
// Make newform 6720.2.g.p in Magma, downloaded from the LMFDB on 01 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6720_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_6720_g_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_6720_2_g_p();".
// 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_6720_2_g_p();".
// 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 := [13, -38, 70, -74, 63, -34, 16, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [5, -2, 2, 0, 0, 0, 0, 0], [29, -41, -71, 117, -115, 67, -21, 6], [-3, 11, -18, 15, -10, 3, -1, 0], [-223, 647, -661, 639, -315, 161, -35, 10], [15, -23, 32, -19, 12, -3, 1, 0], [121, -439, 690, -803, 560, -315, 91, -26], [435, -1503, 2154, -2315, 1420, -771, 203, -58]];
    Rf_basisdens := [1, 1, 37, 1, 37, 1, 37, 37];
    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_6720_g();"
function MakeCharacter_6720_g()
    N := 6720;
    order := 2;
    char_gens := [1471, 3781, 4481, 5377, 1921];
    v := [2, 1, 2, 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_6720_g_Hecke();"
function MakeCharacter_6720_g_Hecke(Kf)
    N := 6720;
    order := 2;
    char_gens := [1471, 3781, 4481, 5377, 1921];
    char_values := [[1, 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, 0, 0, 0, 0, 0], [1, 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, -1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, -1, 0, 0, -1], [0, 0, 1, 0, 1, 0, -1, 0], [1, -1, 0, -1, 0, 1, 0, 0], [0, 0, -2, 0, 0, 0, -1, 0], [-1, -1, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 1, 0, -1, 1], [0, 0, 0, 2, 0, 1, 0, 0], [0, 0, 3, 0, -1, 0, 1, -1], [-2, 0, 0, -1, 0, 1, 0, 0], [0, 0, 1, 0, 1, 0, -2, 0], [-1, 1, 0, 2, 0, -2, 0, 0], [0, 0, -3, 0, 1, 0, 1, 2], [0, 0, -2, 0, 2, 0, 1, 3], [0, 0, 1, 0, -1, 0, 1, -1], [0, 0, -3, 0, -1, 0, 1, -1], [-1, 1, 0, 1, 0, -2, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [0, 2, 0, 0, 0, -4, 0, 0], [0, 0, -6, 0, 0, 0, 1, 1], [-4, -4, 0, 0, 0, 2, 0, 0], [2, 2, 0, 0, 0, 1, 0, 0], [0, 0, 4, 0, 0, 0, -3, 1], [0, 0, 0, 2, 0, -4, 0, 0], [0, 0, 1, 0, 3, 0, -4, 2], [0, 0, -6, 0, -4, 0, 3, -1], [2, 