
// Make newform 6032.2.a.bc in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6032_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_6032_2_a_bc();".
// 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_6032_2_a_bc();".
// 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 := [16, 16, -287, -60, 541, 17, -298, 19, 65, -9, -5, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [636, 10278, -1873, -31635, 8936, 11193, -4619, -202, 493, -186, 27], [-780, -10787, 25769, 26550, -23773, -8175, 6416, -547, -376, 239, -44], [-5608, 15791, 25624, -29837, -13673, 15790, 1801, -3557, 229, 301, -53], [2680, 23289, 19241, -25090, -22785, 4609, 8196, 829, -1312, -185, 92], [790, 15238, -3515, -21653, 10178, 11187, -5507, -2294, 1017, 150, -59], [-11160, -29927, 19388, 67801, 2113, -39026, -1809, 9173, -453, -797, 125], [3884, -32351, -40281, 71520, 44835, -48417, -12070, 12589, 162, -1125, 154], [7504, -52268, -11557, 143815, -8406, -90147, 11755, 21016, -3789, -1672, 373]];
    Rf_basisdens := [1, 1, 1, 2308, 2308, 2308, 2308, 1154, 2308, 2308, 2308];
    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_6032_a();"
function MakeCharacter_6032_a()
    N := 6032;
    order := 1;
    char_gens := [5279, 1509, 1393, 1249];
    v := [1, 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_6032_a_Hecke(Kf)
    return MakeCharacter_6032_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, 0, 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, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 1, 0, 0, -1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1, -1, 1, -1, 0, 1], [1, 1, -1, 0, 1, 0, 0, 0, 0, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, -1, 2, -1, 2, -1, -1, 1], [0, -1, 0, 1, 1, -1, 1, 0, 0, -1, 0], [0, -1, 0, 1, 0, 0, -3, 1, -1, -1, 1], [2, -1, 2, 0, 0, 1, 0, 0, 0, -1, 1], [0, 0, -1, 0, 1, 0, 1, -2, 1, 0, 0], [0, -1, 0, 1, 0, 1, -1, 0, -1, -1, -1], [2, 0, -2, -1, 1, -1, 0, -3, 1, 0, -1], [0, -1, 0, -1, 0, 0, 1, 1, -1, 1, 1], [3, -1, 0, -1, 0, 0, 1, 0, -1, 0, -1], [3, -2, 1, -1, -2, 0, 1, 1, 0, -1, 0], [0, 0, 0, -1, 2, -1, 0, -1, 1, -1, 0], [1, 1, 0, 0, 1, 1, -2, -1, -1, -2, 1], [1, 1, -2, 1, 2, -1, -3, 0, -2, -1, 0], [-1, -2, 2, 1, -1, 1, -2, 2, -1, -1, 3], [0, -1, 2, 1, 0, 1, -1, 3, 0, 0, 2], [-6, 2, -2, 0, 1, 0, 2, -1, 1, 0, -1], [1, 1, 0, 1, -2, 0, -1, 1, -2, 1, 1], [5, 2, 0, 0, -1, 1, -2, 3, -1, -1, 0], [2, 0, 1, -1, 0, 0, -1, -2, 0, 0, 0], [-3, -1, -3, -1, 1, -1, 2, -1, 