
// Make newform 1344.4.c.h in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1344_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1344_c_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_1344_4_c_h();".
// 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_1344_4_c_h();".
// 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 := [525876624, 0, 707138208, 0, 90154296, 0, 3447408, 0, 54793, 0, 386, 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, 517741765875936, 0, 51364871210556, 0, 1820198588256, 0, 26906273167, 0, 174672590, 0, 415171], [0, 1385005612832352, 0, 51364871210556, 0, 1820198588256, 0, 26906273167, 0, 174672590, 0, 415171], [0, -1609312635149940, 0, -103808484356868, 0, -1860503278707, 0, -11496542836, 0, -907043, 0, 150398], [227912499930024, 0, -46608391182636, 0, -3268567476990, 0, -58374438995, 0, -395288260, 0, -897179, 0], [56926306332, 0, -29324763072, 0, -1823053635, 0, -50881052, 0, -543931, 0, -1886, 0], [23304782592, 0, 4564567404, 0, 59669136, 0, -1766423, 0, -34288, 0, -149, 0], [48026973574944, 0, 17320042967436, 0, 713417388048, 0, 15930365615, 0, 180719902, 0, 725843, 0], [0, 308637875104020, 0, 56665055702952, 0, 1485391973733, 0, 6707059802, 0, -88086239, 0, -597544], [0, -11659276188, 0, -2006939520, 0, -80656935, 0, -1316132, 0, -9595, 0, -26], [1529865809906616, 0, 429014116275084, 0, 24505239122262, 0, 503759280859, 0, 4283247980, 0, 12881299, 0], [0, 7952916982828608, 0, 1355342997312936, 0, 58547526756570, 0, 991200314147, 0, 7224960640, 0, 19037951]];
    Rf_basisdens := [1, 433631923478208, 433631923478208, 108407980869552, 20649139213248, 28364202216, 1412003502, 5162284803312, 36135993623184, 660235212, 20649139213248, 108407980869552];
    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_1344_c();"
function MakeCharacter_1344_c()
    N := 1344;
    order := 2;
    char_gens := [127, 1093, 449, 577];
    v := [2, 1, 2, 2];
    // 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_1344_c_Hecke();"
function MakeCharacter_1344_c_Hecke(Kf)
    N := 1344;
    order := 2;
