
// Make newform 1014.2.g.b in Magma, downloaded from the LMFDB on 06 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1014_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1014_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_1014_2_g_b();".
// 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_1014_2_g_b();".
// 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 := [729, 0, 0, -324, 0, 0, 72, 0, 0, -12, 0, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-81, 0, 0, 18, 0, 0, -3, 0, 0, 1, 0, 0], [0, -81, 0, 0, 18, 0, 0, -3, 0, 0, 1, 0], [0, 0, 162, 0, 0, -45, 0, 0, 3, 0, 0, -1], [0, 0, 324, 0, 0, -72, 0, 0, 12, 0, 0, -1], [243, 0, 0, -63, 0, 0, 15, 0, 0, -2, 0, 0], [0, 243, 0, 0, -63, 0, 0, 15, 0, 0, -2, 0], [-324, 0, 0, 90, 0, 0, -15, 0, 0, 2, 0, 0], [0, -324, 0, 0, 90, 0, 0, -15, 0, 0, 2, 0], [0, 0, 135, 0, 0, -27, 0, 0, 6, 0, 0, -1], [0, 0, -486, 0, 0, 153, 0, 0, -30, 0, 0, 4]];
    Rf_basisdens := [1, 1, 81, 81, 243, 243, 81, 81, 81, 81, 81, 243];
    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_1014_g();"
function MakeCharacter_1014_g()
    N := 1014;
    order := 4;
    char_gens := [677, 847];
    v := [2, 3];
    // 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_1014_g_Hecke();"
function MakeCharacter_1014_g_Hecke(Kf)
    N := 1014;
    order := 4;
    char_gens := [677, 847];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 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]];
    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, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1], [0, 0, 0, -1, 1, 0, 0, 1, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, -1, 0, 1, 0, 0, 0, -1, -1, 1], [1, 0, 0, 1, -1, 0, 0, 1, 1, 0, -1, 0], [0, -1, 0, 0, 0, -1, 0, 0, 0, 1, 0, -1], [0, 1, 0, 0, 0, -1, 0, 0, 0, 1, 0, 1], [-1, 0, 0, -1, 1, 2, 0, -1, -1, 2, 1, 0], [-1, 1, 0, 1, 1, 0, 0, -1, 1, 0, -1, -1], [0, 0, 0, 0, 0, -2, -6, 0, 0, 2, 0, 0], [0, -2, 0, 1, 0, 2, 0, 0, 0, 2, -1, 2], [0, -1, 6, 1, -1, 0, 0, -1, 0, 0, 1, -1], [0, -1, 6, 0, 2, 1, -6, 2, 0, -1, 0, -1], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2], [0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, -2], [1, 