1, 1, -1], [5, -4, -1, -1, 1, -3, 2, -1, 1, 0, 0], [4, 0, 0, 1, 0, -2, 1, 1, 0, 1, 1], [-1, 4, -3, -1, 1, -1, 1, -3, 2, 2, -3], [3, -1, -2, 1, -2, 1, 1, 1, 0, 1, 1], [-1, 1, 0, 2, -1, 2, 1, 1, 1, -1, 1], [1, -2, 0, 3, 1, -1, 2, -1, -1, 1, -2], [0, -1, 0, 1, -2, -1, -3, 3, -2, -2, 2], [6, -2, 0, -1, -4, -1, 0, 3, -1, 1, 2], [1, 1, -3, -5, 1, -1, 4, -2, 2, 2, -3], [-3, 2, 2, 0, 0, 2, 0, 1, 0, -1, 3], [8, -1, -3, -3, 0, -1, 2, -4, 0, 2, -3], [1, 1, -1, 1, 1, 0, 0, -2, 1, -1, 1], [0, 0, 0, 5, 1, 0, -1, 0, 2, 0, 1], [6, -5, 0, -4, -3, -2, -1, 1, -2, -1, 1], [-4, 1, 2, 3, 0, -2, 1, 1, 0, 0, 0], [-1, -2, -1, 3, -1, 1, -1, 3, -2, 2, 1], [6, 0, 0, 2, 2, 0, -1, -2, 2, 1, -2], [6, 3, -2, 0, -2, 3, -1, 1, 0, 0, 2], [4, 0, 0, 1, 0, -2, -4, 2, 1, -1, 3], [-1, -1, 0, 3, -3, -2, -3, 2, 0, -2, 3], [5, -2, 2, -2, -2, 1, 0, -2, 1, 0, 0], [1, -1, 1, -1, 1, -1, 0, 0, -1, -1, 0], [4, 1, -1, -4, -1, -1, 2, -2, 0, 1, 0], [6, -2, 1, -1, -2, -1, 2, 0, 1, -2, -1], [0, 2, 1, -4, -3, 3, 5, -1, 3, 2, -2], [3, -2, 2, -5, -3, 7, 1, 0, 0, -1, 2], [-7, 0, 2, 3, -1, 3, -4, 6, -3, -3, 3], [2, -1, -3, 0, 0, -1, -2, 3, 0, -2, 2], [2, 5, -2, 3, 2, 0, -1, 0, 0, 0, 2], [-1, 2, 2, 1, -2, 1, 0, 5, 0, -1, 0], [5, 5, -1, 2, 1, -1, 0, -4, 3, 3, -4], [4, 0, 3, -1, -3, 1, -5, 3, -4, -1, 2], [-1, 0, -3, 2, 3, 3, -3, -2, 0, 0, 0], [-1, 4, 4, -3, -1, 3, -3, 0, -2, -1, 2], [0, 0, 2, -3, 2, -2, 2, -5, -1, 1, 0], [5, -4, 4, -4, -3, 1, 4, 1, -1, 1, 0], [3, 2, -1, -3, 0, -2, 1, 0, 0, 1, 1], [4, -2, 3, 0, 1, 2, 5, 0, 1, 2, -2], [8, -1, 3, -4, -5, 3, -2, 2, -3, 0, 3], [6, 4, 1, -1, -1, -4, -1, 1, 1, 0, 1], [-3, -3, 0, 3, 1, -3, 1, -2, 2, 4, -1], [3, -7, 3, 0, -1, -1, 0, 2, -3, -1, -2], [-3, 1, 4, 0, -3, 2, 0, 1, 2, -1, 2], [4, 4, -2, -1, 2, 0, -2, 0, -1, -1, -6], [7, -2, 2, -1, 2, -1, 2, 1, 0, 1, 0], [-5, 4, 2, 1, 2, 1, 0, -4, -2, 1, -1], [-7, 0, 0, 3, 2, -4, -4, 3, 1, -2, 3], [-1, 0, 3, 6, -3, 0, -3, 6, -3, -3, 3], [6, -3, -3, 1, 2, 1, 0, -4, 3, 1, 0], [6, -1, -1, 4, 1, -1, -2, 2, -2, -1, 2], [-4, -1, -3, 7, 3, -1, -2, -3, 1, -2, 0], [3, -3, -7, -1, 0, -3, 2, 0, 3, 2, -1], [3, 0, 0, 0, 0, 3, 4, -3, -2, 1, -2], [10, 2, 0, 4, 2, 3, -4, -2, -4, 2, 0], [1, -6, 5, 0, -6, 6, -1, 7, -4, -5, 6], [3, 9, -4, 0, 2, -2, 4, -6, 4, 2, -4], [-11, 8, 1, -2, -3, 1, -1, 2, 0, 0, 0], [8, -2, -1, -2, 1, -1, 5, 0, -1, 2, -2], [-1, 4, -6, 0, 2, -3, 2, -4, 1, 4, -6], [-3, -1, 4, 3, 3, 0, -3, 1, 0, -2, 2], [-3, 4, 3, 0, -1, 2, 0, 2, 0, 3, 4], [-1, -6, 1, 1, -4, -1, -3, 0, 0, 1, 