    char_gens := [127, 1093, 449, 577];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 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 := 4;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -11, 0, -1, 0, 0, 0, 0, -1, -1, 0, 0], [0, -3, -1, -1, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, -1, 5, 0, 0, 0, 0, 0, 0], [0, -10, 2, -1, 0, 0, 0, 0, 1, -2, 0, -1], [-6, 0, 0, 0, -2, -5, 2, 2, 0, 0, -1, 0], [0, -20, 4, -5, 0, 0, 0, 0, 1, 3, 0, -1], [50, 0, 0, 0, 2, 10, -2, 1, 0, 0, -2, 0], [0, 62, 0, 3, 0, 0, 0, 0, -5, 9, 0, -3], [114, 0, 0, 0, -11, 1, 6, -6, 0, 0, 0, 0], [0, -39, 5, -4, 0, 0, 0, 0, 7, -1, 0, 2], [122, 0, 0, 0, -5, 8, 0, -10, 0, 0, -5, 0], [0, -246, -12, -6, 0, 0, 0, 0, -1, -13, 0, 0], [0, 8, -10, -13, 0, 0, 0, 0, -6, 0, 0, 9], [0, -263, 5, 11, 0, 0, 0, 0, 22, 8, 0, -7], [0, 73, 3, 18, 0, 0, 0, 0, -13, 11, 0, -6], [-112, 0, 0, 0, 10, -1, -10, -6, 0, 0, 1, 0], [204, 0, 0, 0, -18, -12, 30, -2, 0, 0, 0, 0], [54, 0, 0, 0, -10, -20, -8, 18, 0, 0, 16, 0], [0, -70, -2, 39, 0, 0, 0, 0, 12, 6, 0, -9], [70, 0, 0, 0, 11, 31, 32, 14, 0, 0, 8, 0], [-244, 0, 0, 0, -42, 12, 36, -8, 0, 0, 0, 0], [0, -499, 40, 21, 0, 0, 0, 0, 6, -32, 0, 10], [-592, 0, 0, 0, -6, 24, 2, -3, 0, 0, 18, 0], [0, 43, 20, -27, 0, 0, 0, 0, 25, 25, 0, 0], [0, -224, 36, -18, 0, 0, 0, 0, -22, 29, 0, 6], [310, 0, 0, 0, 63, 8, 48, -26, 0, 0, -19, 0], [-254, 0, 0, 0, -42, -24, -18, -32, 0, 0, 24, 0], [0, -246, 22, 23, 0, 0, 0, 0, -14, 18, 0, -1], [134, 0, 0, 0, 60, -26, 60, -6, 0, 0, -18, 0], [0, 446, -6, 30, 0, 0, 0, 0, -30, 106, 0, -6], [0, 174, -44, 104, 0, 0, 0, 0, -1, 27, 0, 10], [-1184, 0, 0, 0, -68, 44, 58, -18, 0, 0, 8, 0], [0, -445, 31, 49, 0, 0, 0, 0, -30, -26, 0, -29], [0, -229, -11, 70, 0, 0, 0, 0, 13, 129, 0, 34], [-1152, 0, 0, 0, -32, -98, -16, 14, 0, 0, 14, 0], [0, -1303, 14, -7, 0, 0, 0, 0, -64, 94, 0, -16], [0, -617, -50, 97, 0, 0, 0, 0, 1, 109, 0, -6], [0, 739, -95, -65, 0, 0, 0, 0, 18, 56, 0, -17], [-8, 0, 0, 0, 162, 11, 32, -102, 0, 0, -11, 0], [466, 0, 0, 0, -138, 78, 28, -39, 0, 0, -36, 0], [0, 94, 48, -55, 0, 0, 0, 0, 15, 35, 0, -13], [-2534, 0, 0, 0, 184, 2, 44, -52, 0, 0, -4, 0], [0, 1583, 43, 58, 0, 0, 0, 0, 43, 233, 0, -44], [-506, 0, 0, 0, -46, 190, -48, 40, 0, 0, -38, 0], [0, 14, -4, -28, 0, 0, 0, 0, 8, 34, 0, 74], [0, 193, 47, 17, 0, 0, 0, 0, 8, 114, 0, -7], [-866, 0, 0, 0, -49, 94, 220, 16, 0, 0, 31, 0], [336, 0, 0, 0, -164, 43, -162, 62, 0, 0, -29, 0], [294, 0, 0, 0, -34, 22, 74, -18, 0, 0, 112, 0], [0, 582, 0, -210, 0, 0, 0, 0, 94, 22, 0, -24], [-1870, 0, 0, 0, 19, -211, 204, -62, 0, 0, -10, 0], [-446, 0, 0, 0, -218, -75, 8, -26, 0, 0, -15, 0], [0, -3411, -222, 63, 0, 0, 0, 