0, 0, 0, 0, -2, 0, 0, 1, -2, 0, 0], [0, 0, 0, -1, -1, -1, 6, -1, 0, 1, -1, 0], [1, 0, 0, -1, -1, 0, 0, 1, -1, 0, 1, 0], [6, 1, 0, 0, 2, 1, 0, -2, 0, 1, 0, -1], [0, 0, 0, 3, 3, 2, 0, 3, 0, -2, 3, 0], [0, 0, 6, 2, -2, 0, 0, -2, 0, 0, 2, 0], [5, 0, 0, -2, 2, 0, 0, -2, 5, 0, 2, 0], [0, 0, 6, 4, 0, 0, 6, 0, 0, 0, 4, 0], [0, -1, 0, -2, 0, 1, 0, 0, -6, 1, 2, 1], [0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0], [-5, 0, 0, -1, 1, -1, 0, -1, -5, -1, 1, 0], [0, 2, -6, 0, -2, -2, 6, -2, 0, 2, 0, 2], [0, -1, 0, -4, 0, 1, 0, 0, -4, 1, 4, 1], [0, 2, 6, 0, -1, -2, -6, -1, 0, 2, 0, 2], [0, 2, 6, -3, 3, 0, 0, 3, 0, 0, -3, 2], [2, -2, 0, 0, 1, -2, 0, -1, 0, -2, 0, 2], [0, 0, 0, 2, 2, 0, 6, 2, 0, 0, 2, 0], [5, 1, 0, -2, -2, 0, 0, 2, -5, 0, 2, -1], [-2, -2, 0, 0, 4, -2, 0, -4, 0, -2, 0, 2], [-1, -4, 0, -4, -4, 0, 0, 4, 1, 0, 4, 4], [0, 2, 0, -2, 2, 0, 0, 2, 0, 0, -2, 2], [0, -1, 0, -6, 0, -1, 0, 0, 0, 1, -6, -1], [0, 2, 6, 1, 0, 2, 6, 0, 0, -2, 1, 2], [0, -1, 0, -2, 0, 1, 0, 0, 0, 1, 2, 1], [0, 3, 0, 0, 2, -3, 0, 2, 0, 3, 0, 3], [1, 2, 0, 0, 0, 0, 0, 0, -1, 0, 0, -2], [0, 0, 0, -2, -2, 1, 0, -2, 0, -1, -2, 0], [0, 3, 0, -6, 0, -3, 0, 0, 8, -3, 6, -3], [12, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 3, 0, 0, 1, 3, 0, 0], [0, 0, 0, -4, -4, 0, 0, -4, 0, 0, -4, 0], [-5, -1, 0, 1, 1, 0, 0, -1, 5, 0, -1, 1], [0, -1, 6, -1, 0, -1, 6, 0, 0, 1, -1, -1], [0, 0, 0, -1, -1, -5, 6, -1, 0, 5, -1, 0], [-1, -4, 0, 5, 5, 0, 0, -5, 1, 0, -5, 4], [0, -4, -6, 2, 0, -4, -6, 0, 0, 4, 2, -4], [0, 3, -6, -7, 0, 3, -6, 0, 0, -3, -7, 3], [0, -4, 0, 0, 0, 4, 0, 0, 0, -4, 0, -4], [0, 2, 6, 0, 0, -2, -6, 0, 0, 2, 0, 2], [-1, -3, 0, 4, 4, 0, 0, -4, 1, 0, -4, 3], [0, 2, 0, 8, 0, -2, 0, 0, 12, -2, -8, -2], [0, -4, 6, 0, 0, 0, 0, 0, 0, 0, 0, -4], [0, 2, 0, 2, 0, -2, 0, 0, -8, -2, -2, -2], [0, 3, -12, 0, 0, 0, 0, 0, 0, 0, 0, 3], [11, 0, 0, 3, 3, 0, 0, -3, -11, 0, -3, 0], [0, 2, -6, 2, 0, 2, -6, 0, 0, -2, 2, 2], [6, 1, 0, 0, 3, 1, 0, -3, 0, 1, 0, -1], [0, 0, 0, -2, -2, -4, -6, -2, 0, 4, -2, 0], [1, 0, 0, 0, 0, -4, 0, 0, 1, -4, 0, 0], [0, 1, 0, 7, 0, -1, 0, 0, 4, -1, -7, -1], [0, -4, -12, 0, -1, 4, 12, -1, 0, -4, 0, -4], [-11, -1, 0, -3, -3, 0, 