3], [-5, -3, -1, -3, 4, 3, 0, -5, 1, 0, -2], [11, 1, -6, -4, 1, 0, 2, -1, -1, 4, -3], [-1, -5, 4, -3, -3, -1, 3, 1, -3, 1, 3], [-4, 1, 0, 0, 2, 2, -4, 4, -4, -3, -1], [-6, 5, 1, -1, 3, -2, 2, -1, -3, -2, -6], [1, 0, 1, -4, -7, 8, -1, 4, -2, -2, 2], [-2, 4, 2, 0, -3, 4, -4, 7, 3, 0, 3], [-3, -2, -2, 2, 4, -7, 6, -8, 1, 2, -4], [-1, -1, -2, 4, 0, -1, 7, 0, 2, -1, -2], [11, -5, 0, 0, -5, 5, -1, 1, -1, -1, 1], [-3, 4, 2, -4, 1, 3, -1, -3, 1, -4, 2], [-4, 0, 5, -2, 0, 6, 3, -5, 2, 0, -1], [-2, 10, -5, 1, 2, -4, 1, -4, 2, 0, 0], [13, -5, -3, -3, -2, 1, -4, 2, -4, 1, 2], [8, -2, -1, 4, -4, 3, -9, 4, -1, -1, 3], [5, -1, 5, -2, 0, 6, 4, -5, 3, 0, 0], [-1, 2, 4, 9, -2, 1, -5, 7, -3, 1, 3], [2, -4, 5, -1, -7, 4, -5, 7, -3, -2, 3], [-6, 0, 7, -4, -3, 6, 2, 2, -1, -1, 2], [10, 1, 2, 2, -8, 9, -7, 6, -2, -2, 5], [-4, 5, -7, 4, 4, -1, 2, -3, 1, 3, -5], [5, -3, -1, -4, -3, 4, 0, -2, 2, 0, 1], [-3, 2, 6, -4, 2, -1, 4, -5, 0, -1, -2], [12, 4, -5, -7, -2, -1, 9, -4, 0, 6, -6], [3, 3, 1, 1, -4, 9, -6, 9, -3, -6, 4], [7, -7, 3, -9, 0, -1, 6, -4, 0, 3, 2], [-10, 0, 6, -5, -2, 5, -4, 3, -1, -1, 2], [-1, 0, -4, -6, -1, -1, 3, -1, 2, 1, 1], [14, -1, -5, 2, 0, -2, 2, -5, 6, 0, -2], [-9, 3, 1, 3, 2, 3, 0, 1, 4, -1, 2], [-6, -1, 3, 2, 3, 5, -8, 0, 0, -5, 4], [-6, 6, -2, 6, 6, -10, 2, -3, 1, 1, -2], [7, 2, -4, 2, -2, -8, 2, 3, 0, 3, -2], [-7, 1, -6, -3, 1, 1, 1, -6, 4, 0, -3], [4, 2, 2, 1, -1, 0, -4, 2, 1, -2, 0], [0, -4, -1, 0, -5, 5, -1, 0, -3, 0, -2], [-9, -2, -2, -3, 2, -8, 5, -1, 4, 4, -2], [8, -5, 1, -3, -3, 6, 4, -1, 2, -1, 1], [-1, -4, -2, 2, 3, -1, 8, -7, 3, 5, -2], [8, -4, -6, 2, -2, 1, 0, 2, 4, 0, -2], [-5, 6, -10, 8, 6, -8, -5, -2, 0, 2, -3], [5, 0, -1, -6, 3, 0, 7, -6, 3, 1, -3], [-6, 10, -4, 1, 0, -1, -6, 5, -1, 3, 4], [-5, -3, -4, 3, -1, 2, -3, 0, 1, 5, 2], [4, -1, -3, -7, 0, -4, 8, -2, 1, 5, -4], [-5, 4, 3, -5, 0, 2, -3, 1, -3, -4, -3], [3, -3, 1, -7, -5, -2, -4, -1, -2, 2, 2], [-11, 0, -1, 0, -1, -1, -3, 4, -1, 1, 5], [9, -2, 0, 2, 1, 2, 0, 2, 1, -1, 1], [-5, -3, -1, -7, -1, 0, 4, -5, -1, 3, -3], [-7, -1, -6, -1, 1, -6, 2, -6, 0, 3, -5], [-4, -4, 2, 0, 2, -7, 4, -4, 3, 5, -5], [-7, 6, 4, 5, -1, -1, -4, 4, -3, 1, 3], [14, 0, 1, -1, 5, 3, 5, -1, -1, 0, -3], [-1, -9, 6, -4, -5, 0, 3, 1, -2, -2, 4], [-1, -9, -3, 2, -2, -7, 4, 0, 3, 0, 4], [0, -1, 4, 3, 2, 1, -8, 7, -5, -2, 3], [-11, 5, 2, 1, -4, 6, -3, 6, 0, -3, 8], [13, -2, 3, -2, -5, 5, -5, 2, -5, 1, -1], [-5, 5, 4, -1, -1, 3, -3, 8, -1, -5, 2], [8, 3, 2, -5, -1, 2, -1, -6, 