
// Make newform 4032.2.k.g in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4032_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4032_k_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_4032_2_k_g();".
// 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_4032_2_k_g();".
// 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 := [4, 0, 64, 0, 352, 0, 880, 0, 1092, 0, 672, 0, 192, 0, 24, 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], [72, 0, 664, 0, 1440, 0, 918, 0, 4, 0, -100, 0, -20, 0, -1, 0], [-16, 0, -596, 0, -3024, 0, -5334, 0, -3984, 0, -1250, 0, -164, 0, -7, 0], [0, 270, 0, 1648, 0, 2276, 0, -118, 0, -1625, 0, -728, 0, -110, 0, -5], [-172, 0, -1856, 0, -6666, 0, -10038, 0, -6818, 0, -2040, 0, -261, 0, -11, 0], [-68, 0, -804, 0, -3046, 0, -4876, 0, -3502, 0, -1082, 0, -141, 0, -6, 0], [-16, -272, -72, -2004, -1188, -6544, -3276, -9702, -2976, -6672, -1012, -2018, -138, -260, -6, -11], [16, -272, 72, -2004, 1188, -6544, 3276, -9702, 2976, -6672, 1012, -2018, 138, -260, 6, -11], [0, 38, 0, 1046, 0, 4916, 0, 8530, 0, 6311, 0, 1971, 0, 258, 0, 11], [-76, -82, -1496, -1640, -6808, -6554, -11726, -10018, -8638, -6817, -2692, -2040, -352, -261, -15, -11], [-76, 82, -1496, 1640, -6808, 6554, -11726, 10018, -8638, 6817, -2692, 2040, -352, 261, -15, 11], [-428, -82, -4576, -1640, -15248, -6554, -22886, -10018, -16030, -6817, -4904, -2040, -636, -261, -27, -11], [0, -316, 0, -2904, 0, -10328, 0, -16094, 0, -11326, 0, -3460, 0, -448, 0, -19], [0, -372, 0, -4092, 0, -13604, 0, -19070, 0, -12338, 0, -3598, 0, -454, 0, -19], [0, 252, 0, 2412, 0, 7410, 0, 10334, 0, 6862, 0, 2042, 0, 261, 0, 11], [-16, 1888, -72, 21364, -1188, 77296, -3276, 119626, -2976, 84024, -1012, 25666, -138, 3324, -6, 141]];
    Rf_basisdens := [1, 4, 8, 8, 4, 2, 16, 16, 8, 8, 8, 8, 8, 8, 4, 16];
    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_4032_k();"
function MakeCharacter_4032_k()
    N := 4032;
    order := 2;
    char_gens := [127, 3781, 1793, 577];
    v := [2, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4032_k_Hecke();"
function MakeCharacter_4032_k_Hecke(Kf)
    N := 4032;
    order := 2;
    char_gens := [127, 3781, 1793, 577];
    char_values := [[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], [-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]];
    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 := 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, 0], [0, 0, 1, 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, -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, 1, 0, 0], [0, -1, 1, 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, 1, 0, 0, 0, 0, 0, -1, 1, 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, 1, 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, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 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, -1, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -3, 0, 0], [0, 0, 0, 0, 1, 0, -3, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0], [0, 0, 0, 0, -3, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -1, 1, 0, 0, 0, 0], [0, -2, -5, 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, -2, 0, 0], [0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0], [-4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, -1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 3, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0], [0, 0, 0, 0, -2, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 1, 0, 0], [0, 0, 0, 0, -1, 0, -5, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -3, -1, 0, 0, 0, 0], [0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -3, 0, 0], [0, 0, 0, 3, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0], [-4, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, -3, 3, 0, 0, 0, 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, 0, 0, 0, -3, -4, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -2, 0], [0, 0, 0, 3, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -2, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, -4, 4, 0, 0, 0, 0, 0], [0, -3, 9, 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, 1], [-2, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, -1], [0, -3, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 0, -3], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, -4, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 4, 0], [0, 0, 0, 0, 0, 0, 5, -5, 0, 0, 0, 0, 0, 0, 0, 0], [16, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 5, 0, 0], [0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, -1, 0, 0, 0, 0, 0, 0, 0, 4], [-2, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, -3, 0, 0, 0, 0, 0, 0, 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, -5, 0, 0, 0, 