
// Make newform 1024.2.g.a in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1024_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1024_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_1024_2_g_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_1024_2_g_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 := [1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 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, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 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, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0], [0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, -1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0], [0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, -1, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0], [0, -1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0], [0, -1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0], [0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1], [0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1];
    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_1024_g();"
function MakeCharacter_1024_g()
    N := 1024;
    order := 8;
    char_gens := [1023, 5];
    v := [8, 7];
    // 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_1024_g_Hecke();"
function MakeCharacter_1024_g_Hecke(Kf)
    N := 1024;
    order := 8;
    char_gens := [1023, 5];
    char_values := [[1, 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]];
    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, 1], [-1, 0, 0, 1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, -1, 0, 1], [0, 0, 0, 0, 0, -2, 0, -1, 0, 0, -1, 0, 1, 0, 0, 0], [0, 2, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [1, 3, 0, -1, -1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 2, 0, 1, -2, -1, 0], [0, 0, 0, 0, 0, 3, 2, 3, 0, 0, 1, 0, -1, -2, -3, 0], [2, -2, -2, 2, -3, 0, 0, 0, -1, -5, 0, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, -3, -4, 0, 0, -1, 0, 1, 2, 0, -4], [-3, 2, 1, 1, -1, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0], [-1, -2, -2, 1, 0, 0, 0, 0, -6, -2, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 0, -2, 5, 0, 0], [0, 0, 0, 0, 0, -4, 2, 1, 0, 0, -2, 0, -4, 0, -1, 1], [3, 4, -1, -1, -1, 0, 0, 0, -2, -1, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, -5, 0, 2, -1, 1, -1], [2, -2, 1, 2, -4, 0, 0, 0, -2, -3, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0, -1, 0, 3], [0, 0, 0, 0, 0, -2, -3, -2, 0, 0, 0, 0, 0, -2, 2, -1], [4, 3, 3, -4, 0, 0, 0, 0, 2, 5, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 3, -1, -1, 0, 0, 1, 0, 3, 0, 1, -1], [0, 0, 0, 0, 0, 6, 6, 5, 0, 0, 2, 0, 2, -2, -3, 0], [2, 1, -3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 5, 0, -3, 3, 0, 0, 0, 5, 3, 0, 0, 0, 0, 0, 0], [-5, 0, -5, -1, 5, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 