
// Make newform 2025.4.a.be in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2025_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_2025_4_a_be();".
// 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_2025_4_a_be();".
// 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 := [73008, 143136, -96816, -144610, 52669, 45247, -13834, -5130, 1452, 241, -64, -4, 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], [-12, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [10, -20, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [346418736, -19241890608, 6201852960, 12714043102, -3965272965, -2078007262, 533695176, 140077854, -25985670, -4004629, 423747, 37591], [-16398173904, -27249812016, 16298881568, 12441331134, -6182499797, -1135333182, 752424648, 8257662, -35963750, 2246427, 592499, -59961], [-8996491632, -13531173264, 7241968736, 8249829402, -1658564047, -1610120986, 140925528, 134284506, -4652306, -4932735, 48265, 65605], [-264990672, -183919344, 373846496, -36830482, -128683685, 58917130, 13670592, -8238762, -623630, 390691, 10547, -6097], [-4259211024, 6912879120, 4660906144, -6177102186, -1043697913, 1271042346, 115983912, -108583914, -6183454, 4175607, 119023, -59757], [-38296603440, -27707922384, 50098565088, 15043819266, -18229949003, -836392450, 2176503480, -97918782, -103544538, 9232965, 1709165, -196583], [5029485744, 1967766096, -22654098144, 2095604974, 11263200675, -2024460142, -1447343352, 281453742, 70745130, -13940461, -1184469, 232351], [2385982224, 4635050736, 834353632, -4240037334, -1088033191, 1216581942, 170737080, -129243510, -9691522, 5649033, 184561, -86739]];
    Rf_basisdens := [1, 1, 1, 1, 158287872, 158287872, 158287872, 9892992, 79143936, 158287872, 158287872, 39571968];
    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_2025_a();"
function MakeCharacter_2025_a()
    N := 2025;
    order := 1;
    char_gens := [326, 1702];
    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_2025_a_Hecke(Kf)
    return MakeCharacter_2025_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], [-1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [-2, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0], [-1, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1], [-8, 3, -3, 0, 0, 0, 1, 0, 0, 1, 0, 1], [-9, 5, 0, 1, -1, 1, 0, 0, -1, -1, 0, -1], [-24, -1, -4, -1, 0, 0, -2, -2, 1, 0, 1, 0], [-10, 8, -6, 0, 0, 0, -2, -2, -2, 0, 1, 3], [-3, 17, 1, 4, -2, 0, -2, 0, 2, -1, -2, -2], [6, 2, 5, 1, 1, -1, 3, -2, 1, -2, 4, 2], [30, 0, 8, 1, -2, -1, -4, -4, -1, 2, -5, 2], [32, 12, 1, -3, 0, -2, 0, 8, 1, 1, 0, -1], [-105, 12, -11, 6, 4, -3, 3, 4, -2, -1, 3, -4], [-38, 5, 6, -3, 9, -1, 4, 2, -1, 1, 2, 10], [-36, -13, -27, 0, -3, 3, -10, 4, -2, -2, -5, -2], [30, 8, 16, 5, 8, 3, -3, 4, -3, 0, -4, -2], [40, -74, 5, 5, -5, -1, -14, 6, 7, -2, -9, -8], [74, 27, 29, -1, -9, -6, -6, 4, 1, 2, 4, -1], [24, 50, -7, -9, 5, 5, -8, -4, 7, 4, -8, 9], [-5, 80, -6, -14, 1, -7, 15, -2, -2, -5, -2, -1], [-256, -19, -6, -1, -5, 5, -1, 8, 11, -1, -11, -1], [39, 59, -31, 4, 4, 0, 5, 0, 6, -1, 8, 8], [78, -138, 23, 4, 15, -12, 7, 0, -10, 2, 2, 10], [-169, -61, 39, -13, 13, 10, 1, -16, -7, -6, 