
// Make newform 6003.2.a.r in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6003_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_6003_2_a_r();".
// 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_6003_2_a_r();".
// 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 := [-192, 1088, -400, -6032, 6405, 8288, -9405, -5124, 5778, 1665, -1835, -295, 316, 27, -28, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4864, 1952, -50704, -18368, 219845, 62103, -222160, -69732, 95926, 33795, -20506, -7813, 2133, 850, -86, -35], [-7392, 50752, 48528, -251870, -103543, 269185, 95352, -124816, -42120, 29681, 9288, -3713, -985, 226, 40, -5], [7008, -29768, -61040, 162520, 146511, -224535, -120680, 134604, 45498, -41247, -8674, 6765, 815, -566, -30, 19], [-36288, 249856, 182064, -1269304, -208365, 1586665, 114336, -900764, -35982, 272925, 6722, -45683, -693, 3982, 30, -141], [113088, -290848, -632064, 1595216, 791727, -1976055, -451872, 1089876, 138498, -318351, -23626, 51261, 2119, -4302, -78, 147], [108096, -236192, -715664, 1439312, 1282479, -1996211, -988048, 1211508, 381394, -382455, -77254, 65561, 7839, -5794, -314, 207], [-21616, 85400, 118040, -472752, -164508, 648475, 101400, -397008, -32268, 127194, 5583, -22127, -506, 1977, 19, -71], [116832, -346480, -654520, 1823774, 853469, -2245015, -501384, 1238992, 155796, -361895, -26532, 58043, 2343, -4830, -84, 163], [236352, -597312, -1272208, 3041408, 1774663, -4037823, -1126080, 2419716, 374322, -762327, -67610, 130773, 6303, -11550, -238, 411], [298880, -800320, -1775040, 4237280, 2591083, -5542347, -1684848, 3272532, 571338, -1021947, -105146, 174713, 9971, -15438, -382, 551], [164960, -444080, -938472, 2318312, 1281243, -3006939, -788976, 1754796, 256410, -540339, -45618, 91017, 4211, -7926, -158, 279], [-81776, 210816, 501772, -1122980, -811191, 1492239, 569592, -893472, -202962, 282051, 38438, -48613, -3695, 4322, 142, -155]];
    Rf_basisdens := [1, 1, 1, 1, 3904, 1952, 976, 3904, 3904, 3904, 976, 1952, 3904, 3904, 1952, 976];
    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_6003_a();"
function MakeCharacter_6003_a()
    N := 6003;
    order := 1;
    char_gens := [668, 3133, 4555];
    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_6003_a_Hecke(Kf)
    return MakeCharacter_6003_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 := 2;
