
// Make newform 2205.4.a.by in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2205_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_2205_4_a_by();".
// 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_2205_4_a_by();".
// 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 := [-64, 0, 241, 0, -34, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-11, 0, 1, 0, 0, 0], [0, -20, 0, 1, 0, 0], [59, 0, -25, 0, 1, 0], [0, 195, 0, -32, 0, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 2];
    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_2205_a();"
function MakeCharacter_2205_a()
    N := 2205;
    order := 1;
    char_gens := [1226, 442, 1081];
    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_2205_a_Hecke(Kf)
    return MakeCharacter_2205_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], [5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1], [0, 3, 0, 1, 0, 1], [7, 0, 0, 0, 1, 0], [0, 1, 0, -3, 0, 2], [0, 6, 0, -4, 0, 1], [0, 17, 0, -1, 0, 2], [0, 34, 0, 8, 0, -2], [67, 0, -8, 0, -3, 0], [96, 0, 20, 0, -2, 0], [-53, 0, -4, 0, -1, 0], [15, 0, 20, 0, -5, 0], [0, -10, 0, 14, 0, -7], [38, 0, -40, 0, -6, 0], [0, 75, 0, -15, 0, 0], [179, 0, 8, 0, 11, 0], [0, -43, 0, -1, 0, -5], [0, 137, 0, -21, 0, 19], [234, 0, -4, 0, 2, 0], [466, 0, 60, 0, -2, 0], [36, 0, 80, 0, -2, 0], [0, -203, 0, -1, 0, -11], [548, 0, -60, 0, 14, 0], [0, 300, 0, 0, 0, -30], [0, -12, 0, -6, 0, 37], [-96, 0, -36, 0, -18, 0], [0, -260, 0, -44, 0, 11], [684, 0, 12, 0, 8, 0], [-108, 0, 80, 0, 16, 0], [0, -136, 0, 16, 0, 49], [0, -313, 0, 19, 0, 44], [0, 395, 0, 85, 0, -48], [-876, 0, -108, 0, 12, 0], [0, 351, 0, -83, 0, 7], [203, 0, 260, 0, -1, 0], [1059, 0, -200, 0, -13, 0], [-1102, 0, -120, 0, -16, 0], [0, -578, 0, 14, 0, -61], [0, -537, 0, -19, 0, 26], [0, -179, 0, -53, 0, -13], [74, 0, 296, 0, 0, 0], [0, -212, 0, -32, 0, -17], [0, 796, 0, 42, 0, -28], [1222, 0, 100, 0, -30, 0], [0, -256, 0, 148, 0, -52], [3624, 0, -180, 0, 12, 0], [0, 271, 0, -3, 0, -28], [0, -426, 0, -78, 0, 7], [0, 291, 0, 237, 0, -23], [0, 948, 0, -4, 0, 86], [1488, 0, -520, 0, 4, 0], [-1059, 0, -40, 0, 43, 0], [0, 192, 0, 38, 0, -47], [5604, 0, -20, 0, 2, 0], [0, -710, 0, -100, 0, -50], [-3153, 0, -200, 0, -39, 0], [0, 432, 0, -96, 0, -62], [0, -48, 0, 84, 0, -126], [2346, 0, -520, 0, -32, 0], [0, -1086, 0, -162, 0, -42], [5596, 0, -200, 0, -12, 0], [0, -867, 0, 111, 0, 81], [0, 52, 0, 32, 0, -183], [46, 0, -708, 0, 58, 0], [2450, 0, -80, 0, -80, 0], [0, -1028, 0, 202, 0, -113], [0, -791, 0, -157, 0, -152], [5203, 0, -800, 0, -51, 0], [0, -229, 0, 37, 0, 79], [0, -524, 0, -208, 0, -68], [3673, 0, 1016, 0, -113, 0], [-848, 0, 284, 0, -4, 0], [993, 0, 240, 0, -111, 0], [0, -675, 0, -45, 0, 172], [0, -2681, 0, -27, 0, 233], [0, -826, 0, -142, 0, -16], [0, -2034, 