0, 18, -114, 0, -18], [-160, 0, 0, 0, 132, -72, -146, 129, 0, 0, -24, 0], [0, 1700, -114, 69, 0, 0, 0, 0, 27, -117, 0, 3], [-334, 0, 0, 0, -285, 58, 216, -2, 0, 0, -13, 0], [0, -300, 232, -71, 0, 0, 0, 0, -41, 140, 0, 37], [0, -1899, 202, -127, 0, 0, 0, 0, -20, 64, 0, -48], [0, -2738, -178, -241, 0, 0, 0, 0, -33, -70, 0, 11], [2082, 0, 0, 0, 207, 96, -144, -14, 0, 0, 47, 0], [-1012, 0, 0, 0, 52, 122, -40, -38, 0, 0, 68, 0], [0, -562, 192, 118, 0, 0, 0, 0, -29, 45, 0, -76], [0, -797, -77, 220, 0, 0, 0, 0, -3, 1, 0, 118], [1934, 0, 0, 0, 88, -208, -176, -54, 0, 0, -40, 0], [0, 1385, 84, 283, 0, 0, 0, 0, -69, -245, 0, -68], [0, -465, -65, -161, 0, 0, 0, 0, 40, 298, 0, 85], [462, 0, 0, 0, -197, 187, 300, -74, 0, 0, 32, 0], [-3164, 0, 0, 0, -54, 77, 328, 110, 0, 0, 3, 0], [3390, 0, 0, 0, -186, -54, -40, -44, 0, 0, -30, 0], [0, 1778, 494, 8, 0, 0, 0, 0, -36, -207, 0, 44], [0, 757, 29, 38, 0, 0, 0, 0, -131, -53, 0, -16], [5802, 0, 0, 0, 87, -162, -96, -84, 0, 0, 69, 0], [0, -3124, -240, -83, 0, 0, 0, 0, -105, -287, 0, -107], [0, -1591, -159, 549, 0, 0, 0, 0, 154, 344, 0, 51], [3014, 0, 0, 0, 154, -178, 280, -46, 0, 0, 132, 0], [-668, 0, 0, 0, 334, -64, 48, 14, 0, 0, 152, 0], [0, -2812, 174, -473, 0, 0, 0, 0, -104, -432, 0, -79], [0, 404, 242, -589, 0, 0, 0, 0, -19, -93, 0, 29], [8142, 0, 0, 0, 302, -55, -16, 38, 0, 0, -7, 0], [-3054, 0, 0, 0, 236, 424, -146, -18, 0, 0, -104, 0], [-1272, 0, 0, 0, -468, -360, -172, -148, 0, 0, -12, 0], [0, -4121, -86, 325, 0, 0, 0, 0, 25, -499, 0, -86], [1506, 0, 0, 0, -77, 28, 276, 138, 0, 0, -75, 0], [-1254, 0, 0, 0, 358, 188, -556, 180, 0, 0, -94, 0], [0, -279, 224, 139, 0, 0, 0, 0, -98, 286, 0, -84], [3616, 0, 0, 0, 72, -114, -550, -32, 0, 0, -66, 0], [0, -1038, 72, -300, 0, 0, 0, 0, -62, -738, 0, 50], [9306, 0, 0, 0, 47, -76, -52, 92, 0, 0, 59, 0], [-4906, 0, 0, 0, -202, 310, -248, 118, 0, 0, -14, 0], [0, 1551, 156, -219, 0, 0, 0, 0, -167, -803, 0, 12], [0, -4999, 33, -462, 0, 0, 0, 0, 49, -157, 0, 216], [6190, 0, 0, 0, -63, 326, -168, -210, 0, 0, -129, 0], [0, 1897, 48, 491, 0, 0, 0, 0, 236, 586, 0, -124], [5028, 0, 0, 0, -159, -201, -40, 52, 0, 0, -144, 0], [0, 4156, 80, 122, 0, 0, 0, 0, 44, 160, 0, -190], [0, 4808, -222, -549, 0, 0, 0, 0, -89, -119, 0, -183], [0, 11047, 141, -48, 0, 0, 0, 0, 215, 117, 0, 168], [0, 7334, 560, -310, 0, 0, 0, 0, -179, -583, 0, -8], [0, 3868, 854, -345, 