3, 4, 0, 0, 0, 0, 0, 2, -2, -1], [0, 0, 0, 0, 0, -2, 0, -1, 0, 0, -1, 0, 1, 2, 2, 2], [0, 4, 0, 4, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 6, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 3, 7, 4, 0, 0, 3, 0, -1, -4, -6, 2], [0, 0, 0, 0, 0, 7, -7, -3, 0, 0, 6, 0, 3, 6, 0, -5], [-5, 0, 0, 5, 0, 0, 0, 0, -4, -5, 0, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -4, 2, 2], [-5, -4, 2, -7, 2, 0, 0, 0, -5, 2, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, -7, 2, -7, 0, 0, -9, 0, -3, -2, 1, 0], [4, 8, 5, -8, 4, 0, 0, 0, 8, 9, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, -8, 8, -3, 0, 0, -3, 0, 3, -3, 0, 1], [0, 0, 0, 0, 0, 9, -2, -3, 0, 0, 0, 0, 0, 9, 0, 1], [8, -8, -1, -8, 0, 0, 0, 0, -8, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, -4, -4, -6, 0, 0, -3, 0, -5, 3, 1, 0], [-3, -4, 2, -7, 2, 0, 0, 0, -5, 2, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, 11, 6, 0, 0, -3, 0, 7, -14, -2, 2], [1, 7, 8, 3, -3, 0, 0, 0, 7, 1, 0, 4, 0, 0, 0, 0], [1, -8, -8, -11, 8, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, 2, -1, 0, 0, 0, 0, 0, -3, 2, 3], [0, 0, 0, 0, 0, -4, -4, 1, 0, 0, -2, 0, 2, 2, 1, 0], [0, 0, 0, 0, 0, -1, -1, -2, 0, 0, -1, 0, -9, 0, 0, 8], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 2, 0, 2, 2, 0, -7], [13, -6, 4, 7, -4, 0, 0, 0, -3, 3, 0, 0, 0, 0, 0, 0], [-6, 9, 9, 6, 0, 0, 0, 0, 6, 5, 0, 13, 0, 0, 0, 0], [0, 0, 0, 0, 0, -9, -1, 1, 0, 0, 1, 0, -1, 0, -1, 9], [-5, -3, 0, -7, 5, 0, 0, 0, 1, -1, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 5, 0, 0, -2, 0, -5, -4, -9, 9], [2, 4, -6, -2, 2, 0, 0, 0, 4, -1, 0, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 1, 4, 0, 0, 0, 0, 0, 2, -8, -3], [8, -2, -8, -2, 5, 0, 0, 0, -13, 0, 0, -8, 0, 0, 0, 0], [0, 0, 0, 0, 0, 8, -4, 7, 0, 0, 4, 0, -4, 0, -7, -5], [-13, -2, 5, -7, 5, 0, 0, 0, -2, 5, 0, 11, 0, 0, 0, 0], [2, 13, -5, 2, 10, 0, 0, 0, 0, 5, 0, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -10, -7, 0, 0, 2, 0, 0, 5, 5, -5], [-3, -8, -6, 1, 6, 0, 0, 0, -7, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, -1, -5, 0, 0, -5, 0, 2, 5, 7, 0], [0, 0, 0, 0, 0, 0, -7, -6, 0, 0, 7, 0, -1, 7, 3, -6], [9, -6, 4, 9, -8, 0, 0, 0, 0, -4, 0, -4, 0, 0, 0, 0], [-12, 6, -6, -6, -1, 0, 0, 0, -13, -7, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 8, 0, 4, 0, 0, 4, 0, -4, -1, -12, -12], [4, -6, 0, 12, -4, 0, 0, 0, -10, -8, 0, -12, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 9, 0, 0, -2, 0, 5, -4, 1, -1], [-6, -8, 0, 8, -3, 0, 0, 0, -9, -3, 0, -3, 0, 0, 0, 0], [0, -4, 2, 4, -4, 0, 0, 0, -4, -3, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 6, -1, -2, 0, 0, 0, 0, 0, 6, 4, 1], [0, 0, 0, 0, 0, 2, 2, 5, 0, 0, -2, 0, 2, 0, -5, 5], [-7, -8, -9, 19, -9, 0, 0, 0, 10, -9, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -4, 0, 5, 0, 5, -5], [0, 0, 0, 0, 0, -8, -10, -8, 