
// Make newform 1900.4.a.l in Magma, downloaded from the LMFDB on 22 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1900_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_1900_4_a_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_1900_4_a_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 := [73273600, 0, -26837136, 0, 3734252, 0, -255920, 0, 9117, 0, -158, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-26, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-574347568, 0, 131877836, 0, -10868196, 0, 437273, 0, -8674, 0, 61, 0], [-1222414820, 0, 443719372, 0, -57488088, 0, 3248623, 0, -78440, 0, 605, 0], [-1631938000, 0, 520024100, 0, -58210740, 0, 2898659, 0, -63502, 0, 463, 0], [-12753359288, 0, 4022546596, 0, -449918700, 0, 22434631, 0, -491846, 0, 3587, 0], [0, 114707836544, 0, -49183662748, 0, 6987290220, 0, -423294253, 0, 10755722, 0, -85289], [0, 805445727856, 0, -191595366812, 0, 17579731320, 0, -767907557, 0, 15482998, 0, -107881], [0, 315187863964, 0, -94261613648, 0, 9760187235, 0, -451609448, 0, 9307927, 0, -65314], [0, 389805074264, 0, -119179049908, 0, 12994700190, 0, -632439373, 0, 13547312, 0, -97289], [0, -1693350797416, 0, 535318079612, 0, -59609142030, 0, 2956848017, 0, -64592308, 0, 470161]];
    Rf_basisdens := [1, 1, 1, 2222064, 555516, 170928, 1111032, 2377608480, 2377608480, 594402120, 594402120, 1188804240];
    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_1900_a();"
function MakeCharacter_1900_a()
    N := 1900;
    order := 1;
    char_gens := [951, 77, 401];
    v := [1, 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_1900_a_Hecke( : Kf := Kf)
    return MakeCharacter_1900_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]];
    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, 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, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [-9, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 1, 0], [0, -1, 0, 0, 0, 0, 0, 2, 1, 1, -2, 2], [19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, -3, -2, 1, 0, 2], [7, 0, -6, 4, -2, -2, 1, 0, 0, 0, 0, 0], [8, 0, -3, -1, 3, -3, -1, 0, 0, 0, 0, 0], [0, -12, 0, 0, 0, 0, 0, 4, -1, -9, -3, 1], [-54, 0, 3, -11, -3, -1, 1, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, -6, 0, 4, -1, 5], [0, -14, 0, 0, 0, 0, 0, 10, 0, 11, 5, 6], [0, -26, 0, 0, 0, 0, 0, -9, 6, 7, -5, 12], [-85, 0, 3, 13, 1, 7, -4, 0, 0, 0, 0, 0], [-168, 0, 13, -11, -2, 6, -5, 0, 0, 0, 0, 0], [0, -41, 0, 0, 0, 0, 0, -3, -1, -18, 1, -15], [-252, 0, -5, 1, 1, 9, -1, 0, 0, 0, 0, 0], [0, -75, 0, 0, 0, 0, 0, -1, 4, -21, 9, -1], [-47, 0, 23, -21, -6, 9, 11, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, -14, -9, 0, 2, -16], [-213, 0, 2, 32, -5, 2, -10, 0, 0, 0, 0, 0], [0, -84, 0, 0, 0, 0, 0, 10, -1, 29, 13, 11], [-254, 0, 25, -38, 3, -4, 19, 0, 0, 0, 0, 0], [0, 57, 0, 0, 0, 0, 0, 10, -10, 6, 21, -34], [0, -119, 0, 0, 0, 0, 0, -3, 3, 30, -9, -1], [94, 0, 19, 7, 10, -23, -14, 0, 0, 0, 0, 0], [0, -82, 0, 0, 0, 0, 0, -7, -14, 2, -4, -7], [0, -109, 0, 0, 0, 0, 0, -20, -2, 31, 19, -35], [-660, 0, -48, 53, -1, -34, 0, 0, 0, 0, 0, 0], [0, -99, 0, 0, 0, 0, 0, -4, -11, -53, -18, 