19, 6], [-14, 172, 21, -2, 19, 8, 2, 6, 2, -10, 19, 3], [-320, 10, -37, -8, -14, -5, -15, -8, 4, -9, 6, 6], [-32, 165, -21, -28, -2, -8, 17, -18, -8, 1, 9, 3], [-512, -16, 13, -1, -4, 0, 22, 10, -5, 9, 11, -6], [-103, -217, -24, -1, -9, -13, 19, -12, 1, 7, 13, -11], [-140, -43, 38, -6, 8, 15, -25, 22, 10, 10, -22, 19], [-184, -40, 59, -22, 24, 7, -27, 14, 8, -7, 23, 5], [-121, 221, -16, -8, -11, -12, -18, 6, 8, 5, -2, -41], [431, 128, -10, 32, -9, 5, -15, -18, 18, -1, 0, -1], [-259, 21, -85, 3, -11, -4, -47, -28, -11, 10, -16, -3], [207, -163, 50, 53, -6, -30, 1, 26, -13, 2, -23, -18], [-387, 216, -89, 1, 0, -8, -3, 14, 3, -3, -19, -36], [-353, 75, 128, 34, -2, 33, -66, 0, 0, -12, -13, -24], [-886, 19, 145, -10, 8, 14, 31, -8, 4, -13, -4, 26], [764, -242, -38, -84, 25, 27, 11, -20, 4, 7, 27, 28], [170, 77, 54, 36, 7, -4, 59, 28, -4, 17, -19, -5], [-1058, 114, -53, -13, -1, 5, -37, -16, 23, -10, -15, 5], [-307, 215, -97, -36, -13, -27, -22, 10, -10, -4, 10, -16], [-201, -69, 101, -56, -17, -35, 62, -32, -14, 26, -15, 26], [-119, 242, 6, -2, 43, -27, -19, -6, 14, -7, -2, 7], [-459, 33, -43, 4, 10, 34, 41, -42, -10, -13, 49, 29], [425, -18, 85, -66, -14, 53, -19, 8, -14, -11, -25, 12], [-355, -86, 195, -2, -28, 35, -52, 32, -16, 11, -9, -4], [-274, 291, -76, 34, 19, 16, -53, 18, -28, 5, 46, 8], [-1514, -247, 164, -79, 12, 26, -49, -6, 3, 10, 39, 46], [984, 87, -86, -5, -72, -30, 10, -6, -39, -14, 21, -26], [-187, -16, 17, 47, -69, -21, 89, 20, -1, -4, 1, -36], [-1201, 333, 101, 61, 67, 4, -30, 2, -31, 20, 10, 20], [-1129, -498, 129, -61, 22, 2, 1, -30, 15, 9, 32, -39], [-2416, -258, 175, -33, -2, -28, 5, 42, -21, 1, -19, -92], [1075, -106, -9, 31, 56, -42, -102, -44, -23, 5, 32, -19], [-536, -539, -102, -4, 40, -3, -73, -4, 14, -4, -57, 30], [218, -209, 99, -35, -10, -11, 52, 14, -27, 9, 57, 3], [-915, 116, 17, 86, -93, -66, -89, 28, 4, 24, -104, -48], [741, 254, 251, 81, -63, 71, -23, -18, 9, 0, -30, -64], [-2590, -55, 64, -27, -53, -41, -92, -54, 27, 5, -52, -16], [-194, 187, 13, 17, -49, -8, -34, -64, -43, 68, -37, 55], [-1061, -199, 129, -28, -29, 65, 72, -66, 30, -28, 98, 71], [-1402, 387, -268, 81, 3, -19, 179, 52, -31, 5, -48, 16], [-1577, 259, 152, 101, -24, -46, 85, 52, 49, -4, 10, -25], [1075, -1236, 81, -87, -64, 46, -29, 46, 59, 11, -41, -12], [846, -490, 77, -74, 1, 36, -124, -12, -34, -46, -29, -50], [-2581, 343, -42, 88, -31, -64, 36, -26, -92, -5, 2, 31], [472, -814, -157, 74, -95, -26, -42, 20, -16, 74, -75, -11], [-2630, -131, 282, 137, 46, 10, -21, 14, -37, 14, -41, 14], [253, 601, 20, 121, 100, 12, 239, -2, 7, -68, 93, -14], [-1476, -535, -201, -135, 26, 35, -188, 2, 19, -65, 58, -30], [-364, 410, -224, 66, -6, -122, 64, 28, 26, 62, 42, 30], [1775, -1619, 377, 22, 52, 68, -64, -36, -34, -1, -35, -6], [-3073, -759, 189, -40, 34, -36, 14, 60, 30, -25, 32, -16], [-1433, 136, -420, 112, -67, 53, -7, -26, -2, -21, -106, -39], [655, -181, 150, -50, 18, -5, 62, 94, 88, -62, 11, -62], [827, -136, 496, 64, 9, -47, 9, 82, -20, -39, 34, -132], [-68, -1060, 406, 135, 14, 23, 22, -28, -23, -8, 97, -12], [-1131, 492, -374, 194, -121, 5, 282, -18, -12, -13, -14, -25], [-834, -1289, -360, -36, 