
// Make newform 740.2.d.a in Magma, downloaded from the LMFDB on 23 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_740_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_740_d_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_740_2_d_a();".
// 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_740_2_d_a();".
// 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 := [144, 0, 2592, 0, 9408, 0, 14736, 0, 12196, 0, 5841, 0, 1674, 0, 283, 0, 26, 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, 0, 0, 0, 0, 0], [0, -252, 0, 714, 0, 2595, 0, 1475, 0, -273, 0, -459, 0, -148, 0, -20, 0, -1], [-48, 0, -912, 0, -1800, 0, -620, 0, 591, 0, 510, 0, 151, 0, 20, 0, 1, 0], [-144, 0, -1392, 0, -2244, 0, -752, 0, 579, 0, 510, 0, 151, 0, 20, 0, 1, 0], [0, 1368, 0, 4920, 0, 8724, 0, 8794, 0, 4917, 0, 1554, 0, 277, 0, 26, 0, 1], [0, -144, 0, 2184, 0, 7338, 0, 8524, 0, 4899, 0, 1554, 0, 277, 0, 26, 0, 1], [96, 0, 852, 0, 2778, 0, 3931, 0, 2742, 0, 1032, 0, 214, 0, 23, 0, 1, 0], [60, 0, 1092, 0, 3234, 0, 4162, 0, 2787, 0, 1035, 0, 214, 0, 23, 0, 1, 0], [0, -72, 0, -258, 0, 798, 0, 3251, 0, 3327, 0, 1542, 0, 365, 0, 43, 0, 2], [-8, 0, 208, 0, 1244, 0, 2414, 0, 2025, 0, 858, 0, 193, 0, 22, 0, 1, 0], [720, 144, 9864, -1416, 30204, -3090, 38988, -1592, 25794, 264, 9450, 459, 1935, 148, 207, 20, 9, 1], [-720, 144, -9864, -1416, -30204, -3090, -38988, -1592, -25794, 264, -9450, 459, -1935, 148, -207, 20, -9, 1], [-24, 0, -804, 0, -2829, 0, -3958, 0, -2745, 0, -1032, 0, -214, 0, -23, 0, -1, 0], [0, 384, 0, 2940, 0, 6412, 0, 6622, 0, 3771, 0, 1246, 0, 237, 0, 24, 0, 1], [0, -1152, 0, 1824, 0, 15396, 0, 24442, 0, 17661, 0, 6804, 0, 1441, 0, 158, 0, 7], [0, 1440, 0, 8376, 0, 20832, 0, 26368, 0, 17967, 0, 6822, 0, 1441, 0, 158, 0, 7], [504, 432, 9792, -2400, 30204, -11700, 38988, -16582, 25794, -11289, 9450, -4179, 1935, -859, 207, -92, 9, -4], [504, -432, 9792, 2400, 30204, 11700, 38988, 16582, 25794, 11289, 9450, 4179, 1935, 859, 207, 92, 9, 4]];
    Rf_basisdens := [1, 36, 24, 24, 72, 72, 12, 12, 36, 8, 72, 72, 6, 24, 72, 72, 72, 72];
    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_740_d();"
function MakeCharacter_740_d()
    N := 740;
    order := 2;