0, 0, -8, 0, -2, 2, 10, 2], [-15, 2, -5, 1, 5, 0, 0, 0, -6, -8, 0, 0, 0, 0, 0, 0], [6, 0, -8, 0, -3, 0, 0, 0, -3, 0, 0, -8, 0, 0, 0, 0], [-7, 7, 0, -1, 7, 0, 0, 0, -1, 3, 0, 10, 0, 0, 0, 0], [-13, 0, -2, -13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 3, -3, -7, 0, 0, -4, 0, 7, -4, 0, -5], [-9, 10, 2, -17, -2, 0, 0, 0, 19, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -13, 1, 1, 0, 0, -1, 0, -5, 0, -1, 9], [5, 11, 0, -5, -5, 0, 0, 0, -5, 3, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 7, 12, 0, 0, -3, 0, 5, -7, -3, 8], [0, 0, 0, 0, 0, 0, -20, 0, 0, 0, 3, 0, -9, 10, 1, -1], [1, -5, -8, 7, -7, 0, 0, 0, -5, -11, 0, -4, 0, 0, 0, 0], [-7, -8, -8, 7, 0, 0, 0, 0, -18, -6, 0, -10, 0, 0, 0, 0], [-14, -6, 2, -6, 5, 0, 0, 0, 9, 0, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, -11, -1, -15, 0, 0, 1, 0, 5, 0, 15, 1], [0, 0, 0, 0, 0, 1, 1, -2, 0, 0, 4, 0, 1, -4, 3, 0], [0, 0, 0, 0, 0, -5, -1, -4, 0, 0, -5, 0, 7, -4, 2, -6], [0, 0, 0, 0, 0, -9, 2, -5, 0, 0, 0, 0, 0, -9, 2, 7], [0, 0, 0, 0, 0, -14, 0, -7, 0, 0, -7, 0, 0, 5, 2, 2], [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 5, 0, 9, -5, 5, 0], [0, 0, 0, 0, 0, 9, 12, 1, 0, 0, -3, 0, -1, -12, -11, -8], [10, 0, -5, 0, 0, 0, 0, 0, 10, -5, 0, 0, 0, 0, 0, 0], [13, -6, -6, -13, 0, 0, 0, 0, -8, -1, 0, -11, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, -1, 0, 0, -1, 0, -9, 0, -6, -6], [-2, 6, -5, -6, -6, 0, 0, 0, -8, -11, 0, -6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 7, -7, 8, 0, 0, 5, 0, -8, 5, 0, -1], [2, -2, 5, -2, 10, 0, 0, 0, -12, 0, 0, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 11, 11, 20, 0, 0, 10, 0, 15, -10, -5, 0], [15, -2, 2, 15, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 11, 0, 7, -2, 5, -5], [6, -6, 0, -16, 16, 0, 0, 0, -6, 16, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -8, 0, -4, 0, 0, -4, 0, 8, 5, 2, 2], [6, 18, 0, -10, -6, 0, 0, 0, 6, 12, 0, 6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 8, 1, 8, 0, 0, 7, 0, 3, -1, -3, 0], [10, 17, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 8, 2, 10, 0, 0, 8, 0, 2, 6, 0, 0], [-1, -6, -4, 15, 4, 0, 0, 0, -19, -3, 0, 0, 0, 0, 0, 0], [2, -11, -11, -2, 0, 0, 0, 0, -8, -14, 0, -8, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, -1, 0, 0, -1, 0, -5, -6, -12, -12], [0, 0, 0, 0, 0, -7, -2, -1, 0, 0, -5, 0, 5, 2, 7, 6], [7, -19, -8, 9, -9, 0, 0, 0, -19, -13, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, -11, 11, -2, 0, 0, -15, 0, 2, -15, 0, 3], [0, 0, 0, 0, 0, -2, 15, 8, 0, 0, 0, 0, 0, -2, 6, 7], [8, 2, -8, 2, 5, 0, 0, 0, -13, 0, 0, -8, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 4, -1, 0, 0, -10, 0, 0, 10, 1, 0], [1, -12, 2, -3, 2, 0, 0, 0, -1, 2, 0, -13, 0, 0, 0, 0], [-15, 15, 16, 3, -3, 0, 0, 0, 15, 5, 0, 8, 0, 0, 0, 0], [-3, 6, 3, -11, -3, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -20, 0, -10, 0, 0, -10, 