0, 3, 11, 0, 3, 1], [0, 0, 0, -1, -1, 2, 6, -1, 0, -2, -1, 0], [0, -2, 0, -4, 4, 0, 0, 4, 0, 0, -4, -2], [16, -1, 0, 0, 2, -1, 0, -2, 0, -1, 0, 1], [12, -2, 0, 0, -4, -2, 0, 4, 0, -2, 0, 2], [5, 0, 0, -4, 4, 0, 0, -4, 5, 0, 4, 0], [0, 0, 0, 1, 1, 7, -6, 1, 0, -7, 1, 0], [0, 0, 0, -2, 0, 0, 0, 0, 0, 0, -2, 0], [-5, 6, 0, -3, -3, 0, 0, 3, 5, 0, 3, -6], [0, -2, -6, 5, -5, 0, 0, -5, 0, 0, 5, -2], [-5, 0, 0, -4, 4, -2, 0, -4, -5, -2, 4, 0], [0, 2, 6, 0, 7, -2, -6, 7, 0, 2, 0, 2], [7, 0, 0, 3, -3, 5, 0, 3, 7, 5, -3, 0], [0, 0, 0, 1, 1, -5, 18, 1, 0, 5, 1, 0], [0, -9, 0, -3, 0, 9, 0, 0, 16, 9, 3, 9], [0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0], [0, -8, -6, 0, -3, 8, 6, -3, 0, -8, 0, -8], [0, -4, 6, -7, 7, 0, 0, 7, 0, 0, -7, -4], [-1, 0, 0, 1, -1, -8, 0, 1, -1, -8, -1, 0], [0, 0, 0, 6, 6, -9, 0, 6, 0, 9, 6, 0], [-1, -6, 0, -1, -1, 0, 0, 1, 1, 0, 1, 6], [0, -2, 18, 2, 0, -2, 18, 0, 0, 2, 2, -2], [0, 3, -6, 7, -7, 0, 0, -7, 0, 0, 7, 3], [-1, 0, 0, 3, -3, 2, 0, 3, -1, 2, -3, 0], [0, 0, -12, 7, 0, 0, -12, 0, 0, 0, 7, 0], [5, 0, 0, 1, -1, -2, 0, 1, 5, -2, -1, 0], [0, 2, 12, 0, -6, -2, -12, -6, 0, 2, 0, 2], [0, 0, 0, -4, -4, -4, -6, -4, 0, 4, -4, 0], [0, 3, -6, 0, 1, -3, 6, 1, 0, 3, 0, 3], [4, 2, 0, 0, -4, 2, 0, 4, 0, 2, 0, -2], [-5, 9, 0, -5, -5, 0, 0, 5, 5, 0, 5, -9], [26, 0, 0, 0, -3, 0, 0, 3, 0, 0, 0, 0], [0, 1, 0, -4, 4, 0, 0, 4, 0, 0, -4, 1], [0, 10, 6, -3, 0, 10, 6, 0, 0, -10, -3, 10], [0, -5, -6, -5, 0, -5, -6, 0, 0, 5, -5, -5], [0, 2, 0, -7, 0, -2, 0, 0, -10, -2, 7, -2], [-19, 0, 0, 3, -3, 4, 0, 3, -19, 4, -3, 0], [0, 0, 0, -1, -1, -2, 24, -1, 0, 2, -1, 0], [0, 8, -6, -1, 1, 0, 0, 1, 0, 0, -1, 8], [0, 2, -24, 0, -2, -2, 24, -2, 0, 2, 0, 2], [-18, -7, 0, 0, -1, -7, 0, 1, 0, -7, 0, 7], [4, 6, 0, 0, 6, 6, 0, -6, 0, 6, 0, -6], [5, 0, 0, 7, -7, 0, 0, 7, 5, 0, -7, 0], [0, 0, 0, -4, -4, -6, -18, -4, 0, 6, -4, 0], [13, 6, 0, -1, -1, 0, 0, 1, -13, 0, 1, -6], [7, -1, 0, 2, 2, 0, 0, -2, -7, 0, -2, 1], [0, -6, -12, 2, 0, -6, -12, 0, 0, 6, 2, -6], [1, 0, 0, 0, 0, 6, 0, 0, 1, 6, 0, 0], [0, 2, 0, -2, 0, 2, 0, 0, 0, -2, -2, 2], [0, 2, 6, 0, -12, -2, -6, -12, 0, 2, 0, 2], [0, 0, 18, 0, -6, 0, -18, -6, 0, 0, 0, 0], [5, -12, 0, 5, 