    char_gens := [371, 297, 261];
    v := [2, 1, 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_740_d_Hecke();"
function MakeCharacter_740_d_Hecke( : Kf := false)
    N := 740;
    order := 2;
    char_gens := [371, 297, 261];
    char_values := [[1, 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, 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]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, 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, 0, 0, 0, 0, 0, 0, 0, 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, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, -1, 0, 0, 1, 0, 1, 1, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 2, 0, 0, 1, 0, 0, 0, 0, -1, -1, -1, 0, 0], [1, 0, 1, -1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1], [2, 0, 2, 1, 0, 0, 1, -1, 0, -3, -1, 1, 0, 0, 0, 0, 2, 2], [1, 0, 0, 1, 0, 0, 0, -1, 0, 0, -1, 1, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 1, 0, 0, -1, -1, 1, 0, 0, 0, 0, 1, 1], [0, 1, 0, 0, -1, -1, 0, 0, 0, 0, -2, -2, 0, 1, -2, 1, -1, 1], [0, 2, 0, 0, 0, 2, 0, 0, 1, 0, -1, -1, 0, 0, -2, 2, -1, 1], [0, 0, 0, 0, -2, -2, 0, 0, 1, 0, 0, 0, 0, -1, 0, 1, 1, -1], [3, 0, 2, -2, 0, 0, 1, -2, 0, -1, -2, 2, 0, 0, 0, 0, 3, 3], [2, 0, 2, 1, 0, 0, 1, -1, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -2, 2, 1, 1, -1], [-4, 0, -1, 1, 0, 0, 0, 2, 0, 0, 1, -1, 3, 0, 0, 0, 1, 1], [0, 0, 0, 0, 2, -6, 0, 0, -3, 0, 2, 2, 0, 1, 2, -1, 1, -1], [-7, 0, -4, 1, 0, 0, -4, 3, 0, 2, 4, -4, 0, 0, 0, 0, -3, -3], [0, 2, 0, 0, -2, 0, 0, 0, 2, 0, -2, -2, 0, 1, -4, 2, -2, 2], [-2, 0, 0, 1, 0, 0, 1, 3, 0, 1, 2, -2, 2, 0, 0, 0, -2, -2], [0, 0, 0, 0, 4, 0, 0, 0, -2, 0, 2, 2, 0, 0, 1, -1, 1, -1], [-3, 0, -2, 4, 0, 0, -3, 1, 0, 2, 4, -4, -2, 0, 0, 0, -3, -3], [0, -1, 0, 0, -1, -3, 0, 0, 0, 0, 0, 0, 0, -3, 0, -1, 3, -3], [0, -2, 0, 0, 4, -4, 0, 0, -2, 0, 1, 1, 0, 1, 0, -1, -2, 2], [-2, 0, -4, 1, 0, 0, -1, 1, 0, 1, 0, 0, -2, 0, 0, 0, -4, -4], [0, -2, 0, 0, -1, -6, 0, 0, -1, 0, 1, 1, 0, -3, -1, -3, 2, -2], [0, 4, 0, 0, -4, 4, 0, 0, 2, 0, 1, 1, 0, 1, -2, 2, 3, -3], [3, 0, -1, -2, 0, 0, 2, 1, 0, 0, -1, 1, -2, 0, 0, 0, 0, 0], [0, 4, 0, 0, -2, 8, 0, 0, 4, 0, 3, 3, 0, -2, 0, -2, 3, -3], [4, 0, 0, 2, 0, 0, 2, -2, 0, -4, -1, 