0, 42, 0, -108], [5268, 0, 160, 0, -16, 0], [-1756, 0, 752, 0, 122, 0], [0, 2837, 0, -281, 0, 19], [0, -1365, 0, -455, 0, 205], [0, -3430, 0, 340, 0, -160], [0, 1436, 0, 26, 0, 295], [0, -930, 0, -430, 0, -68], [5468, 0, -888, 0, -10, 0], [3242, 0, 560, 0, -144, 0], [-1842, 0, -420, 0, 34, 0], [3814, 0, 1180, 0, -78, 0], [3844, 0, -1040, 0, 12, 0], [-740, 0, -292, 0, 104, 0], [0, 4238, 0, -194, 0, -65], [-6156, 0, 540, 0, 144, 0], [-233, 0, 280, 0, -9, 0], [6140, 0, 660, 0, 50, 0], [8298, 0, 100, 0, 44, 0], [0, -4554, 0, 182, 0, -208], [-5942, 0, -752, 0, 64, 0], [9515, 0, -152, 0, 179, 0], [0, -3828, 0, 336, 0, -59], [4418, 0, 360, 0, 194, 0], [0, -2006, 0, -762, 0, 216], [-972, 0, -828, 0, -16, 0], [0, 1201, 0, 467, 0, 37], [-1442, 0, 120, 0, -146, 0], [5901, 0, 1000, 0, 23, 0], [0, -1495, 0, 575, 0, 107], [0, 162, 0, 334, 0, -146], [0, -2264, 0, 212, 0, -248], [2533, 0, 456, 0, -153, 0], [0, -4544, 0, 400, 0, -125], [0, 2625, 0, 65, 0, -100], [1372, 0, -52, 0, -284, 0], [0, 6614, 0, -382, 0, 322], [0, -4458, 0, 114, 0, -126], [3829, 0, 640, 0, -243, 0], [0, 2068, 0, 640, 0, -1], [0, -4620, 0, 120, 0, 65], [0, -1221, 0, 73, 0, 568], [-13302, 0, 120, 0, -156, 0], [4064, 0, -2200, 0, -18, 0], [0, 16, 0, -514, 0, 291], [0, -2743, 0, -11, 0, -256], [0, 4117, 0, 59, 0, -606], [-9896, 0, 800, 0, 50, 0], [10730, 0, -40, 0, -30, 0], [0, -2972, 0, -924, 0, 446], [0, -71, 0, -757, 0, 373], [13482, 0, -4, 0, -34, 0], [0, 792, 0, -150, 0, 611], [-1834, 0, -60, 0, -226, 0], [12787, 0, -2232, 0, -95, 0], [14974, 0, -1200, 0, -128, 0], [0, -7492, 0, -284, 0, 626], [6400, 0, 2200, 0, 30, 0], [0, -1934, 0, -678, 0, -28], [-15426, 0, -400, 0, 292, 0], [0, -5162, 0, 426, 0, -606], [0, -3225, 0, -165, 0, 480], [0, -197, 0, 1181, 0, -454], [11850, 0, -1804, 0, -82, 0], [0, -5348, 0, 226, 0, -619], [0, 2833, 0, -1269, 0, 206], [-13938, 0, -80, 0, -154, 0], [0, -3877, 0, -359, 0, -439], [-3569, 0, 640, 0, 453, 0], [0, 4269, 0, -827, 0, 658], [0, 4238, 0, -628, 0, -43], [27271, 0, -1520, 0, -47, 0], [-35728, 0, 0, 0, -34, 0], [1931, 0, 2176, 0, 211, 0], [31581, 0, -900, 0, 33, 0], [-7061, 0, 960, 0, -533, 0], [0, 4385, 0, -1345, 0, 499], [-21022, 0, 756, 0, -406, 0], [15990, 0, -1920, 0, -240, 0], [0, -1483, 0, 439, 0, -121], [-10022, 0, 480, 0, -476, 0], [0, 128, 0, -1462, 0, 565], [0, -5798, 0, 438, 0, 303], [24864, 0, -2120, 0, -288, 0], [20434, 0, 960, 0, -218, 0], [0, -3626, 0, 466, 0, 871], [-11465, 0, -20, 0, -285, 0], [13782, 0, 2052, 0, 6, 0], [0, 11595, 0, 465, 0, -495], [-29134, 0, 3124, 0, 248, 0], [31596, 0, 120, 0, 138, 0], [0, -6164, 0, -1468, 0, 549], [0, -9447, 0, 811, 0, -674], [0, -7529, 0, -1283, 0, -197], [5172, 0, -2680, 0, 506, 0], [0, 1818, 0, 1776, 0, 156], [-14184, 0, 2220, 0, -342, 0], [-714, 0, -1148, 0, 378, 0], [0, 