4], [-540, 0, 0, -30, 15, -7, -22, 0, 0, 0, 0, 0], [-149, 0, -15, -3, 3, -14, 3, 0, 0, 0, 0, 0], [-690, 0, -26, 96, -8, 2, -20, 0, 0, 0, 0, 0], [0, -130, 0, 0, 0, 0, 0, -19, 25, -12, -37, 21], [0, 171, 0, 0, 0, 0, 0, 110, 31, -33, -2, -25], [0, 93, 0, 0, 0, 0, 0, -21, 41, 0, -48, 25], [0, -166, 0, 0, 0, 0, 0, 16, -7, 55, -43, 81], [-795, 0, -99, 11, 38, -37, -11, 0, 0, 0, 0, 0], [-574, 0, 44, -48, -14, -12, 42, 0, 0, 0, 0, 0], [-1274, 0, 16, 43, 28, 16, 23, 0, 0, 0, 0, 0], [0, -44, 0, 0, 0, 0, 0, 50, 23, -11, 53, 27], [0, -8, 0, 0, 0, 0, 0, 98, -14, 25, 52, 99], [-474, 0, -72, -91, 10, -100, -1, 0, 0, 0, 0, 0], [-1320, 0, 98, 34, -27, 14, 25, 0, 0, 0, 0, 0], [0, 393, 0, 0, 0, 0, 0, -2, -72, -11, 19, 37], [0, 299, 0, 0, 0, 0, 0, -93, -49, -18, -17, 35], [-159, 0, 145, -33, -49, 56, -31, 0, 0, 0, 0, 0], [0, 235, 0, 0, 0, 0, 0, -81, -68, -97, 66, -17], [-1062, 0, 84, -47, 20, 16, -19, 0, 0, 0, 0, 0], [-1955, 0, -29, -19, -12, 31, 19, 0, 0, 0, 0, 0], [-1115, 0, -14, -117, -10, -108, 12, 0, 0, 0, 0, 0], [0, 374, 0, 0, 0, 0, 0, 83, 60, 40, -78, 147], [0, 344, 0, 0, 0, 0, 0, 51, 37, -21, 96, 111], [-627, 0, 161, 185, 26, 97, 19, 0, 0, 0, 0, 0], [-1884, 0, -208, 267, 71, -51, 1, 0, 0, 0, 0, 0], [0, 289, 0, 0, 0, 0, 0, 9, 50, 125, -82, -41], [-2326, 0, 33, -223, 3, 19, 3, 0, 0, 0, 0, 0], [0, 418, 0, 0, 0, 0, 0, -54, -54, -105, 21, -98], [0, -266, 0, 0, 0, 0, 0, -103, 68, -17, -85, -6], [0, 653, 0, 0, 0, 0, 0, 29, -1, -93, -52, -8], [-3546, 0, -42, -36, -22, 117, -31, 0, 0, 0, 0, 0], [0, 268, 0, 0, 0, 0, 0, 184, 76, 174, -31, 96], [0, 710, 0, 0, 0, 0, 0, 295, -38, 5, 67, -164], [-1024, 0, 248, -54, -47, -148, 39, 0, 0, 0, 0, 0], [0, -180, 0, 0, 0, 0, 0, 37, 8, 60, -24, 93], [0, 198, 0, 0, 0, 0, 0, 9, 61, 143, -36, 59], [-1338, 0, -216, 323, 3, 56, -76, 0, 0, 0, 0, 0], [0, -68, 0, 0, 0, 0, 0, 66, -4, 54, 29, -110], [-1828, 0, 186, 160, -2, 17, -85, 0, 0, 0, 0, 0], [0, 349, 0, 0, 0, 0, 0, -143, -4, -116, -17, 3], [0, 22, 0, 0, 0, 0, 0, 87, -50, -63, -113, -190], [-1844, 0, -148, -26, 37, 106, 57, 0, 0, 0, 0, 0], [0, 311, 0, 0, 0, 0, 0, -360, 56, -166, 31, -18], [1165, 0, -204, 127, 28, -178, 54, 0, 0, 0, 0, 0], [0, -217, 0, 0, 0, 0, 0, -12, -81, -119, 1, -336], [-99, 0, 162, -14, 45, -176, -22, 0, 0, 0, 0, 0], [-28, 0, -161, -299, 89, 65, 29, 0, 0, 0, 0, 0], [-309, 0, 108, -56, -27, -140, 64, 0, 0, 0, 0, 0], [-44, 0, 259, -189, 8, -47, -10, 0, 0, 0, 0, 0], [-1693, 0, -352, -98, 99, -150, -46, 0, 0, 0, 0, 0], [0, -586, 0, 0, 0, 0, 0, 11, -40, 18, 74, -29], [-3278, 0, 351, -361, -71, 83, 9, 0, 0, 0, 0, 0], [0, 1400, 0, 0, 0, 0, 0, 108, 2, 94, 155, -181], [-3673, 0, 23, -155, -42, 289, -131, 0, 0, 0, 0, 0], [0, -783, 0, 0, 0, 0, 0, -239, 172, 223, -45, 197], [555, 0, -262, 177, -46, 52, 94, 0, 0, 0, 0, 0], [0, -44, 0, 0, 0, 0, 0, 65, 99, -295, -84, 279], [0, -616, 0, 0, 0, 0, 0, -214, -16, 22, 191, -369], [-2862, 0, -424, 495, 52, -121, -110, 0, 0, 0, 0, 0], [0, -743, 0, 0, 0, 0, 0, -353, 95, 209, -164, -122], [243, 0, 132, 60, 73, 8, 20, 0, 0, 0, 0, 0], [-129, 0, 34, -756, -40, -235, 12, 0, 0, 0, 0, 0], [0, 1001, 0, 0, 0, 0, 0, 229, -58, -117, -334, 388], [4074, 0, -222, 266, -126, -72, 180, 0, 0, 0, 0, 