5, 0, 0, -5, -5, 0, -5, 12], [0, 3, 0, -5, 0, -3, 0, 0, 30, -3, 5, -3], [0, 3, -6, 0, -6, -3, 6, -6, 0, 3, 0, 3], [0, -6, 12, 1, -1, 0, 0, -1, 0, 0, 1, -6], [-19, 0, 0, -1, 1, 2, 0, -1, -19, 2, 1, 0], [0, 1, 18, 2, 0, 1, 18, 0, 0, -1, 2, 1], [11, -6, 0, -1, -1, 0, 0, 1, -11, 0, 1, 6], [0, -5, -12, 2, 0, -5, -12, 0, 0, 5, 2, -5], [0, -1, 0, 2, 0, 1, 0, 0, -8, 1, -2, 1], [-7, 0, 0, -5, 5, 7, 0, -5, -7, 7, 5, 0], [1, -2, 0, 5, 5, 0, 0, -5, -1, 0, -5, 2], [0, 0, 0, -1, -1, 3, -6, -1, 0, -3, -1, 0], [0, -4, 0, -2, 0, 4, 0, 0, -6, 4, 2, 4], [-24, -3, 0, 0, -2, -3, 0, 2, 0, -3, 0, 3], [0, 0, -30, -4, 4, 0, 0, 4, 0, 0, -4, 0], [5, 0, 0, -9, 9, 2, 0, -9, 5, 2, 9, 0], [0, 0, 0, -6, -6, 11, 0, -6, 0, -11, -6, 0], [-13, 2, 0, -1, -1, 0, 0, 1, 13, 0, 1, -2], [0, -6, 12, -10, 0, -6, 12, 0, 0, 6, -10, -6], [0, -5, 6, 0, 11, 5, -6, 11, 0, -5, 0, -5], [-11, 0, 0, 1, -1, -14, 0, 1, -11, -14, -1, 0], [0, 0, 0, -2, -2, 5, 12, -2, 0, -5, -2, 0], [0, -2, 0, 12, 0, 2, 0, 0, -18, 2, -12, 2], [0, -2, 12, 6, -6, 0, 0, -6, 0, 0, 6, -2], [0, -5, 0, -2, 0, 5, 0, 0, -14, 5, 2, 5], [0, 10, 24, 0, 0, 0, 0, 0, 0, 0, 0, 10], [-13, -13, 0, -1, -1, 0, 0, 1, 13, 0, 1, 13], [0, 10, 6, -10, 0, 10, 6, 0, 0, -10, -10, 10], [-6, -8, 0, 0, 2, -8, 0, -2, 0, -8, 0, 8], [0, 0, 0, 1, 1, -19, -6, 1, 0, 19, 1, 0], [11, 0, 0, 5, -5, -9, 0, 5, 11, -9, -5, 0], [0, 1, 18, 5, 0, 1, 18, 0, 0, -1, 5, 1], [0, 0, 0, 9, 0, 0, 0, 0, -10, 0, -9, 0], [0, -2, 6, 0, 8, 2, -6, 8, 0, -2, 0, -2], [0, -4, 0, -1, 0, 4, 0, 0, 2, 4, 1, 4], [0, -1, 12, 0, 2, 1, -12, 2, 0, -1, 0, -1], [-30, -8, 0, 0, -2, -8, 0, 2, 0, -8, 0, 8], [0, 0, 0, 6, 6, 2, -6, 6, 0, -2, 6, 0], [-12, 2, 0, 0, 16, 2, 0, -16, 0, 2, 0, -2], [0, 0, 0, 4, 4, -3, 36, 4, 0, 3, 4, 0], [0, -12, 12, -1, 1, 0, 0, 1, 0, 0, -1, -12], [0, 7, -18, 5, 0, 7, -18, 0, 0, -7, 5, 7], [13, 0, 0, -10, 10, -11, 0, -10, 13, -11, 10, 0], [0, -6, 24, 0, 9, 6, -24, 9, 0, -6, 0, -6], [0, 0, 0, 7, 7, 4, -18, 7, 0, -4, 7, 0], [0, 11, 6, -5, 5, 0, 0, 5, 0, 0, -5, 11], [-4, 4, 0, 0, -14, 4, 0, 14, 0, 4, 0, -4], [10, 1, 0, 0, -14, 1, 0, 14, 0, 1, 0, -1], [-13, -4, 0, 3, 3, 0, 0, -3, 13, 0, -3, 4], [0, 9, 0, 12, 0, 9, 0, 0, 0, -9, 12, 9], [0, 0, 0, 1, 