10717, 0, 19, 0, -776], [0, 3584, 0, 1728, 0, 118], [0, -1293, 0, -711, 0, -1026], [32410, 0, -872, 0, 74, 0], [-1860, 0, 3480, 0, 360, 0], [-8527, 0, 2144, 0, -49, 0], [17011, 0, 2920, 0, 93, 0], [0, 6692, 0, 1010, 0, 485], [4888, 0, 540, 0, 634, 0], [-39645, 0, 1272, 0, 261, 0], [0, 9162, 0, -1446, 0, 444], [-18962, 0, 1764, 0, -456, 0], [8390, 0, 1160, 0, 750, 0], [0, 4193, 0, 1131, 0, 1081], [0, -7048, 0, -1490, 0, -915], [-9088, 0, 3892, 0, -404, 0], [28602, 0, 1440, 0, 576, 0], [0, -2080, 0, 122, 0, -1343], [593, 0, 840, 0, 179, 0], [-6834, 0, 1908, 0, -492, 0], [-28105, 0, -3616, 0, -43, 0], [-16251, 0, 1680, 0, 27, 0], [23541, 0, 2200, 0, -307, 0], [0, -10423, 0, 1279, 0, -786], [0, 3672, 0, -1946, 0, 64], [0, -329, 0, -43, 0, -1073], [0, 694, 0, -2062, 0, 358], [-21042, 0, 2840, 0, -206, 0], [26640, 0, -560, 0, -370, 0], [0, -2318, 0, -376, 0, 1264], [0, 7524, 0, -1286, 0, 365], [0, 5584, 0, 1488, 0, -1134], [0, -14639, 0, 1697, 0, -1238], [38580, 0, -168, 0, -314, 0], [-8364, 0, -700, 0, -542, 0], [3116, 0, 1984, 0, -860, 0], [32598, 0, -420, 0, 114, 0], [-3378, 0, 400, 0, -34, 0], [0, -4168, 0, 784, 0, 494], [-3476, 0, 5348, 0, 56, 0], [30750, 0, 560, 0, -620, 0], [0, -2516, 0, 1226, 0, -1219], [0, -612, 0, -760, 0, -185], [17044, 0, 3408, 0, -338, 0], [0, -1444, 0, -498, 0, -398], [0, -1146, 0, -302, 0, 1826], [-21062, 0, -3096, 0, 226, 0], [27506, 0, -4440, 0, 218, 0], [-51172, 0, 1460, 0, -234, 0], [1757, 0, -3400, 0, -669, 0], [0, -11303, 0, -221, 0, 1877], [0, -3984, 0, 1572, 0, -78], [0, 10850, 0, 1130, 0, -421], [-635, 0, -4384, 0, -37, 0], [0, -3959, 0, 1677, 0, 1142], [-10358, 0, 1020, 0, 322, 0], [0, -636, 0, -3372, 0, 854], [0, 16690, 0, 1230, 0, -1930], [34401, 0, 3260, 0, 253, 0], [0, 234, 0, 918, 0, -1992], [0, 2636, 0, 1328, 0, -257], [0, 1718, 0, 566, 0, -461], [-13156, 0, -1440, 0, 212, 0], [0, 6100, 0, 2206, 0, -889], [0, -10679, 0, 57, 0, -568], [0, -23852, 0, 1316, 0, 136], [-20702, 0, 3444, 0, 1024, 0], [11385, 0, -3000, 0, 1095, 0], [31128, 0, 1040, 0, 904, 0], [0, 16056, 0, 152, 0, 902], [32514, 0, 2920, 0, 902, 0], [15924, 0, -4660, 0, -456, 0], [0, -8182, 0, 600, 0, -231], [57929, 0, -1360, 0, 267, 0], [-38246, 0, 400, 0, 912, 0], [0, 11706, 0, 3300, 0, -1877], [0, 17388, 0, -84, 0, 2256], [52696, 0, -1200, 0, -122, 0], [0, -17206, 0, 158, 0, 593], [-25384, 0, -388, 0, -482, 0], [0, 7692, 0, 864, 0, 2214], [-43744, 0, 320, 0, -282, 0], [0, -6305, 0, -835, 0, 625], [63898, 0, -2180, 0, -106, 0], [0, 4892, 0, -1438, 0, -253], [0, -17403, 0, 2199, 0, -2016], [0, 16045, 0, -985, 0, 55], [56379, 0, 1240, 0, 797, 0], [0, 6209, 0, 2173, 0, 1988], [0, 8409, 0, -3457, 0, 166], [-3430, 0, -6700, 0, 60, 0], [-10567, 0, 1136, 0, 737, 0], [0, 5982, 0, -2370, 0, 655], [0, 13205, 0, 4495, 0, -1390], [-75463, 