0], [1662, 0, 514, 228, -142, 186, 48, 0, 0, 0, 0, 0], [0, -103, 0, 0, 0, 0, 0, 692, 14, 140, 190, 244], [3516, 0, 119, 673, -12, 14, -107, 0, 0, 0, 0, 0], [0, -805, 0, 0, 0, 0, 0, 366, -192, 103, 179, 77], [0, 425, 0, 0, 0, 0, 0, 9, -66, 353, -30, 595], [0, -245, 0, 0, 0, 0, 0, 270, -6, 167, -107, 19], [247, 0, -151, 193, 122, 41, -327, 0, 0, 0, 0, 0], [3357, 0, 30, -147, -161, -7, 140, 0, 0, 0, 0, 0], [0, -807, 0, 0, 0, 0, 0, -630, 21, -231, 16, 124], [0, -628, 0, 0, 0, 0, 0, 525, 43, 286, 54, 126], [0, -8, 0, 0, 0, 0, 0, -388, -84, -528, 94, -516], [-6018, 0, -39, -169, 49, 189, 195, 0, 0, 0, 0, 0], [8494, 0, 215, 805, 93, -129, 101, 0, 0, 0, 0, 0], [0, 523, 0, 0, 0, 0, 0, -141, -127, 51, -66, 470], [0, 1813, 0, 0, 0, 0, 0, 1047, 256, -216, 38, 114], [0, 1017, 0, 0, 0, 0, 0, -720, -51, -361, 85, 114], [-5157, 0, -186, -179, -9, 61, -114, 0, 0, 0, 0, 0], [2432, 0, 106, -770, -155, -101, 34, 0, 0, 0, 0, 0], [3306, 0, 7, -253, -155, 171, 239, 0, 0, 0, 0, 0], [0, 734, 0, 0, 0, 0, 0, -163, -199, 113, 62, -341], [0, 906, 0, 0, 0, 0, 0, -398, -210, 46, -312, 53], [0, -437, 0, 0, 0, 0, 0, -534, 17, -199, 351, -306], [-165, 0, -665, 1039, -31, -387, -16, 0, 0, 0, 0, 0], [8748, 0, -33, -673, -72, -179, -10, 0, 0, 0, 0, 0], [0, 1836, 0, 0, 0, 0, 0, 882, 163, 155, 29, -109], [0, -1392, 0, 0, 0, 0, 0, -205, 56, -469, 317, -518], [0, -207, 0, 0, 0, 0, 0, -114, 144, -443, 23, -329], [726, 0, -678, -144, -7, 59, -58, 0, 0, 0, 0, 0], [4781, 0, 827, -1114, -110, -8, -55, 0, 0, 0, 0, 0], [4314, 0, -52, -214, -224, -146, -84, 0, 0, 0, 0, 0], [-1068, 0, 250, -334, -34, -93, -13, 0, 0, 0, 0, 0], [0, 1476, 0, 0, 0, 0, 0, -667, -7, -418, 229, -430], [0, 526, 0, 0, 0, 0, 0, 251, -415, -126, 429, -595], [-2233, 0, 346, 755, 170, 344, 30, 0, 0, 0, 0, 0], [0, -939, 0, 0, 0, 0, 0, -463, -83, 200, 554, -595], [5463, 0, -166, -308, -213, 380, 270, 0, 0, 0, 0, 0], [0, 629, 0, 0, 0, 0, 0, -97, 186, 347, -84, 1179], [1790, 0, -169, 437, 153, 354, -320, 0, 0, 0, 0, 0], [13342, 0, 367, -1167, -121, 142, 138, 0, 0, 0, 0, 0], [0, 3366, 0, 0, 0, 0, 0, -925, 26, 559, 33, 26], [0, -1475, 0, 0, 0, 0, 0, -40, 64, 292, -62, -842], [0, 1538, 0, 0, 0, 0, 0, 935, 402, -153, -285, 40], [889, 0, -298, -801, -73, -332, -29, 0, 0, 0, 0, 0], [1105, 0, -359, -1017, 169, 483, 40, 0, 0, 0, 0, 0], [5868, 0, -824, -653, 341, -6, -30, 0, 0, 0, 0, 0], [0, 4244, 0, 0, 0, 0, 0, 239, 315, -430, -323, 434], [0, -767, 0, 0, 0, 0, 0, -766, -184, -252, 90, -1008], [5276, 0, 145, -851, 116, 523, 58, 0, 0, 0, 0, 0], [-11603, 0, 219, 1349, -210, 33, -95, 0, 0, 0, 0, 0], [0, 2290, 0, 0, 0, 0, 0, -950, 84, 144, -428, -674], [0, 1682, 0, 0, 0, 0, 0, -642, -289, 693, 231, 225], [1333, 0, -620, 825, -264, 2, 250, 0, 0, 0, 0, 0], [0, 2215, 0, 0, 0, 0, 0, -181, -315, -150, 20, -577], [0, 2292, 0, 0, 0, 0, 0, 933, -14, -191, -395, 714], [1539, 0, -339, 285, 73, 24, -233, 0, 0, 0, 0, 0], [0, 2148, 0, 0, 0, 0, 0, -282, -254, 411, 613, 184], [0, 2247, 0, 0, 0, 0, 0, 1146, 114, 794, 83, -922], [0, 1659, 0, 0, 0, 0, 0, 673, -27, -452, 275, 411], [-684, 0, -44, 176, -206, 310, -364, 0, 0, 0, 0, 0], [2340, 0, 478, 50, 193, -412, -45, 0, 0, 0, 0, 0], [-5807, 0, -204, 350, -230, 772, 