0, -13, -10, 8, 8], [0, 0, 0, 0, 0, 2, 2, -7, 0, 0, 0, 0, -6, 0, 1, 0], [8, 2, 13, -2, -6, 0, 0, 0, 2, 7, 0, -6, 0, 0, 0, 0], [21, -16, 7, -1, -7, 0, 0, 0, 8, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -5, 5, 13, 0, 0, -5, 0, -13, 0, -13, 5], [0, 0, 0, 0, 0, -6, -11, 2, 0, 0, 5, 0, -15, 11, 1, 8], [-28, -2, -12, 2, 13, 0, 0, 0, -15, 1, 0, 13, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, -2, -9, 0, 0, -1, 0, 9, -1, 0, -7], [0, 0, 0, 0, 0, -1, -8, -5, 0, 0, 0, 0, 0, -1, -8, -3], [0, 0, 0, 0, 0, 9, 3, -5, 0, 0, -3, 0, 9, 0, 5, -5], [-1, 8, 4, -3, 4, 0, 0, 0, 1, 4, 0, 9, 0, 0, 0, 0], [3, -4, 2, 3, 4, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0], [-9, 5, -12, 13, -13, 0, 0, 0, 5, -19, 0, -6, 0, 0, 0, 0], [9, 8, -1, -5, 1, 0, 0, 0, 4, 18, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -22, -22, -28, 0, 0, -7, 0, -11, 7, 17, 0], [6, 20, -1, -20, 4, 0, 0, 0, 10, 3, 0, 4, 0, 0, 0, 0], [5, -4, -4, -5, 0, 0, 0, 0, 10, -10, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 8, 0, 4, 0, 0, 4, 0, 0, -11, -14, -14], [-3, -8, -1, -25, -1, 0, 0, 0, -26, -1, 0, -5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 13, 0, 0, 0, -9, 0, 11, -13, -3, -4], [0, 0, 0, 0, 0, 0, -2, 5, 0, 0, 6, 0, 8, 1, 9, -9], [-14, 20, 11, -20, 0, 0, 0, 0, -14, 11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 11, -2, 11, 0, 0, 0, 0, 0, 11, -6, -13], [3, 16, 11, 3, 11, 0, 0, 0, 14, 11, 0, 13, 0, 0, 0, 0], [-7, -12, -4, -7, 14, 0, 0, 0, 0, 7, 0, 7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 12, 3, 0, 0, 0, 0, 3, -6, -3, 3], [0, 0, 0, 0, 0, 5, -1, 6, 0, 0, 5, 0, -13, 6, 2, 14], [2, 2, 6, 2, 1, 0, 0, 0, -3, 0, 0, 6, 0, 0, 0, 0], [-10, -4, 0, -14, 10, 0, 0, 0, -4, -4, 0, 6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 9, 9, 12, 0, 0, -2, 0, 11, 2, -1, 0], [0, 0, 0, 0, 0, -16, 3, -22, 0, 0, -19, 0, -7, -3, 3, -6], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 8, 8, 8], [0, 0, 0, 0, 0, -8, 10, -13, 0, 0, -10, 0, 16, 0, 13, 11], [0, 0, 0, 0, 0, -13, -16, -17, 0, 0, 3, 0, -3, 16, 13, -4], [5, 15, 8, -13, 13, 0, 0, 0, 15, 17, 0, 4, 0, 0, 0, 0], [-10, 5, 5, 10, 0, 0, 0, 0, 18, 3, 0, 7, 0, 0, 0, 0], [12, -20, -10, -20, -3, 0, 0, 0, -9, 0, 0, -10, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, -3, -9, 0, 0, 1, 0, 10, -1, 19, 0], [-9, 2, 6, -9, -10, 0, 0, 0, 0, -5, 0, -5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5, -10, 3, 0, 0, 0, 0, 0, 5, -14, -13], [0, 0, 0, 0, 0, 14, 0, 7, 0, 0, 7, 0, 26, -13, -6, -6], [-1, -15, 0, 5, 1, 0, 0, 0, 9, -3, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, -5, -4, 0, 0, 7, 0, 19, 5, 11, -6], [0, 0, 0, 0, 0, -4, 24, 2, 0, 0, -4, 0, 4, -28, -18, 2], [-7, 4, -11, 3, 11, 0, 0, 0, -14, 8, 0, 0, 0, 0, 0, 0], [-7, 7, 0, 3, 7, 0, 0, 0, 3, 5, 0, 12, 0, 0, 0, 0], [-11, -10, -2, 19, -2, 0, 0, 0, 17, -2, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -20, -14, 0, 0, 