2, 0, 2, 0, -5, 0, 0], [-2, 0, 0, 1, 0, -3, 0, 0], [0, 0, -10, 0, -2, 0, 1, 1], [-4, 0, 0, 0, 0, -1, 0, 0], [0, 0, -6, 0, -4, 0, 1, -2], [0, 0, -1, 0, 1, 0, -3, 1], [4, 2, 0, 6, 0, -4, 0, 0], [0, 0, 3, 0, -1, 0, 3, 0], [0, 0, -9, 0, 3, 0, 0, 2], [-3, 1, 0, -1, 0, -5, 0, 0], [0, 0, -8, 0, -2, 0, 0, 2], [0, 0, 1, 0, -1, 0, 0, -5], [0, 0, 5, 0, -1, 0, -3, -3], [1, -1, 0, -1, 0, -2, 0, 0], [-2, 0, 0, 2, 0, -2, 0, 0], [0, 0, -3, 0, -3, 0, -3, 0], [0, -4, 0, 2, 0, 1, 0, 0], [0, 0, -12, 0, -2, 0, 1, -1], [0, 0, 0, 6, 0, -4, 0, 0], [0, 0, -4, 0, -2, 0, -5, -1], [0, 0, -1, 0, 5, 0, -1, 3], [-2, 2, 0, 4, 0, -7, 0, 0], [-7, -1, 0, -1, 0, 4, 0, 0], [-2, -4, 0, -6, 0, 6, 0, 0], [0, 0, 0, 0, 4, 0, -5, 1], [-7, -1, 0, 5, 0, -3, 0, 0], [5, 1, 0, 2, 0, 2, 0, 0], [0, 0, 6, 0, -2, 0, -3, 1], [-6, -6, 0, 0, 0, 1, 0, 0], [0, 0, 1, 0, 5, 0, -7, 3], [-18, 0, 0, 0, 0, 4, 0, 0], [0, 0, 0, 0, 4, 0, 2, 0], [0, 0, -12, 0, 2, 0, 2, 4], [0, 0, -12, 0, 0, 0, 0, 6], [-10, 2, 0, -2, 0, -2, 0, 0], [-2, -2, 0, 2, 0, -3, 0, 0], [0, 0, -1, 0, 3, 0, -7, 6], [0, 0, 4, 0, -2, 0, 1, -5], [2, -4, 0, -6, 0, -2, 0, 0], [0, 0, 5, 0, -5, 0, 8, -2], [0, 0, 5, 0, 3, 0, -5, 3], [-7, -5, 0, -3, 0, 3, 0, 0], [9, -1, 0, 1, 0, 4, 0, 0], [6, 2, 0, 2, 0, -2, 0, 0], [0, 0, -3, 0, -3, 0, 3, -5], [0, 0, 10, 0, -4, 0, -3, -3], [9, -3, 0, -1, 0, 3, 0, 0], [0, 0, 5, 0, -1, 0, -1, 5], [0, 0, -13, 0, 3, 0, -1, 2], [4, 6, 0, 2, 0, 0, 0, 0], [-6, 4, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 4, 0, -3, 1], [0, 0, 4, 0, 4, 0, 0, 8], [7, 1, 0, -5, 0, -2, 0, 0], [0, -4, 0, 2, 0, 5, 0, 0], [-6, 6, 0, 4, 0, -7, 0, 0], [0, 0, -1, 0, 1, 0, 4, -2], [2, 0, 0, 2, 0, 4, 0, 0], [10, -4, 0, 0, 0, 6, 0, 0], [0, 0, 16, 0, -6, 0, 0, 0], [6, -4, 0, -3, 0, 1, 0, 0], [0, 0, -12, 0, 2, 0, -5, 3], [6, -2, 0, -4, 0, -4, 0, 0], [0, -6, 0, -3, 0, 5, 0, 0], [0, 0, 9, 0, 3, 0, -4, 9], [0, 0, -10, 0, 0, 0, 1, -3], [-9, -1, 0, 1, 0, 1, 0, 0], [0, 0, 2, 0, -2, 0, -1, -5], [-12, 2, 0, -5, 0, 5, 0, 0], [0, 0, 4, 0, 0, 0, -2, 0], [0, 0, 18, 0, -2, 0, 8, -4], [0, 0, -11, 0, 3, 0, -1, 9], [0, 0, 1, 0, -3, 0, -1, 2], [0, 0, -2, 0, -10, 0, 10, -6], [-14, 0, 0, -10, 0, 0, 