0, 0, 0, 0, -46, -376, 0, 9], [-2846, 0, 0, 0, -305, 66, -28, -116, 0, 0, 27, 0], [0, -1987, -1201, 290, 0, 0, 0, 0, 131, -413, 0, 38], [152, 0, 0, 0, -1072, -170, -858, 180, 0, 0, 16, 0], [0, 5074, -100, 372, 0, 0, 0, 0, -92, -474, 0, -34], [-2024, 0, 0, 0, 779, 223, -302, -160, 0, 0, 42, 0], [640, 0, 0, 0, 246, -97, 170, -86, 0, 0, -171, 0], [-3706, 0, 0, 0, -196, -296, -764, -212, 0, 0, -20, 0], [9626, 0, 0, 0, -140, 590, 44, -224, 0, 0, 44, 0], [0, 1928, 244, 688, 0, 0, 0, 0, 156, -93, 0, -200], [7578, 0, 0, 0, 562, -122, -348, -258, 0, 0, 216, 0], [0, -2360, -96, -354, 0, 0, 0, 0, -144, 48, 0, -342], [-2970, 0, 0, 0, 718, -340, -902, 276, 0, 0, 128, 0], [5426, 0, 0, 0, -626, -46, 92, -96, 0, 0, -8, 0], [0, -7640, 60, 381, 0, 0, 0, 0, -313, 144, 0, 129], [-7988, 0, 0, 0, 174, -526, -252, -234, 0, 0, -336, 0], [0, 3492, -580, -1016, 0, 0, 0, 0, -55, 195, 0, 136], [0, 5869, -412, 501, 0, 0, 0, 0, 283, -633, 0, 194], [0, 2415, -393, 147, 0, 0, 0, 0, 256, -968, 0, -63], [-1010, 0, 0, 0, -964, 106, -280, 115, 0, 0, 34, 0], [0, -7785, -110, -1003, 0, 0, 0, 0, 136, 870, 0, -34], [0, 4319, 1234, -807, 0, 0, 0, 0, -343, -171, 0, 286], [0, 978, 506, -244, 0, 0, 0, 0, -502, 86, 0, -196], [0, 5496, -36, 1014, 0, 0, 0, 0, -311, -881, 0, 210], [0, 6832, -276, -450, 0, 0, 0, 0, -74, 807, 0, -18], [13174, 0, 0, 0, 35, 490, 248, 56, 0, 0, -175, 0], [-5056, 0, 0, 0, -136, -272, 278, 25, 0, 0, 100, 0], [0, -391, -77, 943, 0, 0, 0, 0, -610, 536, 0, -149], [0, -1337, 1293, 906, 0, 0, 0, 0, -203, 237, 0, -54], [-6528, 0, 0, 0, -470, 145, -1568, 166, 0, 0, 43, 0], [2408, 0, 0, 0, 100, -1132, 1172, -248, 0, 0, 416, 0], [0, 8500, -448, 140, 0, 0, 0, 0, 254, -921, 0, -52], [-2552, 0, 0, 0, -413, -243, 284, 476, 0, 0, -328, 0], [-1898, 0, 0, 0, -756, -744, -554, -394, 0, 0, -132, 0], [0, -18427, 184, -261, 0, 0, 0, 0, 106, -668, 0, 144], [0, -1414, -38, 64, 0, 0, 0, 0, 486, 346, 0, -560], [0, -295, -1478, 963, 0, 0, 0, 0, 56, -2, 0, 62], [-3774, 0, 0, 0, 235, 268, 488, -352, 0, 0, 209, 0], [0, -1508, 196, 424, 0, 0, 0, 0, 616, -800, 0, 340], [0, -4250, 48, 920, 0, 0, 0, 0, -393, 1187, 0, -178], [-7424, 0, 0, 0, 84, -78, 64, 70, 0, 0, 6, 0], [0, 8791, -18, -775, 0, 0, 0, 0, -499, -395, 0, 2], [0, -931, 1235, -1207, 0, 0, 0, 0, 188, -716, 0, -55], [-10352, 0, 0, 0, 346, 186, 868, 108, 0, 0, 416, 0], [0, -99, -123, -627, 0, 0, 0, 0, -108, 510, 