    raw_aps := [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0], [1, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, -1], [2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1, 0, 0, 0], [-1, -1, 0, 0, 1, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0], [0, 2, 0, 0, -1, 1, 1, 0, -1, 1, -1, 1, -1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 1, 0, 1, 0], [2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, 0], [2, 0, 0, 0, 1, 1, -1, 0, 0, -1, 2, 1, 0, -1, 0, 0], [-2, 0, 1, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 2, 1], [2, 0, 0, 1, 1, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1], [3, 0, 0, 1, 0, 1, -1, 0, 0, 0, 2, 0, 0, 0, 1, 0], [-2, 1, 0, 0, 0, -1, 1, 2, -2, 0, 0, -1, 0, 0, 1, 1], [0, 2, 0, -2, -1, 2, -1, 0, -1, 1, 0, 1, -1, 0, 0, 1], [2, 1, 0, -1, -1, 0, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0], [2, 3, 0, 0, -2, 2, 0, 0, -1, 1, 0, 0, -1, 2, 1, 1], [2, 2, 0, 0, -1, 1, 1, 2, 0, 1, 0, 1, 0, -1, 0, 0], [1, 0, 0, 1, 0, 0, -1, 2, 0, 0, 0, 0, 0, 2, 0, 1], [2, 0, 0, 0, 1, -1, 0, 0, 0, -1, 2, -1, 0, 2, 2, 1], [0, -4, 0, 1, 3, 0, 0, 0, -1, -1, 2, 0, -1, 2, 1, 0], [4, 1, 0, -1, -1, 1, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0], [2, -1, 2, 1, 2, -1, 0, 2, -1, 0, 0, -1, -1, 2, 0, 0], [-1, 1, 0, -1, 0, -1, 0, 0, 0, 0, 2, -2, 0, 0, 1, 2], [-3, -2, 0, 1, 3, -1, 0, 2, -2, -1, 2, -1, 0, 0, -1, -1], [-1, -3, 0, -1, 2, -2, 0, 0, -1, 0, 0, 0, 1, -1, -1, -1], [-2, 2, -1, 0, 0, -1, 1, 0, -1, 0, 0, 1, -1, 0, 1, 2], [3, 2, 0, -2, -3, 1, 1, 0, 3, 1, -1, 0, 1, -2, -1, 0], [1, -1, 0, -3, 0, -1, 0, -1, -2, 0, 0, -1, 0, 0, 1, 0], [-1, 0, -1, -1, 0, 1, -1, -2, 0, 1, 0, 1, 2, 0, 0, 0], [-4, -4, -1, 0, 1, -3, -1, -2, 1, -1, 0, -1, 1, -2, 0, 0], [3, 3, -3, -1, -3, 2, 0, 0, 1, 1, -2, 0, -1, 0, 0, 1], [2, -1, 2, 2, 0, 1, -2, 2, 1, 0, 2, 1, -1, -2, 2, 0], [-4, -1, -2, 0, 1, 1, -1, -2, 1, 1, -2, 0, 1, -1, -1, -1], [-1, 4, 0, -1, 0, 0, -1, -2, -1, 2, -2, 0, -1, 0, 0, 1], [0, -1, -1, -1, -1, -2, 2, 0, 0, 1, -2, 0, 0, 0, -1, -1], [6, 0, 2, -1, -1, 0, 1, 2, 1, 1, 0, -2, 1, -1, -1, -1], [3, 2, 0, 1, 3, 1, 0, -1, -1, -1, 0, -1, 1, 0, -1, -3], [2, 3, 1, -1, 1, 0, -2, 0, -1, -1, 0, 0, 1, 0, -1, 1], [2, 0, 2, 0, 2, 0, -1, -2, 2, 0, 0, 2, 0, 2, -1, 1], [0, 0, -4, 0, -2, 0, 1, 0, 0, 0, 0, 1, 0, -1, 0, -1], [2, -1, 1, 2, 0, 3, -1, 2, -3, 0, 2, 1, -1, 0, 0, 1], [3, 0, -1, 0, 1, -1, 2, 2, -3, -2, 0, -1, 1, 2, -2, -1], [4, 1, 0, -1, -1, -1, 1, -2, 4, 1, 0, 1, 0, 2, 0, 1], [0, -8, -2, 1, -1, -1, 1, 0, 3, 1, -2, 0, 0, 2, -1, 0], [2, -1, -3, -1, -1, -1, 0, 0, 2, -1, 2, -1, 2, 0, -2, 0], [0, 3, 0, -1, -1, 1, 1, 4, -2, 1, -1, 1, 0, 0, -1, 1], [0, -6, 0, 1, 0, 2, 0, -1, 0, -2, 2, 0, 0, -2, 0, 0], [7, 2, 2, -1, -1, 3, -1, 2, -3, 0, 0, 1, -1, 2, 0, 0], [-4, 1, 3, -1, -1, 0, 2, 1, -1, 1, 2, -2, -1, 0, 1, -1], [-1, -3, -2, 2, 3, -2, 0, -2, 1, -1, 1, 0, 1, 0, -2, -1], [3, 1, 0, 1, -1, 2, -1, -2, 0, 1, 0, 2, -1, -2, -2, 0], [2, 3, 2, 3, -2, 3, 0, 2, -1, 0, 0, 0, -1, 0, 2, 0], [-2, 2, -4, 0, -2, 1, 