275, 0, 0, 0, 0, 0], [0, 1229, 0, 0, 0, 0, 0, -817, -228, -463, -227, 541], [8785, 0, 214, -194, -42, -200, 223, 0, 0, 0, 0, 0], [0, -2257, 0, 0, 0, 0, 0, -541, 146, 696, -305, -7], [0, 4258, 0, 0, 0, 0, 0, -315, 42, 960, 78, 225], [0, 5197, 0, 0, 0, 0, 0, 10, -197, -948, 160, -992], [2747, 0, -1091, 1233, 52, -169, 327, 0, 0, 0, 0, 0], [0, 1468, 0, 0, 0, 0, 0, 149, 630, 432, -634, 49], [0, -3151, 0, 0, 0, 0, 0, 661, 43, -423, 454, -226], [-3645, 0, -284, -1298, 31, -376, -228, 0, 0, 0, 0, 0], [0, 1561, 0, 0, 0, 0, 0, -480, 389, -112, 243, 931], [2990, 0, -6, -192, 202, 6, -220, 0, 0, 0, 0, 0], [0, 5563, 0, 0, 0, 0, 0, -275, 52, -210, -44, 984], [-14964, 0, 164, -1292, -121, -192, -61, 0, 0, 0, 0, 0], [14887, 0, 892, -394, 154, 450, -305, 0, 0, 0, 0, 0], [10884, 0, 718, 1475, -2, -129, 262, 0, 0, 0, 0, 0], [0, 92, 0, 0, 0, 0, 0, -851, -391, 540, -314, -349], [759, 0, 1006, 598, -153, 550, -278, 0, 0, 0, 0, 0], [-3283, 0, 1465, -29, -21, 514, -13, 0, 0, 0, 0, 0], [3087, 0, 858, 382, 327, 254, 14, 0, 0, 0, 0, 0], [8723, 0, -1705, 1453, 97, -280, -169, 0, 0, 0, 0, 0], [0, 1257, 0, 0, 0, 0, 0, -117, 43, 482, 140, 1561], [9664, 0, 143, -1201, 212, 394, -29, 0, 0, 0, 0, 0], [0, -1210, 0, 0, 0, 0, 0, 1784, 648, 312, -196, 353], [-257, 0, -857, 1639, 569, -573, 154, 0, 0, 0, 0, 0], [0, 2394, 0, 0, 0, 0, 0, -570, -267, -63, 21, -79], [0, -1620, 0, 0, 0, 0, 0, -1574, -305, -461, -611, -249], [0, -3098, 0, 0, 0, 0, 0, 815, 95, 756, -581, -542], [-5242, 0, 12, 765, -133, 154, -292, 0, 0, 0, 0, 0], [0, 2470, 0, 0, 0, 0, 0, 1757, -194, -563, 21, -210], [0, -4505, 0, 0, 0, 0, 0, 80, -476, -1247, 35, -265], [26614, 0, 659, -1051, 375, -545, -213, 0, 0, 0, 0, 0], [974, 0, 410, 1670, -177, 88, 355, 0, 0, 0, 0, 0], [0, 2372, 0, 0, 0, 0, 0, -937, 254, -614, 674, 159], [0, -6430, 0, 0, 0, 0, 0, -520, -140, -135, -435, -194], [-3789, 0, -985, 1547, -50, -179, -687, 0, 0, 0, 0, 0], [-6706, 0, -559, -1091, -190, 1069, -96, 0, 0, 0, 0, 0], [0, -5865, 0, 0, 0, 0, 0, -982, -433, -144, -270, -280], [0, -1496, 0, 0, 0, 0, 0, -317, 882, 466, -630, -363], [-12383, 0, -1242, -405, 39, 96, -225, 0, 0, 0, 0, 0], [0, 406, 0, 0, 0, 0, 0, 869, -77, 346, 271, -785], [0, -1488, 0, 0, 0, 0, 0, 1629, 301, 404, 248, 99], [0, 48, 0, 0, 0, 0, 0, 1474, -474, 756, 108, 175], [4316, 0, -1244, 2528, 50, 470, 218, 0, 0, 0, 0, 0], [-8126, 0, 1270, -3280, -122, 86, 354, 0, 0, 0, 0, 0], [0, -4680, 0, 0, 0, 0, 0, -177, 108, -885, -153, -440], [-645, 0, -138, -828, -259, 732, 638, 0, 0, 0, 0, 0], [-33926, 0, -570, -1177, 228, 1003, -458, 0, 0, 0, 0, 0], [0, -2377, 0, 0, 0, 0, 0, 1180, -305, 925, 569, -672], [-18796, 0, -332, 952, 172, 59, -521, 0, 0, 0, 0, 0], [0, -4231, 0, 0, 0, 0, 0, 1409, 307, 346, -689, -607], [-3125, 0, -1686, -671, 501, -1096, 493, 0, 0, 0, 0, 0], [7127, 0, 27, -491, -131, -645, 328, 0, 0, 0, 0, 0], [0, 952, 0, 0, 0, 0, 0, -1427, -207, -262, 228, -995], [1137, 0, -1425, 184, -290, -218, -179, 0, 0, 0, 0, 0], [0, -4140, 0, 0, 0, 0, 0, 31, 701, -771, 162, -1375], [0, -3615, 0, 0, 0, 0, 0, -318, -84, -1746, 22, -1058], [-17231, 0, -1192, -688, -411, 434, -432, 0, 0, 0, 0, 0], [-3828, 0, -693, -517, 245, -1275, 131, 0, 0, 0, 0, 