1, 2, -7], [-10, 6, 2, -2, 4, 3, 7, -4, -2, -3, -2], [14, -10, -1, 5, 2, -1, -2, 2, -2, -1, 4], [0, 4, 2, 5, 8, -7, 3, -5, 4, -1, 3], [0, -2, 3, -6, -2, -4, 1, -7, 1, -1, 0], [-7, 9, -5, 6, 6, -8, 0, -5, 1, 0, -6], [6, -1, 3, -3, 4, -4, 2, 1, -1, -3, -2], [5, -16, 3, -4, -4, 1, 1, 5, -4, 3, 4], [3, -2, -4, -4, 2, -3, 6, -6, -2, 1, -5], [-1, 9, 2, 0, 4, 4, -1, -6, 1, -2, -3], [4, -12, -6, 1, 1, -11, 4, -10, 7, 6, -4], [1, 6, -9, 9, 2, -6, -6, 0, 0, 0, -5], [-8, 3, -4, -2, 0, 2, -3, 2, 1, -1, 4], [-17, -1, 1, -4, -3, -4, 6, 4, 7, 3, 0], [-8, 5, -2, -5, 0, -2, -11, 1, -3, 1, 5], [11, -7, 4, 5, -7, 3, -5, 4, -2, 0, 9], [-10, 7, 5, 4, -6, 8, -2, 9, -6, 0, 2], [-11, 0, -4, 2, 0, -10, 2, 0, -1, 4, 2], [-2, 4, -1, 2, -6, 2, -3, 6, -4, 4, -2], [-5, 4, -9, -1, 3, -7, 9, -7, -1, 7, -12], [4, 0, 0, 3, -5, 3, -6, 4, -4, -3, 5], [10, 7, 3, -4, 1, 4, 0, -4, 3, 2, 5], [-17, 2, 3, 1, -1, -6, -1, 1, 1, -3, 8], [-16, -9, 9, 0, -4, 2, -8, 6, -4, -6, 6], [4, -5, 0, -5, -6, 3, -9, 2, -3, 3, 4], [-2, -1, -1, 5, 8, -4, 0, -4, -2, -4, -3], [9, -4, -1, 2, -6, 3, -12, 11, -6, -4, 8], [-17, -1, 5, 4, -3, 3, -2, 5, -6, -4, 5], [2, 0, 5, -4, -3, -3, 1, 2, -1, 4, -2], [7, -3, 1, 1, 5, -5, 9, -13, 6, 5, -6], [-12, 1, 1, 7, 0, -3, 0, 0, 3, 1, 4], [7, -1, 0, -8, -1, 3, 5, -7, 6, 2, -4], [-5, -5, -1, -4, 7, -7, 6, -1, -6, 0, -4], [3, -3, 9, -4, -4, -4, 6, 2, -2, -3, -3], [-3, -6, -3, 5, 4, -5, -1, -2, -3, -4, -2], [-5, 6, -1, 4, 6, -9, -1, -3, 5, 6, -3], [10, -9, 2, -11, -1, 0, 5, -2, 0, 3, 2], [5, 13, -6, 0, -1, -6, -1, -6, 0, 4, -1], [-8, 5, 1, 4, 3, 10, -8, 5, -7, -4, 1], [4, 5, 2, -5, -3, 7, 11, -8, 3, 4, -7], [15, -8, 0, -6, -9, 1, 1, 7, -1, 2, 2], [4, 1, -7, 8, 4, -2, 0, -1, -4, 4, -4], [6, 7, -4, 3, 5, -3, 2, -6, 5, 9, -1], [7, 3, -3, -5, 2, -5, 8, -10, 0, 5, -6], [-4, 6, -3, -3, -2, 0, 3, -8, 3, 1, -7], [14, 0, -3, 1, -2, 7, -11, 5, -7, -1, 5], [-11, 9, -8, 3, 3, -6, 3, -8, 9, 3, -12], [1, -2, 4, -1, -5, -5, 6, 0, 5, -3, 1], [3, 3, 4, -7, -1, 2, 1, -2, -6, -2, 3], [-7, -3, -1, -3, -1, -2, 8, -1, 4, 10, -2], [10, -7, -4, 0, 6, 1, -3, 2, -5, 3, 2], [-22, 3, 6, 0, -6, 0, 3, 0, 2, 2, -3], [2, -6, 1, -1, -1, -8, -1, 3, -4, -5, 2], [-11, 4, -6, -2, -1, 1, 2, -5, 1, -1, -2], [-13, 7, -4, 3, 2, -3, -7, 5, -4, -3, -3], [6, 8, -10, 1, 7, -8, 4, -6, 1, 4, -8], [-9, -2, 3, 1, -6, 4, 11, -4, 2, 3, -3], [2, -4, -1, 1, 5, -7, 0, -1, -5, -5, -5], [-1, 1, 5, -3, -1, 1, 8, 3, 3, 3, -3], [0, -10, -1, 4, -6, 1, -1, 6, -3, 3, 2], [12, 1, 3, 3, -3, -4, -9, 