
// Make newform 7569.2.a.bf in Magma, downloaded from the LMFDB on 26 December 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7569_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_7569_2_a_bf();".
// 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_7569_2_a_bf();".
// 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, -31, 0, 36, 0, -11, 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, 1, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0], [1, 0, -5, 0, 1, 0, 0, 0], [-7, 0, 16, 0, -8, 0, 1, 0], [0, -7, 0, 16, 0, -8, 0, 1], [0, 27, 0, -40, 0, 12, 0, -1], [0, -43, 0, 44, 0, -12, 0, 1]];
    Rf_basisdens := [1, 1, 1, 1, 4, 4, 4, 4];
    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_7569_a();"
function MakeCharacter_7569_a()
    N := 7569;
    order := 1;
    char_gens := [5888, 1684];
    v := [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_7569_a_Hecke(Kf)
    return MakeCharacter_7569_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]];
    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, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, -1, 1], [-1, 0, 0, -1, -1, 0, 0, 0], [0, 0, 0, 0, 0, -1, -1, -2], [1, 0, 1, 0, 1, 0, 0, 0], [0, -2, 0, 0, 0, 2, -2, 2], [3, 0, 0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 1, -4, 1], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 1, -1, 0, 0, 0], [2, 0, 3, -1, -1, 0, 0, 0], [0, -2, 0, 0, 0, -1, 1, 4], [4, 0, 4, 0, -1, 0, 0, 0], [0, 1, 0, 0, 0, 0, -3, 3], [0, 4, 0, 0, 0, 0, 0, -2], [0, 1, 0, 0, 0, -5, 4, -3], [7, 0, 1, -1, 1, 0, 0, 0], [5, 0, 3, -2, 1, 0, 0, 0], [0, -7, 0, 0, 0, 1, -2, 5], [-3, 0, 2, -4, 1, 0, 0, 0], [5, 0, -1, 0, 7, 0, 0, 0], [0, 2, 0, 0, 0, 7, -3, 0], [0, -2, 0, 0, 0, 5, -3, 4], [-6, 0, 1, -2, -9, 0, 0, 0], [0, -5, 0, 0, 0, -3, 0, 7], [-1, 0, -1, 3, -1, 0, 0, 0], [0, 1, 0, 0, 0, 2, 5, 3], [-1, 0, -1, -1, -1, 0, 0, 0], [0, 6, 0, 0, 0, -2, -2, -2], [3, 0, -1, 3, -5, 0, 0, 0], [0, -4, 0, 0, 0, -2, 6, 0], [0, -2, 0, 0, 0, 5, -9, -2], [5, 0, -7, 1, 5, 0, 0, 0], [0, 2, 0, 0, 0, 3, 3, -6], [-5, 0, 3, -2, 3, 0, 0, 0], [3, 0, -3, -4, -5, 0, 0, 0], [4, 0, -2, -4, 1, 0, 0, 0], [0, 4, 0, 0, 0, 6, 0, -6], [0, 2, 0, 0, 0, -5, -5, -8], [0, -10, 0, 0, 0, -2, 0, 8], [18, 0, -4, 1, 5, 0, 0, 0], [0, -6, 0, 0, 0, 4, 0, 2], [8, 0, -4, 0, 5, 0, 0, 0], [0, 7, 0, 0, 0, -2, 5, 1], [-1, 0, 1, 0, -7, 0, 0, 0], [7, 0, -2, 3, 4, 0, 0, 0], [0, 0, 0, -4, 13, 0, 0, 0], [0, 0, 0, 0, 0, 5, -7, 2], [-4, 0, 1, -2, 8, 0, 0, 0], [0, -4, 0, 0, 0, -2, -4, 12], [0, -3, 0, 0, 0, 6, -5, 9], [1, 0, -7, -3, -1, 0, 0, 0], [0, 9, 0, 0, 0, 9, -6, 1], [0, 7, 0, 0, 0, 7, -2, 1], [0, 