0], [0, -29, 0, 0, 0, 0, 0, 1372, 29, -451, -1045, 389], [-17142, 0, 1296, 188, -214, 418, 656, 0, 0, 0, 0, 0], [0, 2677, 0, 0, 0, 0, 0, -2078, -250, 1017, 19, -809], [0, 6743, 0, 0, 0, 0, 0, -293, 152, 661, -834, 199], [-7029, 0, 1861, -1291, 436, 1091, -859, 0, 0, 0, 0, 0], [334, 0, 614, -192, 288, -878, 292, 0, 0, 0, 0, 0], [-19875, 0, 1082, 970, 169, 562, -28, 0, 0, 0, 0, 0], [0, 3930, 0, 0, 0, 0, 0, -944, 236, -528, 926, 1615], [0, -6323, 0, 0, 0, 0, 0, 431, -463, 604, 355, -49], [6692, 0, 966, 1759, -807, -158, 98, 0, 0, 0, 0, 0], [0, -1276, 0, 0, 0, 0, 0, -600, -483, -617, -499, -355], [-35135, 0, 478, 870, 255, -362, 188, 0, 0, 0, 0, 0], [0, 79, 0, 0, 0, 0, 0, 2760, 240, -866, 29, 288], [-31214, 0, 662, 1369, -261, 980, 56, 0, 0, 0, 0, 0], [0, 7769, 0, 0, 0, 0, 0, -1275, 364, -642, 282, -1306], [19946, 0, 472, 1366, -205, -272, 587, 0, 0, 0, 0, 0], [656, 0, -2040, -986, -520, -762, -176, 0, 0, 0, 0, 0], [-15041, 0, -1129, -1117, 192, -811, 625, 0, 0, 0, 0, 0], [0, -10362, 0, 0, 0, 0, 0, 108, 356, 1015, 319, 284], [0, -1364, 0, 0, 0, 0, 0, -1961, 139, -50, -489, 1348], [11033, 0, 975, 319, -393, -760, 49, 0, 0, 0, 0, 0], [0, 5597, 0, 0, 0, 0, 0, 1696, 567, 926, -379, 1685], [-60888, 0, -600, 1293, -200, 17, -428, 0, 0, 0, 0, 0], [-16911, 0, -60, 2018, 571, -1278, -92, 0, 0, 0, 0, 0], [0, 4965, 0, 0, 0, 0, 0, -1813, 603, 2215, 382, 880], [-20765, 0, 1984, -443, -246, 1030, -878, 0, 0, 0, 0, 0], [0, -423, 0, 0, 0, 0, 0, -446, 617, -179, -285, 707], [7807, 0, -880, 596, -202, -1982, 201, 0, 0, 0, 0, 0], [0, -8456, 0, 0, 0, 0, 0, -571, -650, -1780, 548, 1095], [-11576, 0, 2118, -2783, 459, 1212, 346, 0, 0, 0, 0, 0], [0, 9370, 0, 0, 0, 0, 0, 2563, 1069, 813, -434, -283], [-28746, 0, 1360, 254, 454, -126, -68, 0, 0, 0, 0, 0], [-8052, 0, 1670, 26, 598, 1218, -238, 0, 0, 0, 0, 0], [0, 6248, 0, 0, 0, 0, 0, 817, 253, -862, 635, -1576], [0, -4912, 0, 0, 0, 0, 0, 1426, 398, 37, 734, -1157], [-19671, 0, 626, 1999, 333, 386, 75, 0, 0, 0, 0, 0], [0, 6083, 0, 0, 0, 0, 0, 245, 128, -883, 1450, 1202], [13088, 0, 1293, 2803, 937, 227, -391, 0, 0, 0, 0, 0], [0, -1129, 0, 0, 0, 0, 0, 43, 220, -672, -1232, -200], [-57499, 0, -2230, 1066, 571, -410, -468, 0, 0, 0, 0, 0], [-10754, 0, -772, -1543, -491, 216, 1052, 0, 0, 0, 0, 0], [0, -5793, 0, 0, 0, 0, 0, -103, -141, -1557, -1820, 1222], [0, 2872, 0, 0, 0, 0, 0, -698, -133, 2557, -283, 61], [0, 5624, 0, 0, 0, 0, 0, 624, 722, -1400, 70, -836], [0, -4735, 0, 0, 0, 0, 0, -740, 64, -574, -355, -1668], [0, 4717, 0, 0, 0, 0, 0, -290, -532, 459, 1381, -2337], [17274, 0, -147, -2713, 28, 1114, 1, 0, 0, 0, 0, 0], [0, -8240, 0, 0, 0, 0, 0, 1757, 1066, 905, -1295, 730], [0, 7367, 0, 0, 0, 0, 0, -1949, 296, -1755, -274, -1411], [14891, 0, -3193, 1507, 111, 147, -8, 0, 0, 0, 0, 0], [18797, 0, 344, -2438, 36, 526, 779, 0, 0, 0, 0, 0], [-32656, 0, 3881, -249, 415, 1122, 394, 0, 0, 0, 0, 0], [0, 11103, 0, 0, 0, 0, 0, -1433, 167, -504, 71, 715], [0, -449, 0, 0, 0, 0, 0, -1065, -1068, 1910, 302, 42], [-4385, 0, -2466, 5560, 394, 20, -693, 0, 0, 0, 0, 0], [0, 2131, 0, 0, 0, 0, 0, 1452, -1064, -786, 1482, 14], [0, 12167, 0, 0, 0, 0, 0, 201, -212, -909, -59, -2877], [5006, 0, -2474, 