0, -1, -3, 2, -2, -1, 1, 2, 1, 1], [-4, 4, -2, 2, -1, 2, 1, 2, -1, 1, -2, 0, -1, 0, 0, 1], [2, 2, 0, 2, 1, 3, -1, 2, 0, 1, -2, 1, -2, 2, -1, 0], [1, 2, -2, -1, -2, 1, 2, 2, -1, -1, 0, -1, -1, 0, 0, -1], [3, 3, 2, -1, 0, 1, -1, 4, -2, 0, -2, 1, 0, 2, 0, 1], [3, -5, 0, 1, 1, 0, -1, -2, 1, -1, 4, 1, 1, -3, -1, 0], [1, -1, -1, 3, 1, 0, 0, 4, -1, -1, 0, 0, -1, 0, 0, -3], [6, 1, 1, 0, -1, 0, -1, 2, 0, -2, 2, 0, -2, 2, 0, 2], [6, -1, 0, -2, 2, -2, -2, -4, 1, 0, -2, 0, 1, 2, -1, -1], [-5, -1, -2, 2, 5, -1, 1, 0, -3, -1, -1, 0, -1, 2, 0, -1], [2, 0, 0, 0, -1, 2, 1, 0, -1, -1, 2, 0, -1, -2, 0, 1], [3, -8, 0, -2, 1, -1, 1, -2, 3, -1, 0, -1, 3, -2, 0, -4], [1, -1, 2, 0, -1, 0, 2, 2, -2, -1, -2, 1, -2, 2, 3, 1], [-1, 0, 0, -1, 0, 0, 0, 2, 1, -2, 2, 0, 1, -2, 1, 0], [-1, 0, 0, -2, -2, 1, 0, 0, -3, 0, -2, 0, 1, -2, 0, -1], [3, -4, 2, -2, 3, -1, 0, -2, 2, -1, 2, -1, 2, 4, -1, -1], [-3, -1, -2, 1, 0, -3, 2, 0, 0, 0, 0, -1, 1, 0, 1, 0], [0, -3, 0, 1, 0, 0, 1, -2, 2, 0, -2, -2, 0, 0, 0, 0], [-5, 4, 0, -2, 1, 0, 1, 0, -4, -1, -2, -2, 0, -2, 1, -1], [-3, -1, -4, 1, -1, 1, 0, -4, 3, 1, -2, 0, 1, 3, 1, 2], [-6, -2, -4, 0, 1, 0, -2, -6, 1, -1, 0, 0, 1, -2, -1, 0], [0, 2, 0, -1, -3, 4, 0, -2, 1, 1, -1, 0, -1, 0, 2, 2], [3, 0, 1, 0, -1, 1, 2, 2, 3, 3, -2, 1, 1, 0, -2, 1], [5, -1, 2, 1, 1, 1, 0, 2, -2, -1, -2, -1, 0, 0, -4, -2], [3, -3, 2, 3, 5, -1, -2, 1, 3, -3, 4, 1, 1, 2, 0, 0], [4, -3, 2, -1, -2, 2, 1, 2, -2, 2, 2, 0, -2, 0, 2, 2], [4, -7, 4, 2, 2, -1, -1, 1, 1, -2, 4, 1, -1, -2, 1, -1], [-5, -4, -3, 0, -1, -1, -1, 2, -3, 1, 2, 1, -1, -2, 3, 2], [-4, 3, 0, -1, -1, 3, -1, 0, 1, 3, 0, 2, -1, -5, -1, 1], [9, 5, -2, -1, -2, 2, -2, 0, 2, 0, -2, 2, 1, 0, 0, 1], [-3, 2, 4, -1, -3, 1, 0, 0, 0, 1, -2, 1, 0, -2, 1, 1], [-4, 1, -2, -2, -1, -1, 1, 0, -1, 3, 0, 1, -1, 0, 2, 1], [-1, -5, -3, 0, 1, -4, -1, 2, 2, -1, -2, -2, 4, 2, -1, 0], [4, 1, 6, 0, -1, -1, -1, 2, 2, 1, 1, 2, 2, 2, 3, 3], [7, 5, -1, 2, -1, 3, -1, 0, -3, 3, 0, 3, -1, 0, 2, 3], [-6, 2, -4, -2, -1, 3, 1, -4, 0, 3, -5, -1, 0, 0, 0, -2], [4, 6, 2, -1, -2, 2, -2, 2, -4, 2, 0, 2, -2, -1, -1, 2], [1, 2, 4, 0, 4, 0, 0, 2, -3, 0, -2, 0, -1, 0, 1, -2], [4, -7, 3, 2, 0, 0, 1, 0, 0, 2, -4, 0, -2, 2, -1, 0], [1, 2, -2, -3, 0, -2, 5, 0, 2, 0, 0, -2, -3, 0, -2, -1], [10, -3, 4, 2, 0, 2, -1, 0, 1, -2, 4, 1, -2, 2, 0, 0], [0, 0, 0, 0, 1, -3, -1, -2, 1, -3, -4, -1, 1, 2, -3, 0], [-6, 0, 2, 0, 1, 0, 0, 2, -1, -1, 0, 0, -1, 5, 2, 0], [9, -3, 4, 0, 4, 0, 0, 2, -3, -2, 2, 0, 1, 0, -2, -3], [-1, 5, 3, 1, -2, 1, -3, 2, -4, 2, -2, 1, -2, -2, 1, 1], [0, 3, 1, 2, 3, 0, -1, -2, -6, 1, 6, 0, -4, 4, 4, 4], [0, -7, -8, 1, -1, -3, 3, -2, -1, 1, 0, 1, 1, -4, 0, 1], [3, -1, 4, 1, 0, -1, 0, 6, 0, 0, -2, -3, -2, -2, -1, 0], [5, 