1, 0, -24, 1, 0, 0, 1, 0], [11, -7, 0, -5, -5, 0, 0, 5, -11, 0, 5, 7], [0, -8, 18, -8, 0, -8, 18, 0, 0, 8, -8, -8], [-1, 0, 0, 9, -9, 10, 0, 9, -1, 10, -9, 0], [0, -5, 0, 8, 0, 5, 0, 0, -30, 5, -8, 5], [0, 1, -24, 0, -1, -1, 24, -1, 0, 1, 0, 1], [13, -2, 0, -7, -7, 0, 0, 7, -13, 0, 7, 2], [0, -4, 6, 10, -10, 0, 0, -10, 0, 0, 10, -4], [0, -2, 0, -6, 0, 2, 0, 0, -6, 2, 6, 2], [-12, 5, 0, 0, 2, 5, 0, -2, 0, 5, 0, -5], [-1, 10, 0, -3, -3, 0, 0, 3, 1, 0, 3, -10], [0, -8, 12, 6, 0, -8, 12, 0, 0, 8, 6, -8], [12, 7, 0, 0, -8, 7, 0, 8, 0, 7, 0, -7], [0, 0, 0, 6, 6, 4, -30, 6, 0, -4, 6, 0], [0, -10, 12, -2, 2, 0, 0, 2, 0, 0, -2, -10], [0, -6, 0, -6, 0, -6, 0, 0, 0, 6, -6, -6], [0, -6, 0, -2, 0, 6, 0, 0, 36, 6, 2, 6], [23, 0, 0, -3, 3, 8, 0, -3, 23, 8, 3, 0], [11, 10, 0, -3, -3, 0, 0, 3, -11, 0, 3, -10], [0, 4, 12, -8, 8, 0, 0, 8, 0, 0, -8, 4], [-16, -3, 0, 0, 3, -3, 0, -3, 0, -3, 0, 3], [0, 0, 0, 2, 2, 12, 12, 2, 0, -12, 2, 0], [-26, 0, 0, 0, 15, 0, 0, -15, 0, 0, 0, 0], [0, 16, 18, -4, 4, 0, 0, 4, 0, 0, -4, 16], [0, -6, -6, -7, 0, -6, -6, 0, 0, 6, -7, -6], [0, -2, -6, 10, 0, -2, -6, 0, 0, 2, 10, -2], [7, 0, 0, 1, -1, 2, 0, 1, 7, 2, -1, 0], [0, -15, 0, 0, 0, 15, 0, 0, 14, 15, 0, 15], [0, -14, 6, -7, 7, 0, 0, 7, 0, 0, -7, -14], [0, 2, 18, 0, 10, -2, -18, 10, 0, 2, 0, 2], [0, -17, -12, 8, -8, 0, 0, -8, 0, 0, 8, -17], [-20, 3, 0, 0, 12, 3, 0, -12, 0, 3, 0, -3], [5, 0, 0, 7, -7, 3, 0, 7, 5, 3, -7, 0], [-20, -9, 0, 0, -9, -9, 0, 9, 0, -9, 0, 9], [0, -8, -24, 2, -2, 0, 0, -2, 0, 0, 2, -8], [0, -8, 12, 0, -6, 8, -12, -6, 0, -8, 0, -8], [-13, 0, 0, 6, -6, 13, 0, 6, -13, 13, -6, 0], [0, 2, -36, 0, 8, -2, 36, 8, 0, 2, 0, 2], [0, 0, 0, 9, 9, -8, -18, 9, 0, 8, 9, 0], [0, 2, 0, -7, 0, -2, 0, 0, 14, -2, 7, -2], [0, 6, 0, 6, 0, -6, 0, 0, -26, -6, -6, -6], [0, 3, 6, 0, -8, -3, -6, -8, 0, 3, 0, 3], [6, 1, 0, 0, 2, 1, 0, -2, 0, 1, 0, -1], [0, -10, 12, -2, 2, 0, 0, 2, 0, 0, -2, -10], [0, 0, 0, -7, -7, 3, -6, -7, 0, -3, -7, 0], [-23, 2, 0, -4, -4, 0, 0, 4, 23, 0, 4, -2], [-18, 2, 0, 0, 2, 2, 0, -2, 0, 2, 0, -2], [0, -11, 0, 0, 1, 11, 0, 1, 0, -11, 0, -11], [0, 0, 0, -2, -2, -2, -12, -2, 0, 2, -2, 0], [0, 7, 12, 8, -8, 0, 0, -8, 0, 0, 8, 7], [-14, -10, 