1034, -558, 287, 893, 0, 0, 0, 0, 0], [0, 2962, 0, 0, 0, 0, 0, -79, -912, 170, -1196, 803], [0, -10447, 0, 0, 0, 0, 0, -1583, -52, -865, 680, -1652], [0, -6438, 0, 0, 0, 0, 0, -2343, -169, -1106, -560, -1176], [-31379, 0, -1446, -1472, -604, 900, -313, 0, 0, 0, 0, 0], [6969, 0, 2339, -663, -896, 867, 163, 0, 0, 0, 0, 0], [-50468, 0, 1052, 666, -575, -33, 50, 0, 0, 0, 0, 0], [0, 9479, 0, 0, 0, 0, 0, 1955, 835, 3407, -442, 1416], [-15818, 0, 1628, -1575, 699, -765, 247, 0, 0, 0, 0, 0], [0, -734, 0, 0, 0, 0, 0, -2783, -947, -1974, -495, 516], [8171, 0, 400, 3854, 150, 40, -1763, 0, 0, 0, 0, 0], [0, 8662, 0, 0, 0, 0, 0, 1013, 1049, 1593, -958, 1073], [-13370, 0, -2092, 1102, -1333, 307, 450, 0, 0, 0, 0, 0], [0, 2831, 0, 0, 0, 0, 0, -1132, -1911, -1557, 1034, -1782], [0, -2024, 0, 0, 0, 0, 0, 1904, -103, 3201, -681, 2561], [-16712, 0, 3676, -4731, -578, 36, 685, 0, 0, 0, 0, 0], [-20167, 0, -1851, 1837, 554, 693, -813, 0, 0, 0, 0, 0], [-7003, 0, -4018, 1870, 529, 1410, -1032, 0, 0, 0, 0, 0], [0, -2928, 0, 0, 0, 0, 0, -3375, 1069, 1329, -90, -93], [0, 6930, 0, 0, 0, 0, 0, -887, 392, -2056, -420, -511], [-17912, 0, -1388, -1560, 546, -1342, 1080, 0, 0, 0, 0, 0], [0, -13502, 0, 0, 0, 0, 0, 3186, 411, 425, -609, -159], [-56912, 0, 1855, -2182, 385, 64, -1053, 0, 0, 0, 0, 0], [3123, 0, -2419, -421, -390, 985, 457, 0, 0, 0, 0, 0], [0, 77, 0, 0, 0, 0, 0, 992, 547, -171, 495, 1118], [-42220, 0, -2166, 224, 405, -2726, 99, 0, 0, 0, 0, 0], [0, -1986, 0, 0, 0, 0, 0, 4544, -338, -2570, -519, 255], [0, 839, 0, 0, 0, 0, 0, 2671, -698, 1895, 1708, -1557], [0, -1918, 0, 0, 0, 0, 0, -1363, 986, -1441, -827, -952], [-25322, 0, -1128, 3276, 398, 585, 597, 0, 0, 0, 0, 0], [0, -5537, 0, 0, 0, 0, 0, -3664, 1422, 600, -1186, 1046], [16743, 0, 1836, -2463, 696, -1566, 108, 0, 0, 0, 0, 0], [0, 2722, 0, 0, 0, 0, 0, -2168, 1337, -3073, 33, -273], [0, -823, 0, 0, 0, 0, 0, -68, 690, 2152, 82, 722], [79797, 0, -378, -1416, 868, -346, -7, 0, 0, 0, 0, 0], [-80916, 0, -1006, -709, -153, -2612, 552, 0, 0, 0, 0, 0], [0, -3062, 0, 0, 0, 0, 0, -539, -1997, -2558, 1827, 535], [0, -10534, 0, 0, 0, 0, 0, -1758, -488, 1901, -609, 1270], [-30110, 0, 613, -1190, -217, -400, 847, 0, 0, 0, 0, 0], [-28629, 0, 386, -1958, -1495, 264, 204, 0, 0, 0, 0, 0], [0, -12333, 0, 0, 0, 0, 0, -5230, -1690, 435, -231, 1123], [0, -7767, 0, 0, 0, 0, 0, -258, -73, -768, -936, -1768], [41157, 0, -3628, -100, -465, -490, 224, 0, 0, 0, 0, 0], [-302, 0, 3910, 1880, -305, 1537, 1278, 0, 0, 0, 0, 0], [-35199, 0, 242, -4012, 443, 1048, -28, 0, 0, 0, 0, 0], [0, -17293, 0, 0, 0, 0, 0, 1402, 1719, 835, 282, -700], [13278, 0, 2680, 3995, -829, 888, -560, 0, 0, 0, 0, 0], [-4853, 0, 865, -1409, 401, -1275, -332, 0, 0, 0, 0, 0], [0, -2929, 0, 0, 0, 0, 0, -595, 100, -39, 1137, -4715], [34854, 0, -31, 597, -134, -1983, 956, 0, 0, 0, 0, 0], [0, -10141, 0, 0, 0, 0, 0, -3418, -234, 1116, -503, 2894], [0, 5536, 0, 0, 0, 0, 0, -1632, 1402, 2212, 435, 4842], [-33510, 0, 439, 1933, 673, 1817, 647, 0, 0, 0, 0, 0], [60423, 0, 831, 1323, 1071, -193, -792, 0, 0, 0, 0, 0], [0, -7263, 0, 0, 0, 0, 0, 364, 810, -34, -2534, -86], [0, 5341, 0, 0, 0, 0, 0, 2423, -805, -1004, -1614, -499], [0, 823, 0, 0, 