0, 0, -7, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 9, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -3, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -8, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, 0], [18, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -9, 0, 0, 0, 0, 0, 5, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -2, 0, 0], [0, 0, 0, 0, 0, 0, -3, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 3, 0], [8, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -1, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, -5, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 5, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -4, -1, 3, 0, 0, 0, 0], [0, -4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, -4, 0, 0, 0, 0, 0, 0, 0, -2], [14, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -7, 0, 5, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -19, 0, 0, 0, 0, 0, -4, 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, 9, 0, 0, 0, 0, 0, -6, 0], [0, 0, 0, 0, -1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 5, 1, 0, 0, 0, 0], [0, 5, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 9, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 4, 0, 0], [0, 0, 0, 0, 0, 0, -3, 1, 0, 0, 0, 0, 0, 0, 0, 4], [-6, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 5, 0], [0, 0, 0, 0, 0, 0, -10, -10, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, -7, 7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 8, 0], [0, 0, 0, 0, 0, 0, 4, 7, 0, 0, 0, 0, 0, 0, 0, 3], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 2, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 7, -7, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 11, 0, 0], [20, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0, 0, 0, 5, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, -2, 0], [-4, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 6, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -7, -10, 0, 0, 0, 0, 0, 0, 0, -3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, -9, 0, 0], [0, 0, 0, 0, 3, 0, -9, 9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, 3, -3, 0, 0, 0, 0, 0, 0, 0, 0], [12, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, 7, 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, -12, 6, 0, 0], [0, -6, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, -4], [0, 5, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -25, 0, 0, 0, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, -8, -8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0], [0, 0, 0, 0, -7, 0, -3, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 5, -1, -6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0], [0, -1, -13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, -1, 0, 0, 0, 0, 0, 0, 0, 5], [0, 0, 0, -3, 0, 0, 0, 0, 0, 5, -5, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 8, 0, -7, 7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 9, -9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 7, 0, 0, 0, 0, 0, 2, -2, 0, 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0, 0, -12, -11, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 9, 0], [0, 0, 0, 0, 0, 0, 0, 0, -21, 0, 0, 0, 0, 0, 4, 0], [0, 0, 0, 0, -6, 0, -17, 17, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, -6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 9, 1, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, 0, 0, 0, -7, 7, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -9, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 15, 0], [0, 0, 0, 0, 0, 0, -10, -13, 0, 0, 0, 0, 0, 0, 0, -3], [-20, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -7, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 10, 0, -9, 9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -45, 0, 0, 0, 0, 0, 5, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -22, 0, 0], [0, 0, 0, 0, 19, 0, -13, 13, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 10, 17, 0, 0, 0, 0, 0, 0, 0, 7], [0, 0, 0, -19, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 0, 0, 4, 0], [20, 0, 0, 0, 0, 22, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -15, 0, 0, 0, 0, 0, 13, 0], [0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -9, -5, 4, 0, 0, 0, 0], [0, 0, 0, 0, 11, 0, 7, -7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -6, -8, -2, 0, 0, 0, 0], [-18, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -15, 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, -7, 8, 0, 0], [0, 0, 0, 0, 0, 0, -1, 8, 0, 0, 0, 0, 0, 0, 0, 9], [10, 0, 0, 0, 0, -20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, -20, 0], [0, 1, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -14, 0, 7, -7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, -3, 