0, 0], [0, 0, -12, 0, 6, 0, 3, 3], [0, 8, 0, 10, 0, -3, 0, 0], [0, 0, 14, 0, 2, 0, -4, 8], [-3, -1, 0, 11, 0, -7, 0, 0], [15, -3, 0, -1, 0, 0, 0, 0], [-8, 2, 0, -6, 0, 0, 0, 0], [-12, 0, 0, 4, 0, 0, 0, 0], [0, 0, -3, 0, -3, 0, -3, -5], [-8, 8, 0, 0, 0, -5, 0, 0], [0, 0, -2, 0, -8, 0, 7, -2], [-2, 8, 0, 2, 0, -10, 0, 0], [4, -6, 0, -10, 0, 6, 0, 0], [0, 0, -6, 0, -6, 0, -2, -8], [-17, 3, 0, 5, 0, 1, 0, 0], [0, 0, -1, 0, 7, 0, -3, 6], [0, 0, -7, 0, 3, 0, -8, 7], [0, 0, 9, 0, 7, 0, -5, 5], [2, -4, 0, 4, 0, -2, 0, 0], [0, 0, -8, 0, 2, 0, -10, 4], [0, 0, 17, 0, 3, 0, 0, -8], [0, 0, -12, 0, 2, 0, -11, 6], [0, 0, 3, 0, 9, 0, -3, 13], [0, 0, 6, 0, 2, 0, 6, 2], [-18, -2, 0, -6, 0, 4, 0, 0], [-12, 8, 0, 8, 0, -6, 0, 0], [0, 0, -13, 0, -5, 0, 3, -4], [0, 0, -10, 0, 0, 0, -1, -1], [3, 7, 0, 0, 0, -2, 0, 0], [-18, -4, 0, 4, 0, 4, 0, 0], [0, 0, -1, 0, -5, 0, 7, -9], [-2, -2, 0, -4, 0, -6, 0, 0], [-2, 0, 0, 6, 0, 4, 0, 0], [0, 0, -12, 0, 2, 0, -8, 6], [0, 0, -10, 0, -6, 0, -6, -12], [0, 0, 8, 0, 10, 0, -2, 0], [2, 0, 0, -2, 0, 6, 0, 0], [0, 0, 10, 0, 0, 0, -11, 10], [0, 0, -3, 0, 3, 0, 1, 3], [6, -8, 0, 1, 0, 9, 0, 0], [0, 0, 7, 0, 1, 0, -8, 8], [0, 0, 1, 0, 3, 0, 1, -5], [4, 8, 0, 4, 0, -2, 0, 0], [0, 0, -33, 0, 3, 0, 5, 2], [-19, 3, 0, 5, 0, 9, 0, 0], [0, 0, 6, 0, 4, 0, -7, 12], [7, 7, 0, 6, 0, 0, 0, 0], [0, 0, -3, 0, 5, 0, 5, 7], [8, 8, 0, 6, 0, 0, 0, 0], [0, 0, -19, 0, -1, 0, 5, -5], [-1, 5, 0, -4, 0, -12, 0, 0], [0, 0, 17, 0, 3, 0, -1, -7], [-27, 3, 0, 1, 0, 4, 0, 0], [18, -4, 0, -4, 0, 4, 0, 0], [-6, 2, 0, 8, 0, 2, 0, 0], [32, -8, 0, -6, 0, 7, 0, 0], [0, 0, 26, 0, 4, 0, -4, 4], [0, 0, -17, 0, 1, 0, -2, -2], [-8, -4, 0, 10, 0, -3, 0, 0], [-10, -10, 0, 2, 0, 2, 0, 0], [0, 0, 24, 0, -8, 0, -5, -3], [0, -4, 0, -12, 0, 1, 0, 0], [-3, -5, 0, -10, 0, 2, 0, 0], [2, 0, 0, 6, 0, -6, 0, 0], [0, 0, -13, 0, 3, 0, 3, -6], [6, -2, 0, 1, 0, -11, 0, 0], [0, 0, -36, 0, 6, 0, -2, 0], [0, 0, 7, 0, -3, 0, 12, -9], [0, 0, 19, 0, -3, 0, 5, -9], [9, -9, 0, -3, 0, 8, 0, 0], [30, -4, 0, 4, 0, 0, 0, 0], [-2, -2, 0, -4, 0, -7, 0, 0], [-2, -4, 0, -1, 0, 1, 0, 0], [0, 0, 6, 0, 8, 0, -1, 5], [0, 0, 1, 0, 3, 0, -1, -9], [14, -6, 