6, 4, 1, -1, 5, 1, 2, -3, 1, 0, 1, -1, 0, 4, -2], [1, -8, -4, -2, -1, 0, 1, -2, 0, -1, 0, 2, 0, -4, -5, -1], [6, 0, -2, 1, -2, 5, -1, -2, 2, 0, 4, -1, -2, 2, 0, 0], [-5, -6, 0, 0, 4, -2, 0, 2, -4, -2, 4, -1, 2, -2, 5, 0], [13, -4, 4, -2, 1, 0, 0, -2, 2, -1, 0, -2, 1, 2, -2, -4], [-3, 2, -2, -3, -5, 1, 1, -2, 2, 1, -4, 2, -2, -2, 0, 0], [-9, -6, -2, 0, 4, 1, 0, -2, -7, -2, 3, -3, 1, 0, -1, -3], [9, 0, 1, 0, 3, -1, -1, 0, 1, -4, -2, 1, -1, 0, -3, 2], [14, 0, 2, 1, -2, 2, -3, -5, 5, 0, -2, 2, 1, 2, 1, 1], [6, 11, 0, 0, -3, 4, 1, 2, 4, 1, -3, 1, 2, 0, -4, -2], [-5, 4, -6, -3, -1, -5, -1, -2, -3, -1, 2, -3, 1, -2, 0, 2], [-7, -5, 1, 3, 2, 3, -4, 0, 3, 0, 4, -1, -1, 0, -2, 2], [3, 3, 0, -2, -2, 0, 2, 0, 0, 2, 0, 2, 2, -2, 0, -3], [3, -5, 6, 1, 2, 4, 2, -2, -1, -1, 4, 0, -1, 0, 2, -3], [0, 10, 4, -2, -6, 3, -2, 2, -2, 2, -2, 3, -2, -2, 1, 1], [6, -12, -2, 4, 1, -3, -1, -2, 5, -1, -4, 1, 3, -1, -2, 0], [0, -7, 0, 5, 5, -4, 0, -2, 3, -1, 0, -2, 1, 2, -2, 1], [1, -1, 3, -1, -5, 6, 0, -3, 1, 1, 4, 2, -1, -2, 0, -3], [4, 0, 2, -4, -1, -1, 0, -2, -3, 1, -2, 1, -3, 2, 0, 1], [8, 4, 2, 1, -4, 1, 2, 6, -1, 2, 0, 1, -3, -1, -2, 1], [4, 1, 2, 1, 1, 4, 0, 0, -9, 0, 0, 2, -3, 2, 4, -3], [-10, 4, 2, 1, 0, -1, -2, 4, 1, 2, 0, 0, -1, -1, 0, -1], [14, -2, -4, -7, 0, -3, 2, -2, -1, 0, 4, 0, 3, -2, -1, -1], [11, 3, 6, -3, 0, 3, -2, 2, -1, 0, -2, 0, 0, 2, -1, -2], [-3, -4, -1, 0, 5, 2, -1, -4, 0, -1, 0, 0, 2, -4, -2, -5], [-4, -3, -6, 2, -2, 1, 3, 2, 2, 2, 2, 2, -3, -2, 1, 1], [-3, -9, -2, 0, 1, -4, 0, -2, -1, -1, 4, -2, 1, 2, 3, 3], [4, -5, 0, -2, 2, 2, 0, 2, -6, -2, 0, -2, 2, 1, 0, -3], [5, -1, 2, -3, 1, -4, -2, 4, 5, 3, -2, -2, 3, 2, -5, 1], [-2, 3, -4, 0, -2, 4, 2, 0, 2, 1, 0, 4, -2, -4, -1, -3], [-2, 2, -4, 0, 3, 1, -1, 2, -5, -3, 0, -3, 1, -2, 3, -2], [3, 6, 3, 1, 1, 3, 0, 2, 0, 3, 2, -1, 0, -2, 0, -1], [5, 1, 2, 3, 0, 0, -1, -3, 1, 2, -4, 0, -1, -2, 3, 2], [7, 6, -4, -2, -3, 0, 3, 0, 2, 1, -4, 1, -1, 4, 1, 3], [1, 0, 2, 0, -4, 2, 0, 4, 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0, 2, -4, 2], [3, -6, -8, -2, -3, 1, 3, 4, -1, 3, -4, 0, 1, -4, 0, -2], [22, 2, -2, -1, -5, 6, 0, 0, 4, 1, -4, 0, -2, 2, 1, 2], [-5, -6, -2, 2, 3, 1, 2, -2, -1, -1, 2, 1, 0, -8, 1, -5], [-12, 6, -2, -1, 0, -2, 2, -2, -8, -2, -5, 2, 0, 0, 3, 2], [4, 7, -4, 1, -1, 0, 3, -4, 7, 1, 4, 1, -6, 2, 1, 0], [6, 0, 4, 1, 3, 4, 0, 0, -13, 1, 0, 3, -3, 6, 1, 0], [-2, 9, 1, 0, 2, -3, 0, 2, -2, -1, -2, -1, 4, 4, 3, 0], [-3, 5, 6, -4, -5, 5, 0, -4, 1, 3, -6, 1, 1, -2, -2, -4], [10, 7, -4, 0, -1, 1, 1, 4, 2, -1, -11, 2, 0, 4, -5, -3], [-18, 3, 1, -6, 0, -4, 1, 0, 4, 0, 2, -4, 2, -8, 1, 2], [-9, -4, 5, -2, 1, -1, 1, 6, -1, -3, 0, -1, 3, -4, 3, -4], [8, 2, 2, -3, -1, 1, -2, 