1, 0, -4, 5], [-4, 6, 0, 4, -1, 1, -8, -1, -4, -1, 4], [8, -3, -7, 5, 8, -7, -4, -4, 4, 0, 1], [8, -6, 0, -7, 8, -5, 10, -15, 3, 5, -6], [-26, 7, 4, 11, -1, 5, -9, 14, -9, -6, 7], [-23, -2, -6, 13, 10, -12, 2, -3, 2, -1, -6], [0, 9, -5, -1, -3, 6, 0, -4, 4, 1, -4], [-7, 6, 2, -1, -7, 9, -8, 8, -6, -2, 2], [7, -3, -4, 7, 3, 8, -9, 4, -3, 1, 0], [-16, 7, -7, 3, 5, -6, 2, -6, 3, 8, -9], [-19, 8, -3, 0, -2, 5, -5, 3, -3, -6, 3], [-5, 0, 11, -5, -3, 1, -3, -1, -5, 1, 6], [1, -18, 7, -6, -3, 3, 3, 0, -1, 1, 5], [-6, -7, 11, 4, -7, 9, -10, 8, 2, -5, 10], [-23, 4, -3, 0, 3, -3, 3, -2, -3, -1, -5], [2, -9, -3, 5, 3, 1, -1, 3, 5, -3, 0], [-13, 3, -4, 3, -5, -5, -6, 2, -3, -2, 2], [-13, 5, 1, -5, -1, 3, -6, 8, -8, -2, 5], [12, -10, -6, -2, 4, -9, -4, 0, -5, -3, -3], [-1, 4, -9, -3, 6, -6, 7, -4, 4, 1, -7], [8, 3, 4, -5, -6, 5, -3, 7, -4, -8, 6], [-24, 5, 1, -7, 1, -5, 0, -1, -1, 0, 4], [0, 13, 9, -1, -3, 9, -4, -1, 1, 0, 6], [-11, -4, 7, 10, -7, 3, -7, 12, -5, -9, 7], [-7, 4, 2, -4, 5, 1, -4, -1, 5, -3, 0], [-5, -6, -5, -2, 7, -4, 5, -2, -4, 0, -8], [3, 11, -2, 5, -2, 6, 3, 7, 4, 1, -3], [-13, -7, -3, 8, 0, -4, -3, 5, -4, -2, 7], [5, 9, -8, -1, 9, -14, 11, -10, 13, 5, -10], [15, 1, 8, 4, 4, 3, 3, 0, -4, -3, -6], [-14, 0, 6, 0, -1, 11, 0, 3, 4, 3, 0], [-26, -6, 4, 0, 3, 2, 2, -5, 2, -7, 0], [-8, -3, 2, 1, 0, -6, -3, -1, -3, -5, 3], [-16, 19, -6, 5, 6, 1, 1, -3, 2, 2, -8], [-11, -7, -1, 1, -9, 3, 0, 2, 1, 3, -2], [-12, -2, 11, 7, 2, 2, -7, 5, -2, -6, 7], [-1, -9, 10, -1, -1, 9, 1, 1, -1, -3, 5], [-15, 9, -8, -3, 0, -1, -4, 1, 2, -2, 1], [5, -3, 1, -2, -4, 7, 2, 1, 5, 2, -2], [0, 3, 0, 2, 0, 13, -7, 12, -4, -2, 5], [-2, -2, -3, -11, -2, -2, 3, -4, 0, 0, -4], [-30, 3, 2, -8, -2, -4, 7, -4, 2, 2, 1], [1, -10, -2, 1, 4, -5, 8, -5, 4, -1, -6], [11, 16, -10, 2, -1, 2, 4, -2, 3, 1, -5], [0, -2, -9, -4, 9, -13, 11, -20, 5, 6, -10], [2, 2, 4, 9, 0, -4, -10, 11, -4, 0, 9], [-16, 0, -4, 5, 0, -5, 0, -2, 2, 0, 3], [23, 5, 4, 0, -7, 8, 1, 11, -1, -1, -3], [9, 5, 0, 5, 7, -10, 0, -4, 2, 1, -1], [6, -5, -2, 9, -3, -7, -2, -4, 5, -1, 3], [10, 13, -11, 5, 3, -1, -2, 5, -3, 8, -6], [-13, 1, 7, 0, 2, 4, -6, 11, -7, -8, 12], [1, 1, -9, -13, 10, -11, 10, -16, 10, 5, -10], [9, -3, 1, -10, -1, 3, -1, -2, 7, -2, 2], [-18, 3, -8, 1, -1, -5, -9, 2, 0, 1, 6], [-17, -6, 6, 4, -8, 10, -4, 10, -7, 0, 10], [8, 10, 5, 2, -6, 4, -3, 7, 1, -5, 6], [5, 6, 4, 2, -4, -14, -8, 6, -1, -4, 6], [15, -1, 5, 5, -8, 4, -6, 7, 4, -3, 10], [9, -10, 3, 1, -6, 