0, 0, 0, 1123, -982, 1282, 554, 658], [41001, 0, 635, 3243, 263, 1508, -655, 0, 0, 0, 0, 0], [0, -14470, 0, 0, 0, 0, 0, -847, -1146, 1377, 945, -2142], [58744, 0, 1598, 4657, 561, 471, 423, 0, 0, 0, 0, 0], [0, 568, 0, 0, 0, 0, 0, 618, -1146, -781, 2401, -1760], [59640, 0, -412, -2892, 947, -383, -712, 0, 0, 0, 0, 0], [0, -2802, 0, 0, 0, 0, 0, -1426, -1016, -4106, 1297, -3215], [66582, 0, 1335, -1993, 640, 1145, 8, 0, 0, 0, 0, 0], [0, -17718, 0, 0, 0, 0, 0, -1988, -531, 21, -1275, 2731], [-1323, 0, -2184, -1777, 55, 2742, 857, 0, 0, 0, 0, 0], [0, -15841, 0, 0, 0, 0, 0, 4170, -1027, 427, 373, 865], [0, 10188, 0, 0, 0, 0, 0, -73, -962, -1065, -1835, 3444], [0, -9597, 0, 0, 0, 0, 0, 5964, 375, -585, -196, -534], [-40878, 0, 575, 4603, -952, 1207, 470, 0, 0, 0, 0, 0], [-31771, 0, 3715, -5505, 1142, 3077, -389, 0, 0, 0, 0, 0], [0, -14796, 0, 0, 0, 0, 0, 987, -1144, 127, 823, -2666], [-45925, 0, 239, 971, -390, -2139, 605, 0, 0, 0, 0, 0], [36352, 0, -1862, 4792, 176, -872, 982, 0, 0, 0, 0, 0], [0, -12681, 0, 0, 0, 0, 0, -5182, -2206, 450, 1664, -1624], [17972, 0, 3083, -4469, 515, 1545, -1001, 0, 0, 0, 0, 0], [0, -14317, 0, 0, 0, 0, 0, 1620, 947, 3881, -2347, 4248], [59211, 0, 2555, 1279, 532, 1931, -1007, 0, 0, 0, 0, 0], [0, -7348, 0, 0, 0, 0, 0, 3203, -256, -2108, 1462, -1397], [-1591, 0, -2825, -4797, 1243, -456, -7, 0, 0, 0, 0, 0], [0, -6353, 0, 0, 0, 0, 0, 2179, 111, -3319, 1200, -542], [-11570, 0, -2018, -2874, 1386, 472, -1276, 0, 0, 0, 0, 0], [0, 3878, 0, 0, 0, 0, 0, 3078, 921, 841, -321, 231], [684, 0, -252, 4628, 9, 722, -85, 0, 0, 0, 0, 0], [-15566, 0, -3062, 7006, 416, -454, 406, 0, 0, 0, 0, 0], [0, 1007, 0, 0, 0, 0, 0, 1460, 379, 3441, 770, 6452], [0, -27, 0, 0, 0, 0, 0, -1848, -268, 1343, -1733, -1899], [0, -9535, 0, 0, 0, 0, 0, 2958, 2123, 752, -79, -207], [-14869, 0, -1719, -154, 946, 492, -973, 0, 0, 0, 0, 0], [0, -5349, 0, 0, 0, 0, 0, -4348, 1098, -3624, 1739, -1626], [59126, 0, -4840, 1158, 219, -48, -385, 0, 0, 0, 0, 0], [0, -8052, 0, 0, 0, 0, 0, 4441, 919, 232, 1024, -300], [0, 19630, 0, 0, 0, 0, 0, 6111, 476, -2450, 1592, -3709], [0, -9190, 0, 0, 0, 0, 0, -3000, 440, -4566, 1200, -5526], [0, 313, 0, 0, 0, 0, 0, -14, 1584, -232, -1521, -1160], [5253, 0, -1055, 2299, 6, 1699, 743, 0, 0, 0, 0, 0], [-39826, 0, 1566, -503, -828, 2125, -446, 0, 0, 0, 0, 0], [37063, 0, -711, 1141, -229, -841, -154, 0, 0, 0, 0, 0], [0, -4304, 0, 0, 0, 0, 0, -4282, 1178, -1558, -210, 1145], [-113158, 0, -2695, -3739, 110, -2087, -160, 0, 0, 0, 0, 0], [94220, 0, 1486, -175, -17, 1004, 658, 0, 0, 0, 0, 0], [0, 2891, 0, 0, 0, 0, 0, -2997, 838, -281, 894, -611], [-31578, 0, 3962, 2072, -91, 2702, -487, 0, 0, 0, 0, 0], [-2498, 0, -3542, 3606, 624, -286, 894, 0, 0, 0, 0, 0], [0, 7669, 0, 0, 0, 0, 0, 2325, -1184, 59, 346, -537], [-134339, 0, 976, -169, 961, 1484, 717, 0, 0, 0, 0, 0], [0, 3422, 0, 0, 0, 0, 0, -1979, 2496, 680, -2990, 941], [105966, 0, -2408, 4360, 826, 660, -74, 0, 0, 0, 0, 0], [0, -7517, 0, 0, 0, 0, 0, -4342, -1289, 839, -117, 1498], [0, -6888, 0, 0, 0, 0, 0, -1363, 1641, 3002, -3355, 2318], [0, 6211, 0, 0, 0, 0, 0, 2081, -1262, 2231, -1962, 829], [143159, 0, -3197, 2081, -155, -777, -1072, 0, 0, 0, 