4, -4, 3, -4, -1, -4, 4, -6, 3], [16, -5, 2, 2, 5, -2, -2, 4, -10, -7, 4, -4, -4, 2, 2, 3], [-3, 9, 1, 3, 0, -1, -1, 6, -2, -2, 4, -3, -2, -2, 1, -1], [-2, 1, -4, 4, 2, -2, -1, 0, -7, -2, 6, -6, -3, -2, 8, 0], [-4, 0, 4, 2, 0, -2, -3, 6, 7, 2, 2, -4, -3, 0, 5, 5], [3, 8, -2, -2, -3, 2, 2, 0, -6, 5, -12, 2, -2, 3, -1, 4], [-7, -16, 0, 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9, 1, -6, 2, 5, -4, 1, 1], [1, -1, 2, -1, -1, -1, 4, 4, -3, 5, -2, 0, -3, -2, -2, 0], [8, -11, 12, 1, 3, 1, -2, 2, 4, -1, 0, -1, 0, 2, 7, -2], [2, -7, -4, 0, 4, -5, -1, 2, 11, -2, -2, -1, 3, -2, -7, -1], [4, -2, 1, 5, 3, -3, -4, -1, 0, -3, 10, -1, -2, 0, 1, 5], [2, 5, 8, -1, -3, 0, 1, 6, 0, 1, 3, 0, 0, 6, -4, 0], [-12, 1, -2, -2, 4, -3, -6, -4, -3, 2, -6, -1, 5, 0, 0, 2], [27, 16, -2, 1, -5, 4, -4, -2, -1, 1, 0, 5, -1, 0, -2, 4], [-9, 3, -6, -2, -4, -3, 6, 0, 6, 4, -4, -1, -2, 4, 1, -4], [14, 6, -4, 0, -4, 4, -3, -6, 4, 6, -2, 8, 2, 4, -1, 3], [-6, -5, -8, 2, -7, 1, -1, 2, 6, 3, -4, 7, 0, -2, 0, -1], [-2, 0, 6, 0, -1, -3, -5, 4, 3, 1, -2, -3, 5, 8, -3, 2], [-24, 10, -2, -2, 0, 4, -2, -2, 1, -2, 0, 0, -3, 3, 3, 0], [11, -5, 6, 1, 3, -1, 0, -2, 2, -1, 0, 1, 2, 2, -6, 0], [-16, 9, -4, 6, -2, 1, 4, 5, -9, 2, 4, -1, -7, -4, 4, 4], [1, -8, -8, 5, 3, -6, 3, -4, 1, 3, -5, 0, 3, -4, -2, -5], [1, -3, 0, 4, 5, -1, 3, 0, -7, -5, 8, -1, -3, -6, 1, -5], [-18, -3, -2, 3, -2, -4, -1, -4, 0, -4, -2, 0, 6, -2, 2, 2], [-10, -1, -8, 3, 5, -3, 4, -4, 6, -1, 4, 3, 2, 0, -1, -2], [-17, -1, 6, -4, 3, -1, -5, 0, 2, 3, -2, -1, 0, 2, -5, 1], [5, 9, -4, 3, -3, 1, -3, -6, 6, -1, -3, 0, 2, 0, -6, -1], [13, 8, 14, -2, 3, 1, 1, 12, -4, -1, 4, 0, 0, -2, -2, 2], [3, 12, 8, 0, 2, -3, 3, 16, -3, 0, 2, -3, 3, 0, 1, 0], [2, -7, -1, 0, 2, -3, -3, -4, -9, 0, -8, -3, 3, 0, 0, -1], [24, 2, 4, 2, -4, 5, 1, 2, 7, 2, -1, 7, -1, 4, -1, 0], [18, 1, 8, -1, 1, 3, 0, -2, 5, -1, 6, 3, -1, 0, 1, -6], [6, 10, -5, -4, 0, 5, -2, -8, 0, 2, 0, 1, -2, -2, 0, -1], [-7, -11, 6, 1, 0, 6, -5, 4, 4, -2, 5, 1, 2, -4, 2, 0], [-1, -4, 12, 1, 0, 0, -5, -2, 7, 0, 0, -2, -1, -2, 0, 1], [14, 1, 2, 2, 0, 3, -5, 4, 3, -6, 2, 1, 4, 0, -5, -5], [-4, 11, 2, -3, -2, 5, 0, 6, -6, 0, 4, 0, -4, 4, -4, -2], [4, 3, 9, -1, -3, 6, 1, 8, -6, 5, -6, 0, -2, -8, -2, -4], [28, -3, 4, 3, -8, 9, 0, -5, 7, 6, -8, 3, -1, -2, -4, -4], [-11, 11, -2, 0, 8, -7, 2, 0, -10, -4, -6, -1, 0, 4, -1, 2], [1, 6, -8, 0, -12, 3, 5, 4, 4, 8, -4, 0, -3, 0, -1, 2], [10, -3, 0, 4, -5, 0, 3, 0, 4, -3, 6, 2, -4, -3, 0, -2], [3, -11, -6, -6, 4, -8, -1, -4, 4, -4, 0, -2, 4, -2, -3, 2], [6, 7, 4, 5, 0, 5, -2, 4, -5, 4, 2, 5, -3, -2, 6, 4], [12, 10, 2, -2, -2, 2, -1, 6, 10, -2, -4, 4, -2, -2, -1, 1], [-1, -13, 1, 2, 4, 2, 0, -2, 3, -2, 4, 4, -3, 2, -1, 