0, 4, 0, 4, 0, 0], [0, 0, -23, 0, 3, 0, 9, -5], [6, 6, 0, -2, 0, -10, 0, 0], [0, 0, 14, 0, -2, 0, -5, 9], [0, 0, -15, 0, 1, 0, 7, -1], [9, -1, 0, 9, 0, -1, 0, 0], [11, 3, 0, 0, 0, -2, 0, 0], [0, 0, 20, 0, 4, 0, 3, -1], [0, 0, 15, 0, -5, 0, 7, -11], [0, 0, 0, 0, 4, 0, 10, -6], [30, 2, 0, -1, 0, 3, 0, 0], [18, 14, 0, 8, 0, -6, 0, 0], [-8, 8, 0, 8, 0, -17, 0, 0], [0, 0, 23, 0, 1, 0, -2, 6], [0, 0, -34, 0, -4, 0, -5, -5], [0, 0, 4, 0, -2, 0, -4, -8], [0, 0, 7, 0, 5, 0, 4, 6], [7, 1, 0, 3, 0, -11, 0, 0], [-14, 10, 0, 7, 0, 3, 0, 0], [0, 0, 15, 0, -1, 0, 3, -13], [27, 1, 0, -5, 0, -5, 0, 0], [-7, 5, 0, -11, 0, 1, 0, 0], [0, 0, 5, 0, -1, 0, -1, -9], [-40, -4, 0, -4, 0, 1, 0, 0], [0, 0, 7, 0, -1, 0, 11, 6], [18, -4, 0, -6, 0, -2, 0, 0], [0, 0, 24, 0, 0, 0, 1, -5], [0, 0, 8, 0, -14, 0, 16, -6], [-22, -2, 0, 0, 0, 5, 0, 0], [0, 0, 11, 0, 5, 0, 0, -10], [26, -4, 0, -10, 0, 8, 0, 0], [0, 0, 12, 0, -10, 0, 7, -2], [-30, 4, 0, 2, 0, 4, 0, 0], [0, 0, -34, 0, -6, 0, 3, -1], [36, 0, 0, 4, 0, 4, 0, 0], [0, 0, 18, 0, -8, 0, -3, 1], [32, 0, 0, 10, 0, -4, 0, 0], [24, 2, 0, 0, 0, 6, 0, 0], [44, 6, 0, -1, 0, -1, 0, 0], [-2, 2, 0, -4, 0, 2, 0, 0], [9, 5, 0, -2, 0, 0, 0, 0], [0, 0, -29, 0, -5, 0, 9, -14], [0, 0, 38, 0, 6, 0, -4, -8], [14, -6, 0, -6, 0, -3, 0, 0], [-4, 6, 0, -6, 0, 8, 0, 0], [8, 0, 0, 0, 0, 0, 0, 0], [0, 0, -24, 0, 4, 0, -16, 4], [0, 0, 4, 0, -2, 0, 11, -5], [-9, 5, 0, 17, 0, -19, 0, 0], [-1, 1, 0, 5, 0, 8, 0, 0], [-56, 0, 0, -4, 0, 4, 0, 0], [0, 0, 37, 0, 3, 0, 4, 5], [0, 0, 25, 0, 13, 0, -5, 7], [0, 0, -24, 0, 10, 0, -1, 4], [-2, 16, 0, 4, 0, -10, 0, 0], [40, -10, 0, -8, 0, 14, 0, 0], [0, 0, -2, 0, 10, 0, 2, 4], [13, -5, 0, -6, 0, 6, 0, 0], [-12, -6, 0, -4, 0, -2, 0, 0], [0, 0, -23, 0, -3, 0, -1, -4], [0, 0, 19, 0, -7, 0, -4, -11], [0, -4, 0, -10, 0, 12, 0, 0], [0, 0, -5, 0, 13, 0, -2, 4], [0, 0, 8, 0, -10, 0, -7, 8], [28, 12, 0, -4, 0, -10, 0, 0], [0, 0, -20, 0, 6, 0, -3, -3], [3, 1, 0, 15, 0, -13, 0, 0], [6, 6, 0, 4, 0, 6, 0, 0], [-12, 4, 0, -10, 0, -2, 0, 0], [0, 0, -22, 0, 2, 0, -7, 5], [0, 0, 0, 0, 22, 0, -11, 11], [-9, -13, 0, -16, 0, 2, 0, 0], [0, 0, -5, 0, -9, 0, 5, -17], [14, -2, 