15, -114, -99, -50, 26, -91, 66, 32], [-175, -923, 231, 51, -6, -19, 227, -58, -7, 23, 92, 14], [-1850, 136, -397, 253, 100, -86, -15, 142, 19, -29, -44, -110], [-1521, -1474, -143, 168, -59, -26, -11, 122, 16, -80, -109, -127], [-2963, 688, -119, -6, -80, -99, 161, -126, -42, 83, -49, 149], [-146, 251, -60, 1, 146, -146, 1, 130, -71, 118, 53, 37], [1289, 413, -93, 58, 132, 38, 5, -36, 98, 83, 32, -39], [1352, 1315, 148, -99, -23, -25, 150, -82, -13, 75, 28, -36], [-1615, 165, -300, -42, 197, 94, -168, 22, -16, 71, 121, 195], [-5274, 46, -550, -81, -112, 17, -56, 66, 57, -18, -62, -186], [-873, -1588, -60, -127, 46, -21, 206, 34, 13, -8, -84, -50], [741, 712, -224, 155, -20, -33, -88, 146, -133, -34, 96, -72], [295, 1117, -405, -151, -143, 88, 170, 12, 21, 60, -202, 39], [-1842, -449, -782, 211, -147, -175, 57, -172, -93, 9, 53, 13], [-3844, -750, -220, 61, 67, 94, -95, -32, 1, 25, -36, -71], [-2158, 109, -185, -55, -40, 83, -154, -36, -9, -47, -2, 82], [2655, 655, 31, -166, -43, 93, -114, -62, 196, -106, 14, -34], [2528, -1007, 891, -38, 58, 106, -272, 124, 58, 55, -152, -74], [-3947, -815, -528, 131, -20, 8, -45, -30, 25, -2, 123, 82], [-372, -92, 674, 281, 32, 101, 42, -80, -105, -102, -106, -21], [-2479, 1181, 110, 244, -206, -201, 198, 142, -86, 2, -129, -274], [-1522, 1013, 37, 55, 21, -128, 327, -50, -155, -12, 247, -30], [-3554, 1052, -140, -139, 68, 75, -277, -36, 51, -70, 52, 161], [1412, -1372, -461, 177, -157, 3, -188, 268, 131, 56, -155, -139], [-2178, -1727, 139, -341, -132, 7, 53, 60, 95, -19, 41, -52], [-3080, -1082, 224, 56, 243, 233, -54, -46, 98, 117, 89, 105], [2016, 1811, 96, 3, 9, -63, -229, -8, -221, -77, 52, -50], [-440, 976, 449, -200, 22, -47, 242, -4, 56, 121, -29, 334], [3233, -1771, 488, -173, 152, 80, 229, -106, -21, 60, 117, 211], [3571, -60, -73, -160, -177, 128, 143, -134, 132, -60, -29, 153], [2606, 1004, 342, -136, 88, -164, 164, 26, 62, 8, -75, 49], [-1985, -811, 304, -88, 77, 112, 621, -120, -120, 85, -23, 97], [1917, 171, -382, 56, -9, 68, -111, -188, 6, 39, -80, 264], [3744, -2187, 445, -335, 204, 227, -86, -122, 67, -137, 344, 110], [3050, -2132, -781, 265, -59, 15, -212, 156, -105, -80, 62, -164], [-3285, -523, -675, -248, 149, -43, 45, -10, 120, -168, 66, -281], [-890, -1742, 897, 75, 132, 60, -267, 166, 247, -67, 247, 8], [5333, -1514, -58, 47, -81, -34, -246, -108, 99, 89, -33, 170], [-3350, -251, 192, -110, 2, 1, 398, -210, -120, 192, -68, 213], [-4, -2143, 710, -44, -39, 6, -333, -172, 108, 75, -157, -151], [4808, 980, 60, 56, -163, -161, -112, 198, -134, 85, -301, -36], [-3042, 1427, -27, -279, -222, 159, 82, -154, 77, -145, 83, -91], [2925, -2626, -112, -193, 57, 212, -147, 82, -11, -107, -128, 160], [-863, -1197, -1089, 8, 24, -236, 245, 182, -178, 251, -269, -11], [6239, -3371, -580, 92, -108, -289, 448, -226, -118, 34, -97, 148], [-5902, 733, -743, -61, 50, -97, -50, -134, 39, -93, 113, -129], [-7881, 1570, -47, 165, -113, -115, 398, -176, 85, 12, 3, -159], [6035, -3241, -583, -266, -13, -195, 202, 106, -8, -44, -30, -69], [-5415, 2863, -440, 431, -188, -6, 59, -106, -67, -202, -51, -48], [-3227, 878, -66, -80, 216, -230, -254, 334, -32, 106, 41, 