1], [13, -7, -6, -5, 5, -5, 5, -4, -2, 1, -4, -3, 2, 6, -3, -7], [3, 4, -8, -3, 4, -5, -2, -10, 4, -2, 2, -5, 4, -2, 0, -1], [-11, 1, 6, -1, 4, -7, 0, 0, 8, -2, 2, -1, 4, 2, -3, -4], [7, 7, -5, 1, -1, 2, -1, -6, 2, 3, -6, 2, -4, 6, 3, 0], [-2, -11, 0, 2, 4, -4, 0, -2, -2, 1, -2, -4, 6, 2, -5, -3], [8, 1, 2, -5, 3, 9, -2, 0, -4, -3, 0, 1, 0, 6, -1, -2], [-5, -6, -4, 0, 5, -8, 0, -8, 2, 1, 2, -6, 1, 0, 4, 6], [-14, -2, -3, -2, 0, -7, 1, -2, 1, -4, 4, -1, 7, -2, 7, -4], [-6, -8, -6, 0, -1, -2, 4, -2, 2, -5, 4, 0, 0, -8, -1, -2], [5, -1, 0, 3, 9, -7, 3, 0, 1, -5, -2, -1, 5, 10, -3, -1], [10, 5, 6, 0, -1, 3, -5, 6, -1, 3, -4, -1, 1, 6, -4, 3], [-9, -11, 4, -3, 12, -6, 0, 1, 0, -4, 4, -2, 0, 4, -4, -5], [23, 3, 10, 3, -1, 5, -2, 2, -1, -3, 0, 5, -3, 2, 0, 4], [7, 15, 0, -5, -1, 0, 5, -2, -3, 6, -8, -2, -7, -2, -2, -2], [-4, 3, -8, 3, 8, -3, 1, 0, -3, -2, 0, -1, 3, 6, -1, -3], [-11, 1, -6, -3, -5, -1, 6, 6, 7, 3, 0, -3, 1, 0, 0, 0], [3, -6, 7, -3, 2, -9, -1, 0, -1, -4, 2, -1, -1, -2, 2, 0], [-7, 10, 2, -6, 2, -3, 3, 2, -12, 2, 4, -1, 0, 0, 1, 0], [9, -16, 10, 6, 6, 1, -3, 6, 2, -2, 5, 3, -2, -2, 0, -2], [5, 2, 6, 1, 6, -8, 3, 6, -5, -4, 2, -4, 0, 4, 4, -1], [14, 3, 0, 0, 3, 3, -5, 0, -4, 1, 6, -3, 3, 6, -4, 1], [19, 18, 7, -5, -11, 14, -1, 0, 3, 3, -2, 10, -1, 2, 0, 1], [-18, 5, -8, -3, -6, -10, 2, 0, 3, 4, -10, -6, 0, -2, 3, 1], [-1, 13, 2, 4, 0, 6, -4, 8, -7, 2, 0, 2, -3, 9, 5, 7], [-16, 1, -2, 1, -1, 1, -2, 2, -4, -1, 8, -1, -2, -4, -1, -2], [-13, 3, -13, 5, -2, 1, -4, -2, -10, -4, 4, 1, -4, 6, 6, 6], [9, 11, -1, -2, -13, 8, -3, -8, 6, 5, 2, 2, -2, 6, 7, 6], [5, -2, -10, -2, -4, 0, 0, -10, 0, -4, 2, 4, 2, -8, -3, -2], [9, 18, 6, 0, -3, 10, -2, 4, 1, 5, -4, 1, -3, 4, 0, 2], [-2, -2, 0, 2, 1, 3, 2, 4, -3, -5, 4, 5, -7, -4, 0, 1], [-9, 1, 4, -10, -1, 6, 2, 2, -8, 1, -2, 0, -2, 2, 3, 1], [6, -9, 2, 9, 6, 4, 0, -4, -1, -2, 2, -2, 3, -10, -1, -5], [0, 15, -2, -3, -4, 2, 4, 8, 2, 4, 2, 2, 0, 0, -3, -3], [-12, 6, -8, 4, -2, -1, -5, -10, 1, -4, 6, -1, 0, 0, 6, 8], [0, -4, 4, 5, 3, -7, 1, 2, -6, -3, 2, -3, 2, -8, 3, -6], [21, 9, -2, -3, -3, 7, -4, -6, -11, 1, 0, 3, -6, -4, -2, 2], [20, -3, 2, 7, 7, 4, -1, 2, -13, -4, 2, 4, -5, 2, 5, -2], [-2, 2, 2, -3, 1, -6, -2, -4, -3, -5, 2, -2, 4, 4, 1, -2], [1, 5, 8, -2, 1, 4, 0, 6, -6, 3, -7, 4, -2, 4, 2, 1], [9, -11, -6, 1, 7, -9, -2, -2, 5, -5, 0, -1, 3, 2, -8, 0], [2, 7, 2, 6, -4, 9, 1, 8, 5, 4, 4, 7, -3, -6, 3, 1], [-6, -14, 0, 3, 5, -6, 6, 0, -3, -4, 8, 0, -1, -4, 1, -6], [2, -9, 2, -1, -3, -6, 6, -2, 4, 1, -4, -2, 3, -10, -2, -9], [23, -22, -2, 6, 11, -2, -1, -2, 0, -3, 6, -1, 2, -2, -1, -7], [-5, 1, 3, -2, 0, 8, 1, -2, -11, 2, -6, 0, -5, 2, -2, -4], [-5, 9, -1, 5, 3, 0, -7, -1, 2, -1, 