6, -1, -3, 1, -4, -1], [8, 4, -11, -5, 9, -9, 9, -7, -3, 8, -11], [1, -6, -2, -7, -1, -7, 11, -8, 8, 9, -2], [-9, -1, -4, -8, 1, 6, -1, 3, -3, -3, -1], [28, 0, -7, -2, 9, -5, -3, -4, -1, -2, 0], [7, -1, 18, 3, -4, 11, -13, 7, -7, -10, 12], [-25, -6, -3, 2, 4, -12, 3, -3, 4, 1, -2], [16, -6, 2, 9, -6, 0, -10, 5, 0, -4, 11], [-14, -1, 2, -5, 4, -12, 3, -3, 4, 4, -4], [13, 1, -1, -10, 1, 2, 4, -4, -4, 0, -1], [-14, 2, 2, 7, 8, -8, 8, -8, 2, -4, -8], [19, 0, -4, -5, 0, 8, -2, -5, 5, 2, -3], [7, -5, -2, -8, 0, -2, 15, -8, 14, 7, -12], [-27, 6, -1, 3, 5, 4, -3, -5, 1, 1, 0], [-2, -3, 1, -7, -3, 1, 5, -1, -3, -9, 2], [-1, 7, 8, -8, -5, 9, -3, 5, -3, -9, 3], [12, 2, -1, 3, 2, -7, -1, 8, 1, -1, 1], [9, 1, 11, 0, 4, 7, 8, 4, -2, -3, -2], [8, 3, -2, 2, 1, -15, 7, -3, 3, 6, -2], [-11, 3, -1, -10, -2, -4, 12, -7, 5, 10, -12], [-24, 6, -6, -2, 10, -21, 4, -6, 4, 8, -4], [-1, 2, 5, 3, 6, 3, 7, -6, 3, 2, -4], [-14, 9, -12, 2, 4, -13, 5, -6, -2, 4, -7], [-11, 9, -2, -8, 7, -8, 7, -13, 9, 1, -13], [-16, -4, 7, 2, -3, 7, -11, 12, -11, -6, 10], [3, -10, 3, 0, -6, 6, -8, 15, -5, -3, 5], [6, 5, -6, 5, -3, -1, -5, -6, 5, 4, -1], [-4, -3, 4, 4, -7, 6, 7, 3, -3, 4, 0], [13, -5, -9, 0, 6, -4, 6, -1, 7, 2, -6], [21, 2, 3, 2, -6, 11, -11, 3, -10, -7, 2], [-1, 0, -7, 3, 7, -5, 9, -3, 11, 1, -6], [12, -4, -3, -1, -1, -1, 10, 1, 4, -2, -4], [4, -2, 2, 4, -9, 14, -15, 15, -8, -6, 4], [21, -3, 9, -1, 0, -1, 7, -8, 0, 4, -4], [10, 6, 6, 0, -1, 7, 0, -3, -2, 1, -4], [-9, -4, -2, -1, -4, 7, -7, 15, -4, -2, 10], [-17, -2, 3, -5, 11, -8, 1, -8, -2, -2, -1], [23, 4, 10, 5, -6, 5, -2, 12, 3, 2, 6], [-8, -3, 11, -2, 1, 5, 4, 3, -3, -2, -1], [-11, 8, 5, 6, 3, 6, -3, -6, -3, -3, -5], [-8, 9, 3, -3, -3, 2, 10, 1, -2, 5, -3], [-13, -7, 6, 5, -2, -2, -9, 6, 0, 7, 9], [9, -6, 0, 7, 1, 3, -8, 2, 1, -7, 5], [-1, 5, -4, 6, 0, 4, 1, 9, 0, 5, -3], [9, 8, -9, 0, 5, -4, 7, 1, -1, 9, -3], [2, -6, -9, 12, 11, -11, -7, -4, -3, -4, 2], [24, -6, -7, -10, 3, -1, 9, -1, 0, 3, -10], [-1, -6, 2, 4, -7, -5, -6, 9, -9, -5, 6], [-8, -13, 5, -8, -11, -2, 6, 4, 9, 4, -3], [6, 6, 3, -7, 0, 3, -6, 8, -5, -4, 1], [4, 2, -2, 5, -3, -7, -14, 8, 2, -1, 11], [-1, 0, -2, 0, 4, -4, 4, -10, 7, 6, -6], [3, 5, 8, 6, -8, 5, -9, 14, -8, -5, 8], [3, 11, -4, 1, 1, -12, -3, -2, 7, 1, 0], [-3, -7, 6, 2, 0, -4, 13, -8, 8, 3, -8], [-19, 10, 9, -11, -4, 4, 13, 1, 3, 4, 1], [31, 17, -11, 9, 5, -6, 2, 0, 3, -1, -4], [-4, -1, 8, -17, -2, -6, 8, -11, 0, 3, -8], [-16, 5, -5, 5, -3, 4, -2, -8, 4, 