17, 0, -13, 10, -3, 3], [12, 8, 8, -8, 11, 0, 0, 0, 23, 19, 0, 11, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 1, 2, 0, 0, 0, 0, 0, -2, 14, -1], [-6, 5, 5, 6, 0, 0, 0, 0, 32, 8, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, -8, -3, 0, 0, 8, 0, 8, 0, 3, 1], [0, 3, 13, 0, 14, 0, 0, 0, 0, 7, 0, 7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 20, 19, 0, 0, 4, 0, 1, -10, -9, 9], [15, -16, 7, -5, -7, 0, 0, 0, 12, -8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -9, 20, 15, 0, 0, 0, 0, 0, -9, -8, 5], [-2, -4, -3, -4, 8, 0, 0, 0, -6, 0, 0, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, -8, 0, -10, 0, 0, -8, 0, -12, -8, -2, 10], [0, 0, 0, 0, 0, -8, 8, 0, 0, 0, -4, 0, 0, -4, 0, 13], [11, -16, -2, -11, 2, 0, 0, 0, 9, -3, 0, 0, 0, 0, 0, 0], [18, -3, -3, -18, 0, 0, 0, 0, -8, 2, 0, -8, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, -13, 0, 0, 0, 0, 2, 0, 13, -9], [11, 4, -12, -3, -12, 0, 0, 0, -15, -12, 0, -7, 0, 0, 0, 0], [4, 4, 13, -4, -4, 0, 0, 0, 0, 9, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 4, 0, 0, 4, 0, 1], [3, 2, 2, -3, 0, 0, 0, 0, 14, 6, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, -6, 0, -8, 0, 0, -6, 0, 8, -6, -2, -10], [-20, -20, 6, -4, 4, 0, 0, 0, -20, 7, 0, 3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 26, 0, 13, 0, 0, 13, 0, 4, 11, 4, 4], [0, 0, 0, 0, 0, -11, -11, -15, 0, 0, 1, 0, 20, -1, 35, 0], [6, 34, 18, -34, 1, 0, 0, 0, 7, 19, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, -4, -3, 0, 0, -3, 0, 3, -3, 0, 5], [6, 21, 21, -6, 0, 0, 0, 0, 16, 28, 0, 14, 0, 0, 0, 0], [6, 14, 0, 18, -6, 0, 0, 0, -34, -10, 0, -16, 0, 0, 0, 0], [4, -12, -15, 12, -8, 0, 0, 0, -4, -23, 0, -8, 0, 0, 0, 0], [-28, 2, -12, 2, -2, 0, 0, 0, 2, -8, 0, -6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 29, 18, 0, 0, 0, 0, 0, 0, -18, 11], [2, -10, 13, -10, -14, 0, 0, 0, 12, 0, 0, 13, 0, 0, 0, 0], [0, 0, 0, 0, 0, -11, 7, 1, 0, 0, -7, 0, -7, 0, -1, 5], [1, 20, 11, -13, 11, 0, 0, 0, -2, 11, 0, 19, 0, 0, 0, 0], [2, -32, -2, -6, 6, 0, 0, 0, -32, 5, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -16, 11, 8, 8], [20, -12, 4, 12, -21, 0, 0, 0, -1, -17, 0, -21, 0, 0, 0, 0], [0, 0, 0, 0, 0, -6, -26, -14, 0, 0, -6, 0, 6, 20, 18, -14], [0, 0, 0, 0, 0, -8, 8, 8, 0, 0, 10, 0, -8, 10, 0, 21], [-6, 1, 1, 6, 0, 0, 0, 0, -30, -11, 0, 13, 0, 0, 0, 0], [0, 0, 0, 0, 0, -4, 0, -2, 0, 0, -2, 0, 10, 11, -2, -2], [7, 9, 0, -15, -7, 0, 0, 0, 13, 11, 0, 4, 0, 0, 0, 0], [-7, 4, 10, 1, 10, 0, 0, 0, 11, 10, 0, 11, 0, 0, 0, 0], [0, 0, 0, 0, 0, 18, 5, 20, 0, 0, 13, 0, -11, -5, -17, 2], [0, 0, 0, 0, 0, 0, -14, -17, 0, 0, 4, 0, 8, 7, 15, -15], [0, 0, 0, 0, 0, -9, 18, 15, 0, 0, 0, 0, 0, -9, -2, 3], [15, -4, -4, -15, 0, 0, 0, 0, -28, -11, 0, 3, 0, 0, 0, 0], [0, 0, 0, 0, 0, -27, 1, -9, 0, 0, -1, 0, -7, 0, 9, 11], [5, -1, 0, 3, -5, 0, 0, 0, -13, -7, 0, -12, 0, 0, 0, 0], [0, 0, 0, 0, 0, 22, 5, 22, 