10, 0, 0, 0, 5, 8], [7, 11, 4, -5, -4, 2, -3, -4, 1, 3, -2, 2, 5, -4, 3, 4], [-15, -8, -6, 2, 3, 1, 1, -4, 1, 3, 8, 1, 0, -2, 0, 2], [7, 1, 0, 12, 2, -4, -1, -4, -4, 0, -3, -4, -2, 6, 6, 2], [16, -9, 0, -2, 0, -5, 4, 3, -6, -4, 0, -5, 4, -2, -2, -4], [-17, -1, 0, 0, 2, 1, -4, -8, 8, 2, 2, 2, -1, -6, 1, -2], [0, 1, -4, 1, 0, -3, 1, 4, -9, 2, 2, -1, 3, -8, -5, -5], [-2, 4, -6, 1, 0, -6, 1, -7, 1, 2, 2, 2, 1, 6, 3, 1], [-17, -6, -9, 4, 0, 2, -5, -1, -1, 0, 12, 0, 3, 0, 3, 5], [17, -3, 6, -8, 2, 0, 2, 0, -3, 4, 2, 4, -3, -2, -2, 3], [30, -13, 4, 5, 3, -2, -3, -2, 10, -1, -4, 0, 4, -2, -1, -4], [3, -5, -6, -2, 0, -3, 3, -2, -7, -6, -2, -1, -1, 6, 4, 3], [-4, 8, 0, 1, -5, 4, -6, 0, -6, -1, -8, 0, 2, -2, 1, -2], [6, 15, 4, -7, -6, 5, -5, 3, -3, 2, -6, 5, 5, -2, -1, -1], [-19, 2, -6, 2, -6, 2, 3, 6, -9, 0, -2, -2, -5, -2, 4, 1], [-4, -1, -6, 3, -1, 3, 1, -2, 6, 1, -6, 5, 0, 2, 2, -1], [-23, -1, 2, 3, -4, 4, -2, 2, -7, 0, 8, 8, -2, 0, 2, 1], [-3, -17, -8, 3, 0, -6, -1, 0, 1, -2, -1, -5, 3, 2, -4, 2], [7, -11, -10, -3, -6, 3, -1, -4, -5, 2, -2, 7, -1, 4, 0, 3], [14, -3, 3, -1, 1, -4, 0, -6, 7, -5, -2, -2, 3, 2, 1, 1], [-14, 13, -1, 0, -1, 7, 3, 2, -8, -3, 2, 1, -6, 4, -1, 3], [-18, 3, -8, -4, 1, -5, 1, -2, -2, 1, 2, -8, -2, -6, 2, 3], [9, 2, 0, 1, 1, 3, 4, 10, 3, -3, -2, 1, -9, 1, 0, 3], [-12, 7, -3, 5, 5, -4, 4, 3, -5, -3, 0, -6, 3, 2, 1, -5], [20, -4, 6, 2, -2, 4, -2, 0, 19, 2, -4, 2, 5, -2, -4, -2], [-19, 16, 2, 1, -4, 5, 5, 10, -9, 2, 0, 1, -2, 10, 9, 2], [3, 2, -4, -6, 1, -2, -4, -8, -4, -1, -6, -5, -2, 5, 1, 4], [-5, -2, -6, -12, -1, -4, 2, -8, -6, -1, -6, -4, -1, 0, -2, -4], [-27, -7, 7, -1, 10, -1, -3, -6, -3, 0, 2, -1, -1, -2, -1, -9], [-9, 2, -10, 0, 0, -3, 3, -6, 1, -2, -2, 0, -3, 0, -1, -2], [9, 12, 8, -1, -4, 5, -1, 4, 3, 2, 4, 1, -1, -9, -2, 0], [-29, -16, -6, 1, 1, -3, -4, 1, 3, -1, 6, 1, 9, -10, 1, -5], [12, -5, -10, -3, 6, -2, 1, -2, 7, 5, 2, -2, 1, 4, -6, -4], [-11, 8, -12, -3, -6, 4, 7, 4, -6, 10, 1, 0, -2, -2, -3, -3], [-15, 4, -13, -3, -5, -6, -2, -5, 8, -1, 4, -2, 0, -8, 3, 0], [3, -15, -2, -2, 10, -10, 0, 0, 5, -4, 2, -10, 5, -4, -4, -5], [26, 13, 0, -2, -2, 10, -1, 4, -7, 6, -8, 2, 1, 6, -8, -4], [-11, -8, -4, 8, 9, -10, -2, 6, -7, -3, 2, -3, -1, 0, -2, 4], [2, -3, -8, 3, -5, -3, 1, -4, -5, 3, -4, -2, -3, -9, 7, -1], [12, -10, -6, 4, 6, 0, 2, -8, -3, 4, -2, 5, -4, 0, 0, 0], [-3, 3, 10, 1, -5, -2, -1, 2, 7, 1, -8, 0, -3, 0, 2, 2], [-20, -18, 6, -2, 5, -4, -3, 6, 3, 3, 4, 0, 5, -12, -4, -7], [1, 2, 6, 2, 0, -2, -3, -4, 3, 7, -6, 4, -1, 0, 2, -1], [-26, -7, 2, 4, 7, 3, 2, 14, 2, -1, 6, -4, -1, 4, -1, -4], [5, -10, 8, 5, -1, 6, -2, -2, -2, -1, -2, 0, 4, -2, 4, -2], [-6, -4, -14, -3, -3, 3, 0, -6, -3, -1, 3, 5, 