0, -129], [15100, 0, 0, 0, -107, 621, -300, -376, 0, 0, -470, 0], [0, -14274, -82, -823, 0, 0, 0, 0, 83, 434, 0, -163], [12352, 0, 0, 0, -532, -887, -256, 486, 0, 0, -503, 0], [0, 5460, -184, -2266, 0, 0, 0, 0, 444, 449, 0, 158], [-752, 0, 0, 0, 1601, -401, 926, -288, 0, 0, 100, 0], [0, 12709, -1775, 22, 0, 0, 0, 0, 17, -457, 0, 136], [3652, 0, 0, 0, 1290, -106, 884, -14, 0, 0, -288, 0], [0, 14043, -469, -628, 0, 0, 0, 0, -275, 437, 0, -94], [13270, 0, 0, 0, 914, -1319, -852, -554, 0, 0, -19, 0], [-2770, 0, 0, 0, 1098, 882, 586, -130, 0, 0, -414, 0], [-15694, 0, 0, 0, -1587, 91, 934, 226, 0, 0, -242, 0], [-1772, 0, 0, 0, 168, 762, 110, 966, 0, 0, -60, 0], [0, 11615, -806, -237, 0, 0, 0, 0, -42, -76, 0, -148], [0, 6473, 1390, 645, 0, 0, 0, 0, -93, 587, 0, -520], [18578, 0, 0, 0, -430, -1032, 1140, -322, 0, 0, 502, 0], [-19810, 0, 0, 0, 1046, 280, -1556, -248, 0, 0, -356, 0], [0, -13652, 1172, -1200, 0, 0, 0, 0, -250, -292, 0, 84], [9602, 0, 0, 0, -1101, -230, 448, 254, 0, 0, -213, 0], [-10264, 0, 0, 0, -1238, -376, -1492, 766, 0, 0, -90, 0], [24088, 0, 0, 0, -1128, -684, -1138, 22, 0, 0, -612, 0], [0, -1441, 843, 795, 0, 0, 0, 0, -420, -388, 0, -123], [17230, 0, 0, 0, 186, 678, 592, -941, 0, 0, 324, 0], [0, -12341, 372, -2845, 0, 0, 0, 0, -98, 630, 0, -118], [0, -7089, -604, -485, 0, 0, 0, 0, 793, -251, 0, 114], [0, -15007, 1589, 149, 0, 0, 0, 0, -522, 732, 0, -337], [-11040, 0, 0, 0, -354, 1935, -414, -282, 0, 0, -99, 0], [-11966, 0, 0, 0, -398, 230, 1504, -99, 0, 0, -172, 0], [-5012, 0, 0, 0, -1200, -576, -612, 564, 0, 0, -204, 0], [88, 0, 0, 0, 1355, 1633, 954, -448, 0, 0, 256, 0], [0, 12667, 693, -1062, 0, 0, 0, 0, 465, 133, 0, 258], [0, -26008, 88, 1296, 0, 0, 0, 0, -781, -459, 0, 304], [-14294, 0, 0, 0, 298, -1030, 622, -721, 0, 0, -106, 0], [0, 6645, -95, 3577, 0, 0, 0, 0, 592, 1344, 0, -185], [4510, 0, 0, 0, 1382, 1264, 602, 298, 0, 0, -548, 0], [0, -27704, 984, 134, 0, 0, 0, 0, -4, 304, 0, -70], [0, -190, 876, -1413, 0, 0, 0, 0, -655, -2059, 0, 189], [-13548, 0, 0, 0, -97, 137, 438, -1092, 0, 0, -216, 0], [740, 0, 0, 0, -808, 519, -282, -94, 0, 0, 415, 0], [0, -4609, -1188, -797, 0, 0, 0, 0, -66, 778, 0, -746], [0, -19678, 882, -2778, 0, 0, 0, 0, -994, 523, 0, -210], [0, -686, -2282, -896, 0, 0, 0, 0, -270, 1198, 0, -332], [-1854, 0, 0, 0, 910, 650, -644, -1637, 0, 0, -292, 0], [13262, 0, 0, 0, 77, -342, -2696, 