0, -2300, 0, 781, 0], [0, 10827, 0, 3609, 0, -1071], [-28888, 0, 2916, 0, -544, 0], [0, 11199, 0, 2933, 0, 1373], [0, 132, 0, 644, 0, -1306], [0, 11612, 0, 386, 0, 2741], [38300, 0, -2720, 0, 718, 0], [-7530, 0, -4272, 0, -420, 0], [30574, 0, 5040, 0, -1358, 0], [12411, 0, -4980, 0, -57, 0], [5050, 0, -5060, 0, -206, 0], [61057, 0, -1260, 0, 61, 0], [0, 11967, 0, -411, 0, -1626], [0, 11898, 0, -4534, 0, 1316], [0, 20863, 0, -2199, 0, 1875], [49664, 0, 4816, 0, -230, 0], [0, 4982, 0, -3698, 0, 1237], [0, 4978, 0, -2344, 0, 1006], [-25490, 0, 9040, 0, 600, 0], [0, -12191, 0, 3063, 0, -66], [36948, 0, -12020, 0, -856, 0], [0, 29022, 0, 2250, 0, -1925], [106710, 0, -1160, 0, -550, 0], [7823, 0, 1504, 0, 1421, 0], [43886, 0, -880, 0, 1668, 0], [0, 20862, 0, -516, 0, -516], [-80, 0, -2120, 0, -1710, 0], [-17104, 0, 1280, 0, -972, 0], [0, 13290, 0, 1730, 0, -1000], [0, 25109, 0, 1903, 0, 383], [0, -11216, 0, -260, 0, -73], [16696, 0, 6076, 0, 1568, 0], [0, 5992, 0, 1618, 0, 433], [15668, 0, -2708, 0, 1904, 0], [25364, 0, -10360, 0, -78, 0], [0, -1800, 0, -5358, 0, -173], [0, -1083, 0, 3039, 0, -2136], [0, -20959, 0, -293, 0, -3545], [-60929, 0, 6640, 0, -987, 0], [67179, 0, 10020, 0, -633, 0], [0, -6195, 0, 315, 0, -2350], [0, 28482, 0, 2774, 0, -1652], [28171, 0, 5840, 0, 1163, 0], [0, 7630, 0, 2492, 0, -2323], [0, -7490, 0, 50, 0, -2690], [85014, 0, -3480, 0, 402, 0], [0, -22757, 0, -2559, 0, 1041], [0, 5981, 0, -4673, 0, 1397], [0, 7188, 0, -1404, 0, -1012], [0, 5273, 0, -1079, 0, -2074], [20758, 0, -10320, 0, 784, 0], [-67026, 0, 8640, 0, -108, 0], [-10796, 0, 1064, 0, -76, 0], [0, 20198, 0, 1562, 0, -2303], [0, 13976, 0, -4568, 0, 2462], [-106180, 0, -4276, 0, -80, 0], [0, 11836, 0, -1888, 0, -818], [0, -16134, 0, 84, 0, 2179], [0, -20404, 0, -1192, 0, -1127], [-35110, 0, -6260, 0, -1796, 0], [0, 34304, 0, -1764, 0, 2991], [0, 10080, 0, -1370, 0, 70], [39280, 0, 7960, 0, 720, 0], [-93930, 0, -4140, 0, -854, 0], [6408, 0, 6920, 0, -896, 0], [-29374, 0, 14864, 0, 328, 0], [6783, 0, -800, 0, -211, 0], [0, 9818, 0, 5446, 0, 626], [0, 11864, 0, -2312, 0, 338], [83269, 0, 3280, 0, -1373, 0], [0, 21436, 0, 3284, 0, -321], [9660, 0, -7060, 0, 850, 0], [-87066, 0, 1668, 0, -1818, 0], [0, 33478, 0, -2738, 0, 1747], [0, 32246, 0, 302, 0, -1091], [0, 15563, 0, 6121, 0, -1804], [5665, 0, -7792, 0, 89, 0], [37836, 0, -7200, 0, 708, 0], [-103566, 0, 360, 0, -1848, 0], [0, 27235, 0, -1125, 0, 1370], [-70878, 0, -11312, 0, 1020, 0], [0, -25961, 0, 1633, 0, -2362], [0, 9296, 0, -368, 0, -3928], [22034, 0, 5684, 0, 872, 0], [-114225, 0, 1360, 0, -415, 0], [96150, 0, -1840, 0, 430, 0], [-2606, 0, -12360, 0, 1102, 0], [0, 2508, 0, 3428, 0, -65], [0, 19052, 0, -642, 0, 1947], [22473, 0, 2048, 0, 663, 0], [-56250, 0, -5060, 0, -1180, 0], [0, 43272, 