1, 0, 0, 0, 0, 1, 1], [-7, 0, -2, 0, 0, 0, -3, 3, 0, 4, 4, -4, 4, 0, 0, 0, -1, -1], [2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, -1, 0, 0, 0, 0, -3, -3], [0, -2, 0, 0, -2, 4, 0, 0, -1, 0, -1, -1, 0, 1, 0, 3, 2, -2], [0, 2, 0, 0, -2, 0, 0, 0, 4, 0, 3, 3, 0, -4, -1, -3, 3, -3], [0, -6, 0, 0, -2, 4, 0, 0, 0, 0, -3, -3, 0, 0, 1, -3, -1, 1], [0, -2, 0, 0, 2, -8, 0, 0, 1, 0, -1, -1, 0, -1, 0, -1, 0, 0], [-1, 0, 3, 2, 0, 0, -2, -1, 0, -4, -2, 2, 2, 0, 0, 0, 4, 4], [-7, 0, 0, -2, 0, 0, -3, 1, 0, 4, 1, -1, 6, 0, 0, 0, 0, 0], [5, 0, 2, -4, 0, 0, -1, -2, 0, -1, 0, 0, 4, 0, 0, 0, 3, 3], [0, 1, 0, 0, -5, 1, 0, 0, 0, 0, -2, -2, 0, -1, -1, 2, 4, -4], [0, 0, 0, 0, 2, 8, 0, 0, -3, 0, 0, 0, 0, -3, 2, 3, 1, -1], [5, 0, 1, -1, 0, 0, 1, -2, 0, 5, 1, -1, 0, 0, 0, 0, -4, -4], [-4, 0, -5, 3, 0, 0, -2, 2, 0, 0, 5, -5, -5, 0, 0, 0, -5, -5], [0, 2, 0, 0, 2, -6, 0, 0, 0, 0, -3, -3, 0, 1, -2, 2, 1, -1], [0, 5, 0, 0, -3, 5, 0, 0, 2, 0, 1, 1, 0, -1, -2, -1, -1, 1], [1, 0, 3, 6, 0, 0, 1, -2, 0, -1, -2, 2, -2, 0, 0, 0, 0, 0], [0, 2, 0, 0, -2, -2, 0, 0, 2, 0, 5, 5, 0, -2, 4, -2, 3, -3], [5, 0, 3, -1, 0, 0, 5, -2, 0, 1, -5, 5, 2, 0, 0, 0, 2, 2], [4, 0, 4, -3, 0, 0, 1, -3, 0, -5, -1, 1, -2, 0, 0, 0, 2, 2], [-3, 0, 1, -4, 0, 0, 2, -1, 0, 4, -1, 1, 2, 0, 0, 0, 2, 2], [0, -3, 0, 0, -2, 9, 0, 0, -1, 0, 2, 2, 0, 2, 4, 1, 0, 0], [0, -2, 0, 0, -2, -2, 0, 0, -2, 0, -2, -2, 0, 1, 0, 4, 0, 0], [3, 0, -3, 5, 0, 0, 1, -3, 0, -2, -3, 3, 1, 0, 0, 0, 2, 2], [2, 0, 3, -1, 0, 0, -2, -2, 0, -6, -1, 1, -1, 0, 0, 0, 1, 1], [0, 1, 0, 0, 2, 5, 0, 0, -1, 0, -2, -2, 0, 4, 2, 1, 0, 0], [2, 0, -2, -1, 0, 0, 1, -1, 0, -1, 1, -1, -2, 0, 0, 0, 2, 2], [0, -3, 0, 0, 1, 5, 0, 0, 0, 0, -1, -1, 0, -1, 4, 3, -1, 1], [0, 6, 0, 0, -2, -8, 0, 0, 0, 0, 0, 0, 0, 6, -4, 0, -4, 4], [0, 2, 0, 0, 0, -2, 0, 0, -1, 0, 1, 1, 0, -2, 2, 4, 1, -1], [-9, 0, -6, -2, 0, 0, -1, 4, 0, -1, 1, -1, -2, 0, 0, 0, -1, -1], [0, -1, 0, 0, -3, 1, 0, 0, 6, 0, -2, -2, 0, 1, -8, 1, -3, 3], [0, 4, 0, 0, -2, 6, 0, 0, 6, 0, 1, 1, 0, 0, -4, -2, -1, 1], [5, 0, 3, -2, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, -2, -2], [0, 2, 0, 0, 0, -2, 0, 0, 3, 0, 5, 5, 0, -3, -2, -3, 2, -2], [0, 2, 0, 0, 4, -2, 0, 0, 0, 0, 0, 0, 0, -2, 3, -1, -3, 3], [15, 