19], [2069, 466, -617, -449, 81, 47, -2, -164, 67, 56, 85, -37], [-3626, -684, -553, 85, -292, -18, 145, 146, -279, -81, 3, -142], [1734, -342, -1105, 151, -99, 103, 213, -46, 3, -84, -82, 114], [10014, 4693, -174, -135, 150, -86, 51, -350, -173, 90, 301, 168], [1623, 1054, -868, -33, 163, -56, -325, 322, -175, 35, -115, -60], [7475, 462, -197, 228, 10, -71, 39, 10, -20, 105, 321, 144], [-8835, -3474, -163, -304, -98, -141, 75, 16, 12, -15, 117, -236], [-4130, -1452, -192, 171, -90, 261, -135, -74, 97, -16, 109, 5], [-7437, -1350, 1068, 305, 127, 238, -569, 150, 147, 35, -223, -64], [-5902, -296, -427, -177, 415, 163, 166, 160, 77, 242, 59, 429], [15762, -1320, -268, -312, -88, -76, -14, 192, 312, 300, -38, 246], [6537, 1481, 499, -357, -241, 38, -185, -148, -131, 172, -170, 225], [-3727, -479, -392, -432, 109, -58, -727, -60, 30, 43, -40, 40], [-1888, 2680, -334, 90, -198, -40, -96, -130, 100, -98, -93, -177], [-11802, 1613, -123, -46, 7, -405, 301, 170, 8, 188, -50, -457], [-7936, 3186, -1226, 26, 148, 34, 106, -168, -146, 106, -270, 150], [-4305, -2272, -416, 47, -163, 168, -679, 134, -67, -63, -135, 120], [1196, 3328, -362, 9, -88, -103, 672, -92, -97, 314, -170, 109], [4125, 4032, -560, 91, -108, -437, 638, 150, 255, 186, -468, -216], [9093, -3685, 596, -54, 378, 17, 460, -66, -312, 48, 189, 20], [8495, 3066, -698, 432, 94, 264, -735, 120, -84, -66, -505, -81], [7, 885, -431, -4, -419, 355, 252, 36, 22, 94, -150, -96], [-8127, -1404, -910, -257, -215, 234, -188, 100, 175, 21, 78, 135], 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-2106, 1050, 1141, 2558, 502, -346, 59, 888, 1545], [32790, -6503, -4536, 539, 1093, 29, 40, -884, 415, -43, 225, -1063], [104806, -895, 1937, -866, 88, -304, 3975, 176, 782, 33, 2284, 879], [61872, -14852, -2538, -2156, -144, -846, -294, -1592, -404, -950, 2552, -666], [53535, 4490, -534, 1221, 2415, -808, -2321, -434, 637, -337, 1903, -235], [67925, -3611, 2666, 1707, -666, -2728, 3144, 242, -881, 1738, -831, 720], [-3943, -3639, -4091, -2430, 1861, -1427, 410, -1270, 500, 994, -986, 1567], [-26104, -20765, 5205, -290, 550, -1048, 1649, 1476, -640, 1211, -968, -1308], [-75398, 3846, 1963, 2772, -355, -192, 570, 740, -1790, 718, -2321, -239], [-38454, -9758, -496, 1261, -900, -1333, 1579, -286, -687, 1386, -832, 748], [-2075, 26252, -2648, 3626, -1291, -1341, 2171, -870, 90, -679, -368, -565], [13943, -641, 1412, 57, -1786, -24, -3016, 486, 525, 1152, -1400, 2262], [90025, -1541, 2276, 1769, 168, -904, 2669, -678, -921, 58, 1245, 94], [23927, 19325, 6891, -941, 503, -1756, 4036, -738, 603, -378, 1168, -924], [-13981, 15795, 5078, -3271, -420, -856, -1263, 100, -285, -378, -1463, -729], [20489, -6167, 4595, -1802, -311, 459, 590, -418, -1516, -1500, 1354, -273], [-18534, 17861, -2090, 214, 1779, -1160, 1876, -1748, -1646, 427, 1484, 322], [-19596, -14866, -5566, 143, 672, 1055, 397, -1326, -125, 1096, -158, 2992], [-21157, 8456, 3361, -3274, -2401, 2008, -178, -1456, -706, -452, 63, 922], [53070, -2817, -5619, -745, -371, 104, -768, -1432, 779, -22, 1185, 653], [36223, -7725, -6925, -725, -497, -2724, 1509, 264, -711, -576, -1231, -2934], [19406, -2648, 2444, 1297, -1270, 