0, -6, 0, -16, 0, 0], [-23, -3, 0, -12, 0, 12, 0, 0], [24, 2, 0, -2, 0, 0, 0, 0], [0, 0, 16, 0, 14, 0, -14, 12], [0, 0, 7, 0, -3, 0, 16, -13], [0, 0, 14, 0, 0, 0, 15, -9], [0, 0, 18, 0, -6, 0, -2, -4], [0, 0, 0, 0, 10, 0, 6, -4], [-10, -14, 0, -6, 0, 11, 0, 0], [26, -4, 0, 5, 0, 5, 0, 0], [0, 0, 7, 0, -15, 0, 4, -4], [0, 0, 7, 0, -7, 0, -1, -9], [0, 0, 11, 0, 11, 0, 3, 16], [15, 1, 0, 3, 0, 5, 0, 0], [0, 0, 0, 0, -6, 0, -11, 10], [0, 0, -19, 0, 11, 0, 5, 5], [-48, -6, 0, 5, 0, 7, 0, 0], [0, 0, 11, 0, 5, 0, 11, -11], [0, 0, -29, 0, -9, 0, 9, -6], [0, 0, -17, 0, 9, 0, -11, -1], [0, 0, 3, 0, -5, 0, 4, -10], [26, 6, 0, 8, 0, -13, 0, 0], [-24, -6, 0, -14, 0, 10, 0, 0], [2, -10, 0, -8, 0, 13, 0, 0], [-24, -4, 0, 6, 0, 8, 0, 0], [0, 0, -9, 0, -7, 0, -2, -12], [0, 0, 10, 0, 6, 0, 2, 14], [26, 4, 0, -4, 0, 4, 0, 0], [0, 0, -22, 0, 2, 0, -21, 13], [7, -15, 0, -2, 0, 10, 0, 0], [-12, 2, 0, 8, 0, -12, 0, 0], [-23, 7, 0, 8, 0, 8, 0, 0], [0, 0, -11, 0, 3, 0, 1, 13], [0, 0, -6, 0, 6, 0, -16, 20], [-9, -3, 0, -7, 0, -2, 0, 0], [-50, 0, 0, -10, 0, 0, 0, 0], [0, 0, 29, 0, -3, 0, 3, -2], [-12, -4, 0, 12, 0, 13, 0, 0], [-14, -6, 0, -12, 0, 6, 0, 0], [0, 0, 17, 0, -1, 0, 21, -3], [0, 0, -10, 0, 8, 0, 9, 17], [-30, -10, 0, 4, 0, 13, 0, 0], [0, 0, -2, 0, -10, 0, 3, 7], [0, 0, -33, 0, 11, 0, 5, 9], [0, 0, 0, 0, 6, 0, 4, -4], [-4, -12, 0, -10, 0, 11, 0, 0], [0, 0, -46, 0, 8, 0, -24, 4], [0, 0, 20, 0, 8, 0, -3, 11], [0, 0, 18, 0, 2, 0, -12, -4], [42, 2, 0, 10, 0, -3, 0, 0], [0, 0, -15, 0, -3, 0, -19, -10], [26, 0, 0, -10, 0, 4, 0, 0], [0, 0, -19, 0, 3, 0, -10, 12], [0, 0, 16, 0, 2, 0, -13, 3], [-6, 16, 0, 18, 0, -6, 0, 0], [0, 0, 49, 0, 3, 0, 0, 4], [0, 0, 23, 0, -15, 0, -3, -7], [43, 9, 0, 3, 0, 0, 0, 0], [0, 0, 7, 0, -9, 0, 9, -3], [-31, 7, 0, 0, 0, 12, 0, 0], [0, 0, 43, 0, -3, 0, -7, -1], [0, -18, 0, 0, 0, 20, 0, 0], [0, 0, -13, 0, 1, 0, 13, -13], [-53, 11, 0, 6, 0, -10, 0, 0], [4, -6, 0, 6, 0, 0, 0, 0], [0, 0, -20, 0, -8, 0, 15, -13], [-11, -5, 0, 3, 0, -8, 0, 0], [4, -4, 0, 4, 0, -1, 0, 0], [0, 18, 0, 12, 0, -2, 0, 0], [0, 0, 12, 0, -2, 0, 1, -10], [2, 12, 0, -10, 0, 0, 0, 0], [0, 0, 24, 0, 10, 