7, 2, 1, 1], [-8, 11, 8, -2, 1, 2, 0, 0, -6, -7, 8, 1, -2, -2, 8, 1], [9, -5, 10, -1, -1, 2, -3, -2, 4, 1, 0, 2, -3, 2, 2, 4], [-1, 15, 4, -2, 4, 9, -1, 2, -9, 2, 8, -1, -5, -2, -2, -3], [1, -3, 2, 1, 4, 3, -6, 1, 0, 0, -6, -3, 4, -4, -5, 0], [24, -17, 4, 0, -6, 1, -1, -3, 14, 2, -2, -1, 6, -8, -7, -5], [2, -11, -2, 4, 11, -5, 2, 2, -5, -3, -6, 4, 1, -4, -7, -4], [-2, 0, 4, -2, 7, 1, 5, 0, -5, -3, 14, 1, -3, -1, 2, -2], [7, -5, -4, -6, 5, 0, 0, 0, 1, 1, -8, -6, 7, 8, -5, -5], [0, 8, 1, -5, 0, 2, 0, 4, -1, -2, 0, 0, 3, -8, -3, -2], [23, -10, -10, -1, 1, -1, 2, -14, 13, -3, 4, 3, 1, 0, -1, -3], [-6, 0, 8, 7, 4, 11, -3, 12, -7, 1, 4, 1, -1, -2, -4, -8], [6, 13, 6, -1, 1, 3, -3, -8, -5, 7, -8, 0, -5, 2, -2, -3], [20, 6, -1, 0, -4, 1, 1, 7, 2, 8, -6, 1, -2, -4, 1, -4], [2, 5, 3, -7, -2, 2, 0, -14, 1, 3, -2, -2, -3, 0, 4, -1], [10, -4, -2, -3, -1, 3, -1, 2, 11, -5, 8, 1, 1, 4, -7, 2], [-22, -23, 0, 3, 4, -5, -1, -8, 7, 2, 2, -1, 1, -4, 5, -1], [-21, -5, -6, -2, -1, -4, -2, -8, -5, -1, -9, -4, 1, -4, 4, 1], [15, 1, 0, -1, 4, -6, 0, 6, -2, 0, 0, -3, 8, 1, -3, -1], [30, 15, 4, -4, -2, 9, -4, 2, -11, 2, -2, -3, -3, 4, -6, -4], [-12, 3, 10, 1, 1, 7, -5, -2, -8, -3, -2, 5, 2, -2, 4, 1], [18, 9, 3, 5, 0, 3, 1, 0, 0, 6, 2, 1, -8, 0, 0, 1], [-13, -4, -2, -3, -1, 2, 0, -8, 6, 1, 6, 3, 0, -2, 2, 2], [-12, 4, 10, 4, 4, -2, -4, 2, 9, -6, 4, -2, 1, 1, 1, -2], [0, -4, -2, 1, -4, 2, -1, 2, 18, 2, 4, 4, 7, -8, -5, 3], [-11, 15, 8, 7, 5, 1, -5, 6, -10, -1, 1, -4, -6, 4, 2, 7], [-23, 10, -4, 0, -4, 2, 3, -4, 3, 4, -6, 4, -5, 6, -2, 1], [0, 19, -2, -4, -4, -3, -2, 4, 3, 0, 0, -1, 5, 2, 0, 4], [-22, -14, 3, 1, 0, -2, 8, 10, -4, 1, 6, -4, -4, -8, 1, 0], [17, 9, 0, 4, 7, -3, -7, 4, 3, -5, 12, 1, 3, 6, 1, 11], [1, 5, 4, -4, -5, 5, -2, -2, -5, 3, 4, 5, -7, 2, 4, 10], [-3, 2, 10, 3, 2, 2, 1, 8, 1, -4, 2, 0, 1, 0, 4, -9], [-17, -4, -8, 6, 0, -5, 3, -4, 11, -4, 2, -1, 7, -8, 1, -2], [5, -4, 2, -2, -3, 4, -3, 4, 8, -1, 6, 0, 10, -6, -7, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6003_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6003_2_a_r();".
// 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_6003_2_a_r(:prec:=16)
    chi := MakeCharacter_6003_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_6003_2_a_r();".
// 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_6003_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6003_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-192, 1088, -400, -6032, 6405, 8288, -9405, -5124, 5778, 1665, -1835, -295, 316, 27, -28, -1, 1]>,<5,R![3072, 47632, 206600, 271940, -82804, -313392, -45374, 127665, 32879, -25446, -7691, 2674, 862, -142, -47, 3, 1]>],Snew);
    return Vf;
end function;