1363, -459, 1118, 3, -276, -2199, 777], [110071, -3543, 3310, -725, 1821, 1391, -2024, 1668, 561, 21, 345, 12], [19227, -11177, -8081, 304, 343, -1319, 1040, -422, -1214, 70, 142, -1259], [44785, -7285, -305, 1125, 1984, 317, 2062, 624, 903, -735, 80, -108], [-85103, -18131, 586, -987, 412, -1624, 1833, -914, -65, 64, 2139, 556], [9079, 5372, -7011, 1100, -18, 445, -4251, -920, 468, -111, 97, -1120], [-45766, -8543, 3939, 1949, -1452, -377, -258, -1092, -329, 685, -1790, 18], [11142, 12568, 769, 2267, -1489, -887, 58, -2132, -191, 760, -2020, -770], [-9552, 19753, -4389, 635, -581, 324, 3161, 242, 569, 92, 933, -744], [47778, 5776, -2486, 1763, 38, -2507, -1072, 666, -375, 432, 526, -2160], [36943, -23526, -2294, 1734, -214, -1048, 5374, 96, -1608, 306, 1786, -97], [35706, -8582, -3180, 2669, -1754, 2359, -1031, 204, -507, -1258, -1660, 206], [9823, -2765, -594, -1835, 1667, -17, 4359, -232, -679, -649, 2238, 1049], [-14390, -13790, -1696, 2766, -416, 1026, 25, -220, -1124, -1022, 818, -518], [17737, 29200, -625, 1329, -3215, -3027, 2999, 572, -1001, 874, -2785, -3173], [50683, 20234, -3025, -2635, -432, 1488, 2710, -1288, 959, 977, 200, 2057], [-27145, -188, -2729, 2674, -843, 1420, -783, 46, 1162, -822, -749, -2169], [80118, 8217, -4791, -2075, 1860, 867, 3526, -162, 791, -709, 2138, 3238], [54369, 15832, -7148, -1158, 265, -895, -940, -1070, -890, -465, 3105, 1605], [-10660, -15267, -975, 1039, -667, -666, 496, -1496, -167, -324, -982, -225], [-6757, -20132, 2302, -101, 1821, -1192, 1966, -1610, 793, -919, 1696, 569], [50569, 4576, -6820, -571, -1193, 400, 1304, -608, 309, -177, -2841, 968], [39784, -2664, 2589, 122, 379, 1644, 1080, 2758, 766, -202, 1629, -523], [15957, 24802, 7038, 3502, -394, -164, 2217, 880, -1640, -342, -1465, -2802], [-37213, -4870, 2603, -4785, 92, 258, -1987, -1982, 431, 77, 178, 1715], [13361, 17571, -4052, 2968, -126, 179, -1738, 174, 978, -140, -123, -2096], [-1158, -19552, -2525, -246, -2339, -1594, 695, -422, 556, -456, 267, -3405], [-14532, 11961, -4241, 394, 3993, -297, 3858, 418, -1682, 632, 1912, 1223], [-43484, -16382, 1474, -629, 1805, -1210, 1906, -2, 1939, 121, -69, 1154], [20380, 4240, 8339, -472, 1347, -532, 3652, 1822, 1288, 1784, -1223, 935]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2025_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2025_4_a_be();".
// 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_2025_4_a_be(:prec:=12)
    chi := MakeCharacter_2025_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_2025_4_a_be();".
// 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_2025_4_a_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2025_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![73008, -143136, -96816, 144610, 52669, -45247, -13834, 5130, 1452, -241, -64, 4, 1]>,<7,R![284111126595504, -49069079780724, -12328245550998, 857927085426, 159370316127, -5001274746, -849150162, 12946194, 2090268, -14914, -2373, 6, 1]>,<11,R![27256136074433856, -12280529942857944, -3157793940816144, 34317620761625, 18363263926723, -190920855323, -33136722397, 337214331, 25135263, -189035, -8485, 29, 1]>],Snew);
    return Vf;
end function;