126, 0, 0, -97, 0], [3976, 0, 0, 0, 1414, 932, 1742, -376, 0, 0, 296, 0], [0, -7029, 518, 1417, 0, 0, 0, 0, 877, 1993, 0, 78], [0, 13827, 1355, -934, 0, 0, 0, 0, -1077, -131, 0, 488], [0, -4791, 616, 923, 0, 0, 0, 0, 878, 1002, 0, -260], [0, -11317, -288, -1451, 0, 0, 0, 0, 677, 1729, 0, 332], [-8872, 0, 0, 0, 1467, -575, 410, 1516, 0, 0, 576, 0], [16094, 0, 0, 0, 2056, 314, 368, -463, 0, 0, -190, 0], [0, -19874, -204, -87, 0, 0, 0, 0, -441, 915, 0, 279], [-19824, 0, 0, 0, -461, -1967, -666, 544, 0, 0, 260, 0], [-28506, 0, 0, 0, 121, -752, 2500, -846, 0, 0, -127, 0], [0, 23102, -1528, -2263, 0, 0, 0, 0, -713, -927, 0, 371], [14344, 0, 0, 0, 1120, 692, 948, 824, 0, 0, -124, 0], [0, 36691, 117, -1869, 0, 0, 0, 0, 152, 1472, 0, 435], [17660, 0, 0, 0, 230, -404, -1090, 354, 0, 0, 628, 0], [0, -11652, -936, -150, 0, 0, 0, 0, -600, 760, 0, -326], [0, 10895, 986, -1441, 0, 0, 0, 0, -496, 240, 0, 904], [11840, 0, 0, 0, -718, -308, 2680, 588, 0, 0, 490, 0], [0, -3179, -2238, -1585, 0, 0, 0, 0, 819, 1247, 0, 194], [31770, 0, 0, 0, -1202, -824, -636, 538, 0, 0, -322, 0], [0, 17730, 122, 2753, 0, 0, 0, 0, 487, 230, 0, -67], [-13142, 0, 0, 0, 316, -556, 616, 824, 0, 0, 332, 0], [0, 8545, 830, 555, 0, 0, 0, 0, -730, 272, 0, 210], [-5840, 0, 0, 0, 2440, 956, -2226, -470, 0, 0, 212, 0], [0, -29240, 280, 1228, 0, 0, 0, 0, -424, 2152, 0, -112], [-27762, 0, 0, 0, 817, 220, -3248, -108, 0, 0, -115, 0], [-22966, 0, 0, 0, 744, 60, 1334, -68, 0, 0, 348, 0], [-14416, 0, 0, 0, -1568, 404, -2562, 670, 0, 0, 428, 0], [-9614, 0, 0, 0, -1895, 1197, -314, -1462, 0, 0, 60, 0], [-16422, 0, 0, 0, 1052, -2443, -28, 502, 0, 0, 675, 0], [0, 8060, 1440, 691, 0, 0, 0, 0, -159, -3297, 0, 51], [0, -4836, 160, 472, 0, 0, 0, 0, 638, 967, 0, -440], [-1030, 0, 0, 0, -3294, -690, 1458, 1069, 0, 0, -246, 0], [-31130, 0, 0, 0, 1365, 392, -640, -530, 0, 0, -93, 0], [10784, 0, 0, 0, 36, 2964, -644, 404, 0, 0, 552, 0], [0, -15518, -122, -557, 0, 0, 0, 0, -1092, 2738, 0, -709], [0, 37894, 1040, -1267, 0, 0, 0, 0, -1133, -2523, 0, -169], [-3020, 0, 0, 0, 901, -261, -1462, 128, 0, 0, 356, 0], [-2146, 0, 0, 0, -724, -573, -1808, -222, 0, 0, -835, 0], [-11342, 0, 0, 0, -100, 910, 2346, 539, 0, 0, 280, 0], [0, -1143, 2792, 739, 0, 0, 0, 0, -109, -125, 0, 32], [0, -22924, -520, -1060, 0, 0, 0, 0, 1238, -151, 0, -172], [0, -4076, 2572, -3101, 0, 0, 0, 0, -251, -2080, 