 22251817, 0, -6594119, 0, 1016064, 0, -86197, 0, 4066, 0, -100, 0, 1]>,<5,R![-682535158703125, 0, 695574291912500, 0, -215484650567550, 0, 20322031766127, 0, -689387047881, 0, 11320990891, 0, -99614185, 0, 472720, 0, -1117, 0, 1]>,<7,R![-36574832125, -20231277300, -3053104080, 34034223, 29038353, 310197, -68263, -1176, 47, 1]>],Snew);
    return Vf;
end function;