0, 0, -10, -10, 0, 10, 0, -10, 0, 10], [-23, -1, 0, 4, 4, 0, 0, -4, 23, 0, -4, 1], [0, 0, 0, -2, -2, -2, 18, -2, 0, 2, -2, 0], [-13, 0, 0, -14, 14, 7, 0, -14, -13, 7, 14, 0], [0, -12, 12, -7, 0, -12, 12, 0, 0, 12, -7, -12], [0, 8, 0, -6, 0, -8, 0, 0, 0, -8, 6, -8], [0, -1, -24, 0, 15, 1, 24, 15, 0, -1, 0, -1], [0, -9, 6, 0, 8, 9, -6, 8, 0, -9, 0, -9], [0, 0, 0, 6, 0, 0, 0, 0, -20, 0, -6, 0], [0, 0, 12, 11, -11, 0, 0, -11, 0, 0, 11, 0], [0, 6, 0, 2, 0, -6, 0, 0, 36, -6, -2, -6], [-16, -2, 0, 0, -2, -2, 0, 2, 0, -2, 0, 2], [5, 0, 0, -9, 9, -18, 0, -9, 5, -18, 9, 0], [0, -1, -24, -7, 0, -1, -24, 0, 0, 1, -7, -1], [26, 12, 0, 0, 9, 12, 0, -9, 0, 12, 0, -12], [0, 0, 0, -1, -1, 9, 6, -1, 0, -9, -1, 0], [13, -4, 0, -9, -9, 0, 0, 9, -13, 0, 9, 4], [0, -17, 24, -8, 8, 0, 0, 8, 0, 0, -8, -17], [0, 10, 24, -6, 0, 10, 24, 0, 0, -10, -6, 10], [0, 11, 12, 0, -2, -11, -12, -2, 0, 11, 0, 11], [0, 0, 0, 4, 4, -6, 12, 4, 0, 6, 4, 0], [0, 4, 0, 1, 0, -4, 0, 0, -50, -4, -1, -4], [-42, 6, 0, 0, -10, 6, 0, 10, 0, 6, 0, -6], [1, 0, 0, 9, -9, -7, 0, 9, 1, -7, -9, 0], [0, 0, 0, 3, 3, -4, 54, 3, 0, 4, 3, 0], [0, 3, -24, -16, 0, 3, -24, 0, 0, -3, -16, 3], [1, 1, 0, 10, 10, 0, 0, -10, -1, 0, -10, -1], [0, 4, 24, 4, -4, 0, 0, -4, 0, 0, 4, 4], [-17, 0, 0, 6, -6, 6, 0, 6, -17, 6, -6, 0], [0, -6, 18, 2, 0, -6, 18, 0, 0, 6, 2, -6], [5, 15, 0, 3, 3, 0, 0, -3, -5, 0, -3, -15], [0, 0, 0, 13, 13, 4, -6, 13, 0, -4, 13, 0], [0, 5, -18, -13, 13, 0, 0, 13, 0, 0, -13, 5], [0, -5, 0, 13, 0, 5, 0, 0, 20, 5, -13, 5], [0, 1, 24, 0, -6, -1, -24, -6, 0, 1, 0, 1], [0, 0, 36, -5, 5, 0, 0, 5, 0, 0, -5, 0], [12, 11, 0, 0, 10, 11, 0, -10, 0, 11, 0, -11], [-1, 0, 0, 1, -1, 4, 0, 1, -1, 4, -1, 0], [0, -5, -18, -2, 0, -5, -18, 0, 0, 5, -2, -5], [19, 0, 0, 7, -7, 4, 0, 7, 19, 4, -7, 0], [0, -6, 0, 0, 0, 6, 0, 0, -32, 6, 0, 6], [0, 4, -12, 0, 3, -4, 12, 3, 0, 4, 0, 4], [-19, 0, 0, -7, 7, 1, 0, -7, -19, 1, 7, 0], [10, 3, 0, 0, 6, 3, 0, -6, 0, 3, 0, -3], [0, -6, -18, 4, -4, 0, 0, -4, 0, 0, 4, -6], [12, 4, 0, 0, -17, 4, 0, 17, 0, 4, 0, -4], [0, 0, 0, 2, 2, 5, -12, 2, 0, -5, 2, 0], [0, 0, 0, -5, -5, -2, 6, -5, 0, 2, -5, 0], [-23, -10, 0, 7, 7, 0, 0, -7, 23, 0, -7, 10], [0, 4, 30, 8, 