0, 0, 17, 0, 5, -5, -11, 0], [-4, -22, -6, 22, -1, 0, 0, 0, -5, -7, 0, -1, 0, 0, 0, 0], [1, -10, 4, 11, -4, 0, 0, 0, -7, -13, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 23, 14, 21, 0, 0, 9, 0, 3, -14, -17, -2], [-3, -21, -8, -13, 13, 0, 0, 0, -21, 9, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 19, -7, 19, 0, 0, 7, 0, -9, 0, -19, -9], [0, 0, 0, 0, 0, 1, 1, 15, 0, 0, 9, 0, -6, -9, -21, 0], [1, 24, 7, 3, 7, 0, 0, 0, 10, 7, 0, 23, 0, 0, 0, 0], [-12, -9, 3, -12, -4, 0, 0, 0, 0, -2, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 12, -16, 2, 0, 0, 12, 0, -8, 28, 6, -2], [0, 0, 0, 0, 0, -4, -21, -8, 0, 0, 0, 0, 0, -4, 18, -13], [0, 0, 0, 0, 0, -6, 0, -3, 0, 0, -3, 0, 0, 7, -4, -4], [-12, 10, 26, 10, 7, 0, 0, 0, 5, 0, 0, 26, 0, 0, 0, 0], [0, 0, 0, 0, 0, 17, 17, 3, 0, 0, 15, 0, 6, -15, 3, 0], [0, 0, 0, 0, 0, 22, 8, 26, 0, 0, 22, 0, 0, 14, -4, 4], [-8, -5, -5, 8, 0, 0, 0, 0, 0, 6, 0, -16, 0, 0, 0, 0], [0, 0, 0, 0, 0, -32, 0, -16, 0, 0, -16, 0, 3, -6, 16, 16], [0, 0, 0, 0, 0, 27, 13, 7, 0, 0, -13, 0, 3, 0, -7, -17], [-7, -5, 0, 11, 7, 0, 0, 0, -17, -11, 0, -4, 0, 0, 0, 0], [21, 14, -8, 11, -8, 0, 0, 0, 3, -8, 0, -7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 16, 0, 0, 19, 0, -5, -4, -9, 9], [0, 0, 0, 0, 0, 1, -14, -25, 0, 0, 0, 0, 0, 1, 10, 11], [0, 0, 0, 0, 0, 4, 4, 18, 0, 0, 7, 0, 11, -7, -7, 0], [0, 0, 0, 0, 0, 0, 4, -6, 0, 0, -27, 0, 9, -2, 7, -7], [0, 0, 0, 0, 0, -20, -5, -12, 0, 0, 0, 0, 0, -20, -10, 7], [24, -4, -21, -4, -12, 0, 0, 0, -12, 0, 0, -21, 0, 0, 0, 0], [32, 6, 0, -4, -32, 0, 0, 0, -2, 2, 0, -30, 0, 0, 0, 0], [0, 0, 0, 0, 0, -6, -7, 4, 0, 0, 1, 0, -15, 7, -1, 10], [0, 0, 0, 0, 0, 3, -1, 2, 0, 0, 3, 0, -9, 4, 0, 8], [0, 0, 0, 0, 0, -29, 29, -2, 0, 0, -19, 0, 2, -19, 0, 35], [0, 0, 0, 0, 0, -10, 0, -5, 0, 0, -5, 0, 8, 9, -2, -2], [0, 0, 0, 0, 0, 1, -11, -19, 0, 0, 11, 0, 9, 0, 19, -11], [14, -4, -6, 4, 7, 0, 0, 0, 21, 1, 0, 7, 0, 0, 0, 0], [24, -4, -10, -8, 8, 0, 0, 0, -4, 3, 0, -5, 0, 0, 0, 0], [0, 0, 0, 0, 0, -22, -3, 4, 0, 0, 0, 0, 0, -22, 0, -7], [16, -17, -17, -16, 0, 0, 0, 0, -26, -11, 0, -23, 0, 0, 0, 0], [24, 16, 6, 16, -23, 0, 0, 0, -1, 0, 0, 6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 6, 6, 17, 0, 0, 0, 0, 18, 0, 1, 0], [-9, -34, -10, -9, -10, 0, 0, 0, -19, -10, 0, -25, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -24, -3, 0, 0, 6, 0, -21, 12, -9, 9], [-15, 24, 9, 3, -9, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [-19, 14, 4, -19, 32, 0, 0, 0, 0, 16, 0, 16, 0, 0, 0, 0], [0, 0, 0, 0, 0, 14, -8, 8, 0, 0, 14, 0, -4, 22, 2, -2], [0, 0, 0, 0, 0, 18, -18, 3, 0, 0, 7, 0, -3, 7, 0, 1], [-17, -2, -13, -3, 13, 0, 0, 0, -10, -6, 0, 0, 0, 0, 0, 0], [-2, 0, -16, 0, 5, 0, 0, 0, 3, -11, 0, 5, 0, 0, 0, 0], [-15, -17, -4, -9, 9, 0, 0, 0, -17, 7, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, -13, 13, 11, 0, 0, -2, 0, -11, -2, 0, 15], [-12, 8, 8, 8, -1, 0, 