0], [0, 0, 0, -23, 0, 0, 0, 0, 0, 3, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, -9, 0, 0], [0, 0, 0, 0, -4, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, -1, 2, 0, 0, 0, 0], [0, -16, -19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 10, 15, 0, 0, 0, 0, 0, 0, 0, 5], [-10, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -12, 0], [0, 0, 0, 0, 0, 0, 13, 11, 0, 0, 0, 0, 0, 0, 0, -2], [-34, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 13, -15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 5, -1, -6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 1, 9, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -19, 0, 0, 0, 0, 0, 16, 0], [0, 0, 0, 5, 0, 0, 0, 0, 0, -12, 12, 0, 0, 0, 0, 0], [30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 15, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -9, 0, 0, 0, 0, 0, -14, 14, 0, 0, 0, 0, 0], [0, 0, 0, -3, 0, 0, 0, 0, 0, -4, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 14, 12, 0, 0, 0, 0, 0, 0, 0, -2], [44, 0, 0, 0, 0, -17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -7, 0, -7, 7, 0, 0, 0, 0, 0, 0, 0, 0], [-12, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0, 0, 0, -6, 6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -16, -13, 0, 0, 0, 0, 0, 0, 0, 3], [0, 0, 0, 7, 0, 0, 0, 0, 0, -7, 7, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 15, 0], [0, 0, 0, 0, 0, 0, 7, 10, 0, 0, 0, 0, 0, 0, 0, 3], [54, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 16, 2, 0, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 20, 0, 0], [0, 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, -3, -10, 0, 0], [0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -17, 0, 0, 0, 0, 0, 2, 0], [0, 0, 0, 0, -2, 0, 9, -9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 33, 0, 0, 0, 0, 0, 4, 0], [0, 0, 0, 0, 0, 0, -8, -4, 0, 0, 0, 0, 0, 0, 0, 4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 5, -2, -7, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 7, -7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 17, 0, 0], [0, 0, 0, 0, 9, 0, -11, 11, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -8, -2, 6, 0, 0, 0, 0], [26, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -16, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -9, 0, 0, 0, 0, 0, 9, -9, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -18, -17, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, -9, 0, 0, 0, 0, 0, 6, -6, 0, 0, 0, 0, 0], [-56, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, -9, 0, 0, 0, 0, 0, 0, 0, -10], [0, 2, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, -21, 21, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 45, 0, 0, 0, 0, 0, 9, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -7, 21, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 14, 0], [0, 0, 0, 0, 11, 0, -13, 13, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 13, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 1, 0, 0, 0, 0, 0, 0, 0, -4], [-14, 0, 0, 0, 0, -21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -18, 0], [0, 0, 0, 0, 0, 0, -14, -6, 0, 0, 0, 0, 0, 0, 0, 8], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -24, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, -7, 0], [0, 0, 0, -25, 0, 0, 0, 0, 0, 3, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, -9, 9, 0, 0, 0, 0, 0, 0, 0, 0], [42, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 21, 0, 0, 0, 0, 0, -10, 10, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -17, -20, 0, 0, 0, 0, 0, 0, 0, -3], [0, -2, 27, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 12, 8, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -19, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -3, 0, 0], [0, 0, 0, -3, 0, 0, 0, 0, 0, 6, -6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -6, -6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 2, 0, 0], [0, -13, 35, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -19, 0, 0, 0, 0, 0, 11, -11, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -27, 0, 0, 0, 0, 0, 12, 0], [0, 0, 0, 0, 0, 0, -10, -6, 0, 0, 0, 0, 0, 0, 0, 4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 12, -1, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4032_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4032_2_k_g();".
// 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_4032_2_k_g(:prec:=16)
    chi := MakeCharacter_4032_k();
    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_4032_2_k_g();".
// 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_4032_2_k_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4032_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![8, 0, -8, 0, 1]>,<43,R![5488, 0, -152, 0, 1]>,<67,R![448, 0, -208, 0, 1]>],Snew);
    return Vf;
end function;