0, 3, -5, 0, 0, 9, -3, 0, -2, -9, 9, -3, 0, 0, 0, 2, 2], [0, 3, 0, 0, 1, -9, 0, 0, -8, 0, 0, 0, 0, 1, 1, 4, 2, -2], [-4, 0, -3, -1, 0, 0, 0, 4, 0, 4, 6, -6, -3, 0, 0, 0, -8, -8], [0, -4, 0, 0, -2, -4, 0, 0, 1, 0, -6, -6, 0, -2, -6, 3, -1, 1], [0, 0, 0, 0, 0, 6, 0, 0, -3, 0, 1, 1, 0, -3, 2, -5, 0, 0], [-2, 0, 1, -3, 0, 0, 0, 0, 0, -2, -6, 6, 3, 0, 0, 0, 4, 4], [0, 0, 0, 0, -4, 8, 0, 0, 8, 0, 0, 0, 0, -4, -1, -1, 1, -1], [-2, 0, -6, 6, 0, 0, -4, -2, 0, 0, 2, -2, -2, 0, 0, 0, 0, 0], [0, -8, 0, 0, 4, -2, 0, 0, -5, 0, 2, 2, 0, 1, 10, -3, 1, -1], [4, 0, -2, -2, 0, 0, 2, 2, 0, 4, -1, 1, 0, 0, 0, 0, 0, 0], [-6, 0, 2, -4, 0, 0, 2, 6, 0, 4, 0, 0, 8, 0, 0, 0, 2, 2], [8, 0, -1, 3, 0, 0, 4, -6, 0, 4, -3, 3, -3, 0, 0, 0, 1, 1], [-2, 0, 0, 2, 0, 0, 4, 0, 0, 2, -1, 1, -4, 0, 0, 0, 0, 0], [-3, 0, 6, 3, 0, 0, 0, -1, 0, 0, 2, -2, 0, 0, 0, 0, -3, -3], [0, -2, 0, 0, 4, 0, 0, 0, -9, 0, -4, -4, 0, 5, 4, 7, -3, 3], [5, 0, -3, -1, 0, 0, 5, 0, 0, 1, -3, 3, 6, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 8, 0, 0, -7, 0, -3, -3, 0, 2, -2, 8, -1, 1], [4, 0, 2, -5, 0, 0, -1, 1, 0, -5, -4, 4, -2, 0, 0, 0, 2, 2], [0, 7, 0, 0, -3, 1, 0, 0, 6, 0, 0, 0, 0, 1, -4, 1, -1, 1], [-6, 0, -4, 2, 0, 0, -8, 4, 0, 0, 1, -1, 4, 0, 0, 0, 4, 4], [0, 1, 0, 0, 3, 1, 0, 0, -2, 0, -3, -3, 0, 1, 2, -1, 1, -1], [0, 5, 0, 0, -7, 1, 0, 0, 2, 0, -3, -3, 0, 5, -4, 5, 1, -1], [-3, 0, 1, 6, 0, 0, 0, -3, 0, -2, -4, 4, 0, 0, 0, 0, 4, 4], [0, -3, 0, 0, 5, -17, 0, 0, 0, 0, -2, -2, 0, 3, 4, -1, 1, -1], [2, 0, 6, 0, 0, 0, -4, -4, 0, -2, -2, 2, 2, 0, 0, 0, 6, 6], [9, 0, 0, -1, 0, 0, 2, -7, 0, 0, -1, 1, -2, 0, 0, 0, -1, -1], [0, 7, 0, 0, 1, -3, 0, 0, 8, 0, -6, -6, 0, 3, -8, 3, -1, 1], [6, 0, -3, 1, 0, 0, 4, -2, 0, 2, 4, -4, -1, 0, 0, 0, -4, -4], [-7, 0, -4, 6, 0, 0, -7, -3, 0, 0, 5, -5, -2, 0, 0, 0, 0, 0], [0, 1, 0, 0, -1, 15, 0, 0, 6, 0, -2, -2, 0, 1, 0, 1, 1, -1], [-2, 0, 2, 7, 0, 0, 1, 1, 0, -1, 1, -1, 2, 0, 0, 0, 0, 0], [0, -1, 0, 0, 3, -13, 0, 0, -2, 0, -3, -3, 0, 5, -2, 1, -3, 3], [0, 1, 0, 0, -6, 13, 0, 0, 5, 0, 0, 0, 0, 2, -6, -1, 6, -6], [0, -4, 0, 0, 12, 2, 0, 0, -8, 0, 5, 5, 0, 0, 9, -1, -3, 3], [12, 0, 2, -5, 0, 0, 7, -3, 0, -5, -5, 5, 0, 0, 0, 0, 4, 4], [-10, 0, -3, 1, 0, 0, 0, 0, 0, 4, -5, 5, 1, 0, 0, 