0, 0, 13, 0, 0, 8, 0, 0, 0, 0], [-24, -21, 11, -24, -16, 0, 0, 0, 0, -8, 0, -8, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, -25, 0, 0, 8, 0, 15, 4, 19, -19], [-13, 26, 3, 5, -3, 0, 0, 0, -2, 10, 0, 0, 0, 0, 0, 0], [-4, 6, 7, 6, -6, 0, 0, 0, 10, 0, 0, 7, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, -3, -3, 0, 0, 9, 0, 4, -9, 7, 0], [-24, -21, -7, -24, 18, 0, 0, 0, 0, 9, 0, 9, 0, 0, 0, 0], [0, 0, 0, 0, 0, -8, -8, -18, 0, 0, -8, 0, -20, 0, -2, 10], [-12, -18, 0, 6, 12, 0, 0, 0, -12, -15, 0, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 31, 2, -1, 0, 0, 29, 0, 19, -2, -7, -32], [0, 0, 0, 0, 0, 0, 12, -14, 0, 0, -15, 0, 13, -6, 7, -7], [-20, 4, -19, -4, 8, 0, 0, 0, -12, -11, 0, 8, 0, 0, 0, 0], [16, 11, 11, -16, 0, 0, 0, 0, -10, 13, 0, 9, 0, 0, 0, 0], [0, 0, 0, 0, 0, 11, 11, 17, 0, 0, -3, 0, 24, 3, 7, 0], [0, 0, 0, 0, 0, -21, -17, -34, 0, 0, -21, 0, -5, -4, 4, -8], [0, 0, 0, 0, 0, 32, -11, -2, 0, 0, 0, 0, 0, 32, -2, -9], [0, 0, 0, 0, 0, -13, -4, 5, 0, 0, -9, 0, -27, 4, -5, 18], [23, -38, -4, 7, 4, 0, 0, 0, -11, -11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -20, 0, -10, 0, 0, -10, 0, -25, -8, -2, -2], [0, 0, 0, 0, 0, 30, -10, 15, 0, 0, 10, 0, 10, 0, -15, -5], [-13, 29, 0, -3, 13, 0, 0, 0, -11, 9, 0, 22, 0, 0, 0, 0], [3, 0, 1, 15, 1, 0, 0, 0, 16, 1, 0, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, -9, -10, -21, 0, 0, 1, 0, 7, 10, 13, -12], [0, 40, 32, 6, -6, 0, 0, 0, 40, 10, 0, 16, 0, 0, 0, 0], [-24, -10, -2, -10, 7, 0, 0, 0, 17, 0, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, -15, -15, -26, 0, 0, -6, 0, 1, 6, 27, 0], [-18, -15, 9, -18, 20, 0, 0, 0, 0, 10, 0, 10, 0, 0, 0, 0], [0, 0, 0, 0, 0, 22, 0, 11, 0, 0, 11, 0, 19, -10, -4, -4], [-14, -22, 12, -22, 7, 0, 0, 0, 7, 0, 0, 12, 0, 0, 0, 0], [-24, -26, -3, 26, -14, 0, 0, 0, -38, -17, 0, -14, 0, 0, 0, 0], [0, 0, 0, 0, 0, 13, 23, 30, 0, 0, 13, 0, 11, -10, -6, 6], [-17, 12, 12, -19, 12, 0, 0, 0, -7, 12, 0, 29, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, -2, 0, 0, -11, 0, -21, 4, -17, 17], [0, 0, 0, 0, 0, 15, -15, -12, 0, 0, -1, 0, 12, -1, 0, -17], [-39, 14, 14, 39, 0, 0, 0, 0, 8, 11, 0, 17, 0, 0, 0, 0], [6, -22, 1, -22, 10, 0, 0, 0, -16, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 22, -26, 11, 0, 0, 26, 0, -10, 0, -11, -21]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1024_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1024_2_g_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_1024_2_g_a(:prec:=16)
    chi := MakeCharacter_1024_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_1024_2_g_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_1024_2_g_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1024_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 0, 64, 0, 128, 0, -416, 0, 568, 0, -208, 0, 32, 0, 8, 0, 1]>,<5,R![4, -16, 16, 8, 0, 16, 16, 4, 1]>],Snew);
    return Vf;
end function;