0, -6, 18], [-12, -4, 0, -2, 0, -2, 0, 0], [-8, 0, 0, -10, 0, -5, 0, 0], [-18, -12, 0, 10, 0, 8, 0, 0], [0, 0, 44, 0, 6, 0, -5, 15], [0, 0, 12, 0, -20, 0, 6, -12], [15, 3, 0, -8, 0, -4, 0, 0], [0, 0, 3, 0, 3, 0, 5, -12], [0, 0, -18, 0, -4, 0, -1, 3], [0, 0, 17, 0, 13, 0, -3, 14], [-46, -2, 0, 10, 0, -3, 0, 0], [0, 0, 10, 0, 10, 0, -10, 6], [0, 0, -22, 0, 4, 0, -9, -3], [0, 0, 30, 0, 6, 0, 0, -8], [0, 0, -46, 0, -6, 0, -6, -6], [21, -1, 0, 13, 0, -15, 0, 0], [4, -6, 0, -2, 0, 8, 0, 0], [32, 8, 0, 0, 0, -6, 0, 0], [0, 0, -19, 0, -7, 0, 1, 13], [-42, -2, 0, -4, 0, 1, 0, 0], [0, 0, 2, 0, -6, 0, -9, -5], [6, -4, 0, 0, 0, 10, 0, 0], [0, 0, -25, 0, -5, 0, 1, 2], [0, 0, -13, 0, 5, 0, -4, 9], [0, 0, -27, 0, 7, 0, -9, 11], [0, 0, -15, 0, -1, 0, 3, 5], [-24, 0, 0, -4, 0, 6, 0, 0], [0, 0, 0, 0, 2, 0, 0, 14], [0, 0, 14, 0, 0, 0, -13, 10], [34, 16, 0, -8, 0, -6, 0, 0], [0, 0, -5, 0, 13, 0, -1, 7], [16, 16, 0, 12, 0, -10, 0, 0], [0, 0, 44, 0, -6, 0, -9, -5], [-7, -1, 0, 9, 0, -13, 0, 0], [22, 6, 0, -2, 0, -16, 0, 0], [0, 0, 12, 0, 12, 0, -13, 19], [-12, -4, 0, -14, 0, 1, 0, 0], [-19, 9, 0, 10, 0, 6, 0, 0], [0, 0, -31, 0, 9, 0, -5, 3], [-8, 10, 0, 19, 0, -15, 0, 0], [41, -15, 0, -8, 0, 26, 0, 0], [0, 0, 3, 0, 9, 0, 0, -11], [0, 0, -20, 0, 12, 0, 6, -2], [26, 12, 0, -4, 0, -6, 0, 0], [0, 0, 2, 0, -12, 0, 8, -24], [10, 6, 0, -6, 0, -10, 0, 0], [26, 0, 0, 6, 0, -18, 0, 0], [0, 0, -21, 0, 1, 0, -6, 7], [0, 0, -4, 0, 6, 0, -23, 12], [-41, 3, 0, 10, 0, -12, 0, 0], [0, 0, 51, 0, -7, 0, -3, -7], [14, -2, 0, -10, 0, 1, 0, 0], [0, 0, 23, 0, -1, 0, -11, 15], [0, 0, -38, 0, 6, 0, 11, -1], [-13, 5, 0, -11, 0, -10, 0, 0], [-30, -10, 0, -10, 0, 3, 0, 0], [0, 6, 0, -13, 0, -15, 0, 0], [0, 12, 0, 8, 0, 1, 0, 0], [0, 0, 8, 0, -4, 0, -15, 1], [34, -6, 0, 2, 0, -1, 0, 0], [16, 6, 0, 13, 0, -19, 0, 0], [-8, -4, 0, -2, 0, 17, 0, 0], [0, 0, -12, 0, 2, 0, -1, 10], [13, 5, 0, 15, 0, -9, 0, 0], [36, 2, 0, 20, 0, -16, 0, 0], [0, 0, -1, 0, 11, 0, 1, 8], [0, 0, -23, 0, -5, 0, 7, -3], [27, 5, 0, 12, 0, -8, 0, 0], [-6, 6, 0, 19, 0, -17, 0, 0], [0, 0, -2, 0, 16, 0, -6, 2], [0, 0, -27, 0, 15, 0, 10, 7], [0, 0, 14, 0, 6, 0, -8, 