3, -1], [25, -2, 4, 7, 4, 5, 2, 1, -6, -5, 0], [18, -2, 3, -13, -3, 0, 1, -5, 2, 3, -2], [7, -6, -13, -13, 11, -4, 9, -19, 13, 3, -8], [3, -2, 5, 4, -7, 13, -1, 12, -6, -8, 4], [19, 7, -13, -10, 5, -9, 2, -12, -1, -1, -8], [-2, 1, 2, 1, 4, -11, 7, -4, -1, 1, -10], [11, -9, -5, 10, -3, -9, 4, -2, 1, -1, -4], [29, -3, 3, -3, 1, -3, 0, 1, 3, -5, 5], [2, -1, -4, 1, 5, -15, 9, -8, 1, 4, -5], [-5, -14, 0, 2, -4, -1, -12, 16, -2, -5, 13], [15, -3, 3, -4, -6, 1, -8, 3, -3, -6, 8], [-5, 1, 1, 15, 0, -3, -10, 1, 0, -7, 8], [13, -14, -1, -15, 5, -10, 7, -3, 2, 4, -1], [15, -3, 5, 6, -4, 4, -2, 5, -9, 0, 6], [7, -8, 9, 4, -2, -1, -11, 11, -12, -11, 12], [17, -3, -3, -4, -1, -13, -5, 6, 1, 0, 2], [4, -7, 4, 0, -1, -6, 3, 7, -6, 7, 1], [23, 5, -8, 6, -2, -8, 3, 2, 0, -1, -4], [-7, -11, 7, -4, 0, -2, -4, -5, -6, -7, 7], [-2, 2, -12, 5, 9, -15, 2, 0, 0, 5, -3], [-5, -8, 10, 4, -2, 15, -6, 5, 3, -6, 9], [21, 0, -3, 10, -3, 1, -1, 0, 0, 2, 0], [18, -10, -1, -4, -1, -5, 19, -8, 12, 9, -1], [1, 0, 6, -6, 7, -10, -4, 9, -2, -4, 5], [19, -9, 3, 1, 6, 6, -2, -8, -1, 0, 1], [-7, 1, -1, 10, -3, 7, -8, 8, 9, -1, 6], [4, 10, 8, -2, 1, 3, -8, 1, 3, -2, 3], [-16, 11, -18, -5, 8, -26, 17, -22, 14, 14, -19], [-23, 2, -13, -5, 4, -13, 7, -8, 10, 5, -5], [9, 12, -12, -3, -1, -3, 12, 3, 5, 5, -12], [33, 6, -12, -1, 5, -9, -10, 4, -1, -5, 1], [19, -8, 13, 9, 0, 12, -13, 11, -9, -6, 2], [2, 3, -1, 8, -3, 8, -13, 6, -4, -4, 10], [4, 16, -8, -9, 9, -3, 2, -9, 1, 2, -5], [20, 5, 1, 5, -6, 8, -1, 4, 1, 0, 0], [-25, -16, 3, 2, 0, -7, 3, -5, 4, -1, 10], [2, -17, -5, 6, 3, -2, 4, -6, 4, 3, 4], [23, -2, -3, -3, -3, 6, 1, -8, 2, 2, -4], [2, 5, -7, 6, 5, 1, -14, 0, -1, -8, -1], [25, -16, -4, -8, -2, 1, -3, 0, 2, -2, 1], [0, 19, -15, 5, 14, -1, 4, -6, 0, 6, -11], [-11, 23, 0, 2, -6, 4, -9, 5, -6, -1, -3], [8, 5, -6, -10, 4, 4, -9, -4, 5, -3, -2], [7, 4, -13, 4, -5, 2, -1, -2, -3, 7, 1], [-7, -2, 0, -2, -5, -3, 6, -3, -3, 7, 2], [9, -7, 8, 12, -8, 0, -5, 14, 2, 1, 20], [2, 0, -9, 0, 4, -10, 10, -11, 2, 1, -3], [-5, 3, -1, 10, 1, -4, -6, 6, -6, 0, -5], [-19, -2, 2, 0, -2, 5, 8, 3, 5, 10, -2], [7, 10, -3, 2, 0, 2, 1, -3, 2, 5, 8], [17, -5, -15, 7, -3, -5, -4, 5, 5, 5, -3], [5, -5, 10, -2, -4, -1, -3, 14, -7, -6, 5], [-23, 9, 0, 6, 0, 4, 11, -8, 2, 9, -8], [-6, 13, -4, -1, -4, 6, -5, -1, -1, -7, 3], [11, 18, -12, 3, 2, 3, 4, -4, 5, 6, -6], [9, -15, -3, -2, -3, -7, 0, -1, -5, 1, -1], [3, 3, -2, -11, -5, 3, 3, -6, -1, 5, -10], [29, 11, 2, -2, -3, 12, -1, -2, -6, 