0, -5, -5], [0, 0, 0, 0, 3, 2, 0, 0, -3, 0, -4, -4, 0, -7, 3, 1, -2, 2], [0, 0, 0, 0, -2, -2, 0, 0, 2, 0, -9, -9, 0, 2, -7, 9, -3, 3], [0, -4, 0, 0, 2, -8, 0, 0, 3, 0, 2, 2, 0, 1, 4, -5, 1, -1], [-6, 0, -3, -7, 0, 0, 0, 6, 0, 4, -2, 2, 3, 0, 0, 0, -4, -4], [11, 0, 1, 5, 0, 0, 5, -5, 0, -8, -11, 11, -3, 0, 0, 0, 6, 6], [0, -5, 0, 0, -3, -11, 0, 0, -4, 0, 4, 4, 0, 5, 4, -3, 5, -5], [0, 2, 0, 0, -6, 6, 0, 0, -5, 0, 1, 1, 0, -1, -2, -1, 4, -4], [0, 4, 0, 0, 6, -2, 0, 0, -5, 0, -7, -7, 0, 9, -8, 1, -6, 6], [2, 0, -1, 1, 0, 0, 8, 0, 0, 2, -2, 2, -3, 0, 0, 0, -6, -6], [-13, 0, -5, 2, 0, 0, -4, 9, 0, 10, 9, -9, 8, 0, 0, 0, -6, -6], [1, 0, 2, -4, 0, 0, 1, 1, 0, 6, 2, -2, 0, 0, 0, 0, -9, -9], [0, -8, 0, 0, 4, 8, 0, 0, -6, 0, 0, 0, 0, 6, 9, -5, -1, 1], [0, -2, 0, 0, 0, 8, 0, 0, 4, 0, -1, -1, 0, 2, 1, -5, 3, -3], [0, 3, 0, 0, 5, -7, 0, 0, 2, 0, 5, 5, 0, -1, -1, 2, 0, 0], [-6, 0, -1, 1, 0, 0, -6, 6, 0, -2, 7, -7, 1, 0, 0, 0, -1, -1], [10, 0, 2, -1, 0, 0, 1, -3, 0, 1, -4, 4, 2, 0, 0, 0, 4, 4], [0, 4, 0, 0, -4, 16, 0, 0, 1, 0, 2, 2, 0, 3, -2, 1, 1, -1], [0, -2, 0, 0, -2, 20, 0, 0, 1, 0, 1, 1, 0, -5, 4, 1, 0, 0], [0, 0, 0, 0, -2, 18, 0, 0, 6, 0, 7, 7, 0, -2, 1, -5, 5, -5], [-2, 0, 0, -4, 0, 0, -4, 4, 0, -2, 8, -8, 0, 0, 0, 0, -6, -6], [8, 0, 6, 4, 0, 0, 0, 6, 0, -2, 0, 0, 0, 0, 0, 0, -4, -4], [0, 0, 0, -8, 0, 0, 0, -4, 0, -6, -1, 1, -8, 0, 0, 0, 6, 6], [-2, 0, -3, 5, 0, 0, -4, 2, 0, 6, -1, 1, 5, 0, 0, 0, -5, -5], [0, 4, 0, 0, -2, 2, 0, 0, 4, 0, 7, 7, 0, 6, 5, -5, 7, -7], [0, 2, 0, 0, 6, 8, 0, 0, -1, 0, 6, 6, 0, 3, 4, 3, 3, -3], [-2, 0, 1, 5, 0, 0, 0, 8, 0, 6, 6, -6, 5, 0, 0, 0, -8, -8], [0, -4, 0, 0, -6, 4, 0, 0, -5, 0, -7, -7, 0, 2, 0, 2, 1, -1], [10, 0, -3, 3, 0, 0, -4, 0, 0, 2, 2, -2, 1, 0, 0, 0, -4, -4], [0, 7, 0, 0, 1, -7, 0, 0, 0, 0, 3, 3, 0, -3, -9, 4, -4, 4], [-14, 0, -3, -3, 0, 0, 4, 6, 0, 2, 2, -2, 3, 0, 0, 0, -2, -2], [-4, 0, -4, -5, 0, 0, -3, 3, 0, 11, 9, -9, 4, 0, 0, 0, -4, -4], [0, -4, 0, 0, -2, 6, 0, 0, 3, 0, -1, -1, 0, -5, -2, 5, 2, -2], [0, -6, 0, 0, -4, -8, 0, 0, 5, 0, -2, -2, 0, -18, -2, 0, 0, 0], [0, 6, 0, 0, 1, 14, 0, 0, 7, 0, -1, -1, 0, 3, -5, 1, -4, 4], [12, 0, 8, -1, 0, 0, 9, -9, 0, -5, -9, 9, -4, 0, 0, 0, 2, 2], [8, 0, 11, -3, 0, 0, 10, -10, 0, -12, -13, 13, 