10], [5, 11, 0, -5, 0, -20, 0, 0], [-6, 0, 0, 4, 0, 8, 0, 0], [4, 8, 0, 0, 0, -5, 0, 0], [-28, 0, 0, -4, 0, 10, 0, 0], [0, 0, -14, 0, 0, 0, -3, -7], [0, 0, -3, 0, 11, 0, 11, -9], [0, 0, -29, 0, 9, 0, -15, 9], [10, -10, 0, -2, 0, -5, 0, 0], [26, -6, 0, 14, 0, 10, 0, 0], [-1, 5, 0, 13, 0, -11, 0, 0], [0, 0, 30, 0, -8, 0, 6, -2], [-74, -2, 0, -2, 0, 3, 0, 0], [0, 0, 47, 0, -21, 0, -1, -7], [-38, 8, 0, 14, 0, -2, 0, 0], [0, 0, 24, 0, 0, 0, -8, -24], [0, 0, -6, 0, -6, 0, 5, -13], [-28, -12, 0, -14, 0, 9, 0, 0], [0, 0, 57, 0, 13, 0, -13, 16], [0, 0, -25, 0, -7, 0, -8, 4], [0, 0, 6, 0, -8, 0, 5, -13], [-14, -16, 0, 4, 0, -4, 0, 0], [0, 0, -6, 0, 18, 0, -17, 7], [-11, -5, 0, 9, 0, 10, 0, 0], [2, -22, 0, -8, 0, 4, 0, 0], [9, -9, 0, 9, 0, 3, 0, 0], [-15, -17, 0, -8, 0, 12, 0, 0], [0, 0, 37, 0, 7, 0, -19, 21], [0, 0, 1, 0, -3, 0, -23, 2], [17, 13, 0, 12, 0, -6, 0, 0], [0, 0, -40, 0, -8, 0, 7, -11], [12, 12, 0, 10, 0, -3, 0, 0], [0, 0, -12, 0, 10, 0, 8, -2], [0, 0, -7, 0, -13, 0, 10, -2], [-20, -18, 0, -22, 0, 10, 0, 0], [0, 0, 18, 0, -12, 0, 23, -12], [32, 0, 0, 2, 0, -16, 0, 0], [14, 12, 0, 6, 0, -20, 0, 0], [0, 0, 5, 0, -1, 0, -18, 9], [0, 0, -6, 0, 4, 0, -13, 1], [17, 17, 0, 21, 0, -15, 0, 0], [0, 0, 41, 0, -3, 0, 21, 4], [-10, 16, 0, -5, 0, -15, 0, 0], [-14, -16, 0, -10, 0, 14, 0, 0], [0, 0, 18, 0, 16, 0, -13, 23], [0, 0, -17, 0, 5, 0, 5, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6720_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6720_2_g_p();".
// 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_6720_2_g_p(:prec:=8)
    chi := MakeCharacter_6720_g();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(3067) - 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_6720_2_g_p();".
// 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_6720_2_g_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6720_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![64, 0, 1344, 0, 560, 0, 48, 0, 1]>,<13,R![7744, 0, 7360, 0, 1200, 0, 64, 0, 1]>,<23,R![-32, -64, -16, 4, 1]>,<29,R![256, 0, 768, 0, 416, 0, 48, 0, 1]>,<47,R![208, -208, -64, 4, 1]>],Snew);
    return Vf;
end function;