2, -4], [29, 2, -12, 9, -3, 1, -12, 4, -1, 5, -3], [3, -15, 7, 11, -3, 6, -7, -5, 5, -2, 1], [6, 8, -14, -13, 5, 0, 2, -14, 7, 2, -8], [8, 13, -8, 0, 7, 4, 1, -13, 7, 0, -6], [-32, 11, -5, 8, 9, -12, 4, -8, 11, 2, -15], [-9, 6, -1, 2, 9, 2, 8, 2, -1, -2, -3], [12, 6, 5, -4, -7, 8, -3, 0, 1, 0, 6], [15, -4, 2, 8, -9, 9, -18, 16, -9, -7, 12], [-25, 14, 7, 6, 2, -4, 1, 0, 4, -3, 1], [24, -13, -4, -1, -4, -4, -9, 7, 0, -6, 6], [-7, 7, -2, 2, 4, 8, 3, -6, 12, 1, -2], [13, -17, -8, 10, 0, -20, 3, -2, -5, 0, -1], [5, 7, -15, 6, -1, -2, -6, 8, -3, -3, 6], [-4, -5, 5, 6, -12, 0, 0, 19, -4, -8, 14], [-18, 2, 1, -2, 4, 0, -3, -1, -6, -2, 7], [20, -1, 1, -5, -6, -1, -6, 6, 1, 1, 4], [14, -10, -5, -12, 10, 4, 15, -15, 4, 10, -11], [2, -12, -1, -2, 1, 6, 5, 4, -2, 3, -3], [7, 13, -8, 5, 13, -5, -9, -6, -13, -5, -8], [9, 2, 5, 11, 2, 8, 5, -6, -10, 3, -5], [6, -2, -2, -10, -10, -7, 10, 8, 1, 5, -3], [-43, 3, -3, -3, -2, 3, -2, -1, 1, -6, 2], [-7, -11, 7, -18, -4, 9, 10, -8, 6, -1, -3], [14, 1, -18, -5, -2, -16, 15, -4, 2, 14, -12], [-21, 5, 9, 1, -1, -2, 6, -3, -9, 1, 1], [5, -5, -6, 12, -8, -5, -13, 16, -10, 1, 12], [7, 1, -8, 7, 5, 5, -3, 4, 3, -3, -2], [10, -20, 7, 5, 2, 4, 1, 0, 3, -1, 1], [-1, 2, 5, 5, 4, 3, -1, -6, 2, -3, -3], [8, 3, -9, 3, 9, 7, -8, -3, -8, 1, -3], [10, -18, -2, -10, -1, -11, 4, 3, -4, 1, -4], [8, 0, -9, -13, -1, -13, 10, -5, 8, 12, -6], [12, -4, -11, -8, 5, -2, 3, 0, -2, -5, -11], [36, -17, -2, 9, -8, -3, -9, 5, -8, -4, 2], [10, 0, -12, 10, -3, 2, 4, 5, 11, -4, -3], [5, -10, 5, 1, 3, 6, 2, -11, -3, -4, -6], [30, -11, 1, -13, 1, -1, 14, -13, 0, 7, -13], [-8, 1, -1, 6, 8, 8, -2, -3, 0, -6, 0], [20, 6, 13, -1, 8, 10, 3, -8, 4, -2, 2], [0, 10, -2, -14, 1, -2, 10, -13, 3, -2, -5], [-14, -6, 1, 12, -4, 16, -11, 10, -8, 2, 2], [-9, 4, 4, 11, 2, 1, 13, -9, 5, 5, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6032_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6032_2_a_bc();".
// 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_6032_2_a_bc(:prec:=11)
    chi := MakeCharacter_6032_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_6032_2_a_bc();".
// 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_6032_2_a_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6032_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-16, 256, -204, -572, 483, 336, -311, -55, 76, -4, -6, 1]>,<5,R![16, -416, -328, 2336, -751, -1470, 388, 334, -51, -31, 2, 1]>],Snew);
    return Vf;
end function;