0, 0], [0, -3627, 0, 0, 0, 0, 0, 4729, -839, -1285, -308, -2320], [44534, 0, -4332, -3495, -73, -1273, -753, 0, 0, 0, 0, 0], [0, -8436, 0, 0, 0, 0, 0, -9584, 690, 309, -1200, 393], [0, -6480, 0, 0, 0, 0, 0, 5160, -888, 2667, 1511, 3162], [0, -15619, 0, 0, 0, 0, 0, 3447, -69, -3201, 1806, 2836], [-39991, 0, -1251, 3265, 1008, 1627, 1155, 0, 0, 0, 0, 0], [0, -10378, 0, 0, 0, 0, 0, 6767, 336, 4379, 775, 2354], [20890, 0, -562, -5850, -486, -1614, -696, 0, 0, 0, 0, 0], [0, 616, 0, 0, 0, 0, 0, 915, -2507, -3136, 922, -6794], [7625, 0, -1330, 7860, -113, 378, -222, 0, 0, 0, 0, 0], [0, -892, 0, 0, 0, 0, 0, 2624, 1821, -4233, -2083, -2931], [20379, 0, 789, -7367, -1042, -581, -127, 0, 0, 0, 0, 0], [-84100, 0, 1408, 3700, -36, 1676, 682, 0, 0, 0, 0, 0], [27875, 0, -593, 2863, -914, 2011, -941, 0, 0, 0, 0, 0], [22865, 0, 6032, -9170, -199, 2390, 284, 0, 0, 0, 0, 0], [18819, 0, 3819, -8519, -938, 1717, 1445, 0, 0, 0, 0, 0], [0, 5931, 0, 0, 0, 0, 0, 1894, 807, 2041, -743, -6472], [0, 13517, 0, 0, 0, 0, 0, 820, -3210, 635, 1819, -2069], [0, 4518, 0, 0, 0, 0, 0, -3753, 1074, 1874, -1910, -5233], [3821, 0, 1991, -9815, 310, 2029, 111, 0, 0, 0, 0, 0], [64582, 0, 1668, 5689, -1436, -4100, 901, 0, 0, 0, 0, 0], [0, 635, 0, 0, 0, 0, 0, -4579, 1344, -892, -2346, -2842], [18946, 0, -646, -7242, 1408, -1046, 1520, 0, 0, 0, 0, 0], [0, 3289, 0, 0, 0, 0, 0, 1892, -1098, -1720, 1894, -4996], [-28579, 0, 2644, -8177, -688, 2502, 632, 0, 0, 0, 0, 0], [0, -18762, 0, 0, 0, 0, 0, 3913, -426, 1660, 764, -579], [0, 17331, 0, 0, 0, 0, 0, 2795, 320, 1253, 3258, -3365], [0, -3141, 0, 0, 0, 0, 0, -2285, 635, 3560, -1339, 5271], [114659, 0, 1408, 10174, 422, -155, 0, 0, 0, 0, 0, 0], [0, 4788, 0, 0, 0, 0, 0, 7250, -416, 2980, -1705, 1060], [1577, 0, 5598, 664, -1529, 3488, -1330, 0, 0, 0, 0, 0], [-36904, 0, -1477, -313, -693, 5161, -1721, 0, 0, 0, 0, 0], [0, -8405, 0, 0, 0, 0, 0, 265, 2095, 4390, -470, 4559], [0, 5052, 0, 0, 0, 0, 0, 2777, 99, 1752, 1422, -2533], [0, -13625, 0, 0, 0, 0, 0, 6497, 1413, -507, -1946, -2128], [-42199, 0, 2811, 7915, -58, -993, -1485, 0, 0, 0, 0, 0], [0, 14636, 0, 0, 0, 0, 0, -2231, 554, -1747, 1501, 5894], [0, -152, 0, 0, 0, 0, 0, 3100, 1320, 2352, -304, 403], [-43840, 0, -2652, -3486, -1159, 2218, -1427, 0, 0, 0, 0, 0], [0, 11024, 0, 0, 0, 0, 0, -3849, 750, -1668, -2520, 3715], [0, -4600, 0, 0, 0, 0, 0, 4627, 630, 3917, 619, -886], [0, 1489, 0, 0, 0, 0, 0, -1310, -1940, -476, -956, 3014], [-41614, 0, -3171, 9717, -1121, -306, -586, 0, 0, 0, 0, 0], [71624, 0, -8796, 7235, -207, -2666, 1944, 0, 0, 0, 0, 0], [-4146, 0, 4094, -3978, -424, -2215, 2977, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1900_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1900_4_a_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_1900_4_a_l(:prec:=12)
    chi := MakeCharacter_1900_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_1900_4_a_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_1900_4_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1900_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![73273600, 0, -26837136, 0, 3734252, 0, -255920, 0, 9117, 0, -158, 0, 1]>],Snew);
    return Vf;
end function;