1, 0, 0, 0, 13, 13], [14, 0, 7, 1, 0, 0, 6, 2, 0, 0, -5, 5, -1, 0, 0, 0, -3, -3], [0, -4, 0, 0, 4, 2, 0, 0, -4, 0, 4, 4, 0, -1, -2, -1, 1, -1], [0, -6, 0, 0, 0, 2, 0, 0, -10, 0, -4, -4, 0, -2, 3, 1, 3, -3], [0, 0, 6, 4, 0, 0, -2, -2, 0, 2, -2, 2, 4, 0, 0, 0, 4, 4], [-4, 0, 2, 8, 0, 0, 2, -2, 0, -2, 4, -4, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -8, 0, 0, -4, 0, 0, 0, 0, 6, -1, 7, -9, 9], [0, -4, 0, 0, 4, 6, 0, 0, -4, 0, -5, -5, 0, -2, 1, -5, -1, 1], [-9, 0, -2, 2, 0, 0, -5, 0, 0, 3, 3, -3, -6, 0, 0, 0, -5, -5], [0, 0, 0, 0, 2, 6, 0, 0, -3, 0, -5, -5, 0, 2, -4, 1, 2, -2], [0, -8, 0, 0, 0, -4, 0, 0, 6, 0, -6, -6, 0, 0, -4, -4, -6, 6], [-11, 0, 7, 7, 0, 0, 3, -5, 0, 0, 0, 0, 1, 0, 0, 0, -3, -3], [0, 13, 0, 0, -3, 21, 0, 0, 6, 0, -1, -1, 0, 3, -8, 1, 1, -1], [0, 6, 0, 0, -6, 6, 0, 0, 7, 0, 4, 4, 0, -12, -6, 2, 6, -6], [0, 4, 0, 0, 6, -8, 0, 0, 0, 0, -6, -6, 0, 10, -3, 3, -3, 3], [-7, 0, -8, -6, 0, 0, -3, 4, 0, 3, 3, -3, -4, 0, 0, 0, -3, -3], [-19, 0, -6, 1, 0, 0, -6, 13, 0, 10, 10, -10, -4, 0, 0, 0, -13, -13], [-7, 0, -10, 9, 0, 0, -3, 2, 0, 1, -3, 3, 5, 0, 0, 0, -1, -1], [0, 0, 0, 0, -2, -2, 0, 0, -3, 0, 0, 0, 0, -11, 2, 3, -5, 5], [-15, 0, 9, 3, 0, 0, -3, -5, 0, -2, -5, 5, 3, 0, 0, 0, 10, 10], [0, -3, 0, 0, 1, -5, 0, 0, -6, 0, -2, -2, 0, -11, 2, -3, -1, 1], [0, -2, 0, 0, 0, -6, 0, 0, 5, 0, -4, -4, 0, -11, 2, -5, 5, -5], [0, -5, 0, 0, 11, -25, 0, 0, -10, 0, -5, -5, 0, 1, 4, 9, -9, 9], [-6, 0, -4, -8, 0, 0, -6, 8, 0, 10, 4, -4, 6, 0, 0, 0, -4, -4], [0, -3, 0, 0, 3, 21, 0, 0, 2, 0, 5, 5, 0, 3, 4, -3, 5, -5], [0, -4, 0, 0, 4, 2, 0, 0, -6, 0, 9, 9, 0, -1, 8, -2, -1, 1], [-15, 0, -3, -13, 0, 0, 1, -2, 0, 1, -2, 2, 2, 0, 0, 0, 0, 0], [0, 7, 0, 0, -3, 9, 0, 0, 2, 0, 5, 5, 0, 7, -5, 4, -4, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_740_d_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_740_2_d_a();".
// 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_740_2_d_a(:prec:=18)
    chi := MakeCharacter_740_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_740_2_d_a();".
// 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_740_2_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_740_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;