0, -1814, 0, -1297], [0, -41704, 0, 1612, 0, 1429], [0, 186, 0, 1362, 0, -2268], [6560, 0, 10960, 0, 1090, 0], [-26004, 0, 1360, 0, 2318, 0], [97866, 0, 10104, 0, -1390, 0], [22138, 0, 7472, 0, -736, 0], [0, -4328, 0, 5644, 0, 223], [0, 9021, 0, 2717, 0, -2938], [0, 3408, 0, 1602, 0, -313], [0, 19189, 0, 3043, 0, -3532], [0, 6822, 0, -1216, 0, -226], [-5013, 0, 1712, 0, 437, 0], [-108420, 0, -1680, 0, -600, 0], [0, -24001, 0, 173, 0, 3997], [0, 5547, 0, 849, 0, 2349], [-134694, 0, -10140, 0, 948, 0], [0, -21801, 0, -1267, 0, -937], [-3768, 0, 11380, 0, -214, 0], [0, -26272, 0, -4888, 0, -953], [-40184, 0, -9348, 0, 2524, 0], [0, 19520, 0, 2552, 0, -163], [0, -14289, 0, -1733, 0, -3368], [-49523, 0, 9120, 0, 901, 0], [-77908, 0, -8908, 0, -1164, 0], [0, -25987, 0, -5729, 0, 4121], [-51753, 0, 2016, 0, -1841, 0], [56741, 0, -10260, 0, -607, 0], [-53538, 0, 1796, 0, -544, 0], [-41724, 0, 6840, 0, 1158, 0], [0, -26944, 0, -3628, 0, 5045], [0, -4742, 0, 1086, 0, -2194], [0, 41005, 0, -65, 0, 2011], [107216, 0, 2584, 0, -170, 0], [0, -33284, 0, -658, 0, -188], [-92676, 0, -2320, 0, -1838, 0], [-91086, 0, 8260, 0, 646, 0], [0, -27457, 0, 7001, 0, -3250], [0, -5429, 0, -663, 0, -1138], [0, 6178, 0, -4378, 0, 3107], [82650, 0, -10688, 0, -1214, 0], [-38315, 0, -14240, 0, 1155, 0], [-44394, 0, -4560, 0, 888, 0], [0, -48872, 0, 266, 0, -3284], [419, 0, 992, 0, 653, 0], [0, -14934, 0, -3018, 0, -3388], [0, -9932, 0, -1444, 0, -3404], [75978, 0, 5080, 0, 974, 0], [0, 24913, 0, 3771, 0, 2231], [0, -7854, 0, -4086, 0, -4091], [0, -47360, 0, 1294, 0, 103], [-87456, 0, -7908, 0, -2028, 0], [14864, 0, 10632, 0, -1242, 0], [-138504, 0, -660, 0, -42, 0], [0, -4881, 0, -2087, 0, 3833], [0, 10828, 0, 2276, 0, -2054], [-204271, 0, -3040, 0, -93, 0], [122493, 0, -16700, 0, -1231, 0], [0, -11263, 0, -11241, 0, 864], [0, -21039, 0, 8787, 0, -3243], [0, -744, 0, -7230, 0, -37], [93462, 0, 1560, 0, 1266, 0], [0, 4617, 0, 2139, 0, -3231], [-128094, 0, 6240, 0, -1032, 0], [0, 46068, 0, -1818, 0, 1067], [0, 50404, 0, 2228, 0, -112], [0, -18509, 0, 3227, 0, 4432], [-143490, 0, 3840, 0, 1350, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2205_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2205_4_a_by();".
// 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_2205_4_a_by(:prec:=6)
    chi := MakeCharacter_2205_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_2205_4_a_by();".
// 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_2205_4_a_by( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2205_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![-64, 0, 241, 0, -34, 0, 1]>,<11,R![-335622400, 0, 2814992, 0, -3512, 0, 1]>,<13,R![-10404000000, 0, 14726800, 0, -6744, 0, 1]>,<17,R![216600, -5860, -22, 1]>],Snew);
    return Vf;
end function;