0, 403], [10478, 0, 0, 0, -2922, -1530, -2232, -1214, 0, 0, 18, 0], [-29890, 0, 0, 0, 699, 244, -612, -916, 0, 0, -1103, 0], [0, 9412, 420, 3591, 0, 0, 0, 0, 15, 628, 0, 111], [-16984, 0, 0, 0, 1958, -372, -3100, 984, 0, 0, 238, 0], [-3074, 0, 0, 0, -3298, -1262, 102, 1668, 0, 0, 124, 0], [0, 6630, 804, 2253, 0, 0, 0, 0, 1159, 2423, 0, -77], [0, 17903, -2006, 1521, 0, 0, 0, 0, 677, 1565, 0, -474], [-1822, 0, 0, 0, 3241, 2582, 2260, 926, 0, 0, -501, 0], [0, 9793, 250, 205, 0, 0, 0, 0, -1629, 2939, 0, -820], [0, -29772, -1620, -324, 0, 0, 0, 0, -1772, -4128, 0, 624], [4344, 0, 0, 0, 3552, 876, 1016, -1228, 0, 0, 156, 0], [0, -12320, -10, -3067, 0, 0, 0, 0, -339, 2409, 0, -625], [-31858, 0, 0, 0, -4687, 1701, -728, 258, 0, 0, -268, 0], [15280, 0, 0, 0, 1608, 86, 636, 1484, 0, 0, -530, 0], [-4176, 0, 0, 0, 136, -2860, 2786, 1409, 0, 0, 392, 0], [0, 178, -2836, -2820, 0, 0, 0, 0, 884, -2218, 0, 702], [0, 15649, 1615, 2734, 0, 0, 0, 0, 263, -545, 0, -554], [-29614, 0, 0, 0, 218, -39, -1350, 2230, 0, 0, -871, 0], [0, -15434, 270, -1494, 0, 0, 0, 0, -1558, -2745, 0, -1446], [-17700, 0, 0, 0, 2059, -2057, -802, 1196, 0, 0, 662, 0], [-3806, 0, 0, 0, 1726, -201, 768, -794, 0, 0, -1117, 0], [11432, 0, 0, 0, -3558, 120, -1246, 422, 0, 0, -444, 0], [0, 40403, 2636, 1037, 0, 0, 0, 0, 1426, 2952, 0, -196], [0, 24347, 222, 109, 0, 0, 0, 0, 1341, 505, 0, 370], [0, 30178, 92, -2593, 0, 0, 0, 0, -1287, 461, 0, -151], [0, -3242, -490, 2333, 0, 0, 0, 0, -21, -2430, 0, 905], [0, -19259, -1338, -679, 0, 0, 0, 0, -96, 1964, 0, 500], [-4294, 0, 0, 0, -1440, 2664, -694, -1712, 0, 0, -192, 0], [-27604, 0, 0, 0, 208, -1366, -524, -578, 0, 0, 110, 0], [0, 4233, 804, -5067, 0, 0, 0, 0, -221, -1093, 0, -650], [0, 11105, -161, 889, 0, 0, 0, 0, -74, -2174, 0, -995], [0, -50993, -1657, 1019, 0, 0, 0, 0, 2070, 2250, 0, -163], [-47530, 0, 0, 0, -3271, -51, -2818, -186, 0, 0, 1216, 0], [0, -8524, 1156, 2800, 0, 0, 0, 0, -916, -2048, 0, 52], [0, 49210, -1492, 75, 0, 0, 0, 0, 1241, -4879, 0, 213], [8726, 0, 0, 0, 743, -3999, -2640, -462, 0, 0, -114, 0], [0, 53749, -677, 2860, 0, 0, 0, 0, -367, 231, 0, -776], [0, -4304, 2736, -1984, 0, 0, 0, 0, -537, -223, 0, -256], [0, 11113, 5089, -455, 0, 0, 0, 0, -1158, 1784, 0, -713], [0, -465, -3097, -4480, 0, 0, 0, 0, -923, -4091, 0, 554], [7246, 0, 0, 0, 670, 782, 1352, -1454, 0, 0, 776, 0], [21098, 0, 0, 0, 