0, 4, 30, 0, 0, -4, 8, 4], [0, -5, 0, -8, 0, 5, 0, 0, -14, 5, 8, 5], [7, 0, 0, -5, 5, 12, 0, -5, 7, 12, 5, 0], [5, 0, 0, -18, -18, 0, 0, 18, -5, 0, 18, 0], [0, -5, 0, 2, 0, 5, 0, 0, 18, 5, -2, 5], [24, 4, 0, 0, -8, 4, 0, 8, 0, 4, 0, -4], [-17, 0, 0, 5, -5, -8, 0, 5, -17, -8, -5, 0], [0, 0, 0, 12, 12, -10, -12, 12, 0, 10, 12, 0], [-43, -6, 0, -3, -3, 0, 0, 3, 43, 0, 3, 6], [-17, 10, 0, -7, -7, 0, 0, 7, 17, 0, 7, -10], [0, -2, -24, -7, 0, -2, -24, 0, 0, 2, -7, -2], [0, -13, -12, 0, -4, 13, 12, -4, 0, -13, 0, -13], [-1, -23, 0, -4, -4, 0, 0, 4, 1, 0, 4, 23], [0, 3, -12, 0, 22, -3, 12, 22, 0, 3, 0, 3], [17, 0, 0, 5, -5, -14, 0, 5, 17, -14, -5, 0], [19, -4, 0, 8, 8, 0, 0, -8, -19, 0, -8, 4], [0, -2, 12, 18, 0, -2, 12, 0, 0, 2, 18, -2], [-6, -3, 0, 0, -13, -3, 0, 13, 0, -3, 0, 3], [0, 0, 0, 8, 8, -9, 12, 8, 0, 9, 8, 0], [19, -11, 0, 4, 4, 0, 0, -4, -19, 0, -4, 11], [0, -2, 6, 16, 0, -2, 6, 0, 0, 2, 16, -2], [0, 2, 12, 0, -4, -2, -12, -4, 0, 2, 0, 2], [0, -6, 12, 0, -25, 6, -12, -25, 0, -6, 0, -6], [0, 0, 0, 8, 8, -6, -18, 8, 0, 6, 8, 0], [0, 12, 0, 3, -3, 0, 0, -3, 0, 0, 3, 12], [2, -13, 0, 0, -10, -13, 0, 10, 0, -13, 0, 13], [0, 10, -24, 12, 0, 10, -24, 0, 0, -10, 12, 10], [8, -5, 0, 0, -2, -5, 0, 2, 0, -5, 0, 5], [0, 8, 24, -4, 0, 8, 24, 0, 0, -8, -4, 8], [0, -4, 12, 0, -17, 4, -12, -17, 0, -4, 0, -4], [41, 2, 0, -4, -4, 0, 0, 4, -41, 0, 4, -2], [0, 3, 0, -13, 0, -3, 0, 0, -30, -3, 13, -3], [0, 13, 12, -10, 10, 0, 0, 10, 0, 0, -10, 13], [0, 5, 0, 8, 0, -5, 0, 0, 6, -5, -8, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1014_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1014_2_g_b();".
// 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_1014_2_g_b(:prec:=12)
    chi := MakeCharacter_1014_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(1999) - 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_1014_2_g_b();".
// 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_1014_2_g_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1014_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1296, 0, 0, 0, 5265, 0, 0, 0, 162, 0, 0, 0, 1]>,<7,R![338, -78, 9, -8, 18, -6, 1]>],Snew);
    return Vf;
end function;