1982, -314, 170, -1524, 0, 0, 1006, 0], [-61480, 0, 0, 0, 1390, 1628, -1358, -1452, 0, 0, -496, 0], [19082, 0, 0, 0, -150, -2682, 5010, 2177, 0, 0, 594, 0], [0, -15207, 1742, 1681, 0, 0, 0, 0, -497, 523, 0, 900], [0, 5660, 994, -443, 0, 0, 0, 0, 569, 113, 0, -521], [-13110, 0, 0, 0, -3794, 2846, 2068, -363, 0, 0, 260, 0], [0, -12006, -480, 2568, 0, 0, 0, 0, 1912, -3854, 0, -354], [-45884, 0, 0, 0, -1284, 1724, -3528, 206, 0, 0, -20, 0], [26644, 0, 0, 0, 148, -46, -412, 2096, 0, 0, 674, 0], [43112, 0, 0, 0, -2810, -388, 1076, -546, 0, 0, 718, 0], [0, -3775, -2475, -1893, 0, 0, 0, 0, -1900, 1230, 0, -339], [0, -24704, 1940, 524, 0, 0, 0, 0, 840, -208, 0, 1004], [34150, 0, 0, 0, -1404, 1721, -3380, 2378, 0, 0, -453, 0], [34854, 0, 0, 0, -3412, 2656, 1084, -174, 0, 0, -728, 0], [0, 28974, 3332, -305, 0, 0, 0, 0, -199, -5929, 0, 141], [21376, 0, 0, 0, -612, 432, 286, 2347, 0, 0, 744, 0], [-6750, 0, 0, 0, 2025, 2229, -3864, 78, 0, 0, -264, 0], [0, 59262, -840, 711, 0, 0, 0, 0, -325, -2051, 0, -55], [0, 19150, -154, 2415, 0, 0, 0, 0, -946, 142, 0, -13], [-4690, 0, 0, 0, 2274, 2518, -3216, -530, 0, 0, -604, 0], [0, 34248, -34, -2033, 0, 0, 0, 0, -58, -1260, 0, -851], [0, -13130, 1890, -396, 0, 0, 0, 0, -1332, -1545, 0, 1200], [0, 42313, -2278, 2703, 0, 0, 0, 0, 60, 174, 0, -198], [18570, 0, 0, 0, 5188, 194, 2736, -2136, 0, 0, 920, 0], [0, -3341, -3834, -2449, 0, 0, 0, 0, 1174, -3546, 0, 488], [0, -1486, 1708, 1260, 0, 0, 0, 0, -1732, 426, 0, 94], [0, -1992, 1988, 2504, 0, 0, 0, 0, -1624, -496, 0, -520], [-38104, 0, 0, 0, -3924, -2154, -116, -992, 0, 0, -228, 0], [0, 81100, -860, -2147, 0, 0, 0, 0, 815, 1887, 0, 385], [11858, 0, 0, 0, 2710, 3464, 268, -284, 0, 0, 488, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1344_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1344_4_c_h();".
// 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_1344_4_c_h(:prec:=12)
    chi := MakeCharacter_1344_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_1344_4_c_h();".
// 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_1344_4_c_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1344_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![129036134656, 0, 84573340800, 0, 3747196176, 0, 58092608, 0, 374136, 0, 1032, 0, 1]>,<23,R![-854202265344, 2746383552, 361912548, -638448, -36968, 40, 1]>],Snew);
    return Vf;
end function;