5, 0, 0, 0, -5, 6, -3], [0, 5, 0, 0, 0, -2, 7, 1], [5, 0, -1, 5, -4, 0, 0, 0], [0, 0, 4, -1, -7, 0, 0, 0], [0, 4, 0, 0, 0, 3, -11, -2], [5, 0, 7, 2, 1, 0, 0, 0], [0, 1, 0, 0, 0, 13, -8, 7], [11, 0, -7, 2, 15, 0, 0, 0], [0, 1, 0, 0, 0, -1, 10, -1], [11, 0, 5, -1, 5, 0, 0, 0], [0, 4, 0, 0, 0, 5, 9, -4], [12, 0, -8, 0, 7, 0, 0, 0], [5, 0, -1, -1, 4, 0, 0, 0], [0, 11, 0, 0, 0, -3, 10, -5], [6, 0, 7, 3, -9, 0, 0, 0], [0, 0, 0, 0, 0, -7, 5, -8], [0, -5, 0, 0, 0, 5, -4, 11], [23, 0, 1, -1, -1, 0, 0, 0], [-11, 0, 3, -3, -11, 0, 0, 0], [1, 0, 4, 1, -4, 0, 0, 0], [0, -13, 0, 0, 0, 2, -3, 7], [0, 1, 0, 0, 0, -8, 15, -5], [-20, 0, -4, 0, -3, 0, 0, 0], [0, 2, 0, 0, 0, 0, 12, -6], [11, 0, -11, 2, 9, 0, 0, 0], [0, 6, 0, 0, 0, -9, 5, 0], [1, 0, -9, 4, 1, 0, 0, 0], [0, -8, 0, 0, 0, 0, 10, 0], [6, 0, 2, -1, -17, 0, 0, 0], [2, 0, 2, 9, 3, 0, 0, 0], [0, -1, 0, 0, 0, 7, -2, 13], [0, -8, 0, 0, 0, 13, -15, 2], [13, 0, -10, 5, 11, 0, 0, 0], [0, -2, 0, 0, 0, -9, -1, -12], [4, 0, 1, -3, -15, 0, 0, 0], [0, 10, 0, 0, 0, -1, 1, 6], [0, 0, 0, 0, 0, 2, 4, -10], [7, 0, 4, -6, -1, 0, 0, 0], [0, -3, 0, 0, 0, 16, -11, 11], [-4, 0, 1, 0, 9, 0, 0, 0], [0, -4, 0, 0, 0, 0, -6, 2], [0, -6, 0, 0, 0, -10, 10, 8], [0, -7, 0, 0, 0, 4, 15, 3], [-15, 0, 4, -8, -1, 0, 0, 0], [-1, 0, -6, -7, 4, 0, 0, 0], [-13, 0, 3, -2, -29, 0, 0, 0], [0, -11, 0, 0, 0, 5, 4, 3], [0, 7, 0, 0, 0, 8, -11, 9], [0, -13, 0, 0, 0, -2, 1, 5], [8, 0, -1, 3, 3, 0, 0, 0], [-3, 0, -9, 1, 17, 0, 0, 0], [0, 2, 0, 0, 0, 8, 10, 6], [0, -15, 0, 0, 0, -14, 1, 9], [0, -5, 0, 0, 0, -14, 11, -15], [4, 0, 6, 0, -7, 0, 0, 0], [6, 0, -15, 2, 12, 0, 0, 0], [-4, 0, -1, -4, -3, 0, 0, 0], [0, 12, 0, 0, 0, 5, -1, -12], [18, 0, 12, -1, 1, 0, 0, 0], [7, 0, -3, -1, 12, 0, 0, 0], [0, -3, 0, 0, 0, -14, 3, -21], [20, 0, -8, 4, 12, 0, 0, 0], [0, -8, 0, 0, 0, 8, 8, 10], [0, 11, 0, 0, 0, 13, -12, 1], [0, -4, 0, 0, 0, 1, -9, 16], [10, 0, -6, 8, 20, 0, 0, 0], [-16, 0, -2, -2, -20, 0, 0, 0], [0, 0, 0, 0, 0, -15, 23, -4], [0, -5, 0, 0, 0, 16, -9, 1], [4, 0, 0, 1, 5, 0, 0, 0], [0, 12, 0, 0, 0, 5, 1, -16], [-3, 0, 3, -8, 9, 0, 0, 0], [0, -23, 0, 0, 0, -4, -1, 13], [13, 0, -1, 10, 1, 0, 0, 0], [8, 0, 1, 4, -1, 0, 0, 0], [13, 0, 2, 5, 13, 0, 0, 0], [0, -26, 0, 0, 0, 7, -5, 20], [13, 0, 1, -3, 5, 0, 0, 0], [-4, 0, 10, 2, 4, 0, 0, 0], [0, 1, 0, 0, 0, -3, -4, -9], [6, 0, 13, -10, -7, 0, 0, 0], [0, 8, 0, 0, 0, 11, 7, -6], [1, 0, 9, 1, 0, 0, 0, 0], [0, -7, 0, 0, 0, 11, -2, 5], [0, -9, 0, 0, 0, -2, 7, 7], [25, 0, 4, -3, 1, 0, 0, 0], [0, -12, 0, 0, 0, -6, 12, 6], [-16, 0, -1, 1, -7, 0, 0, 0], [0, 0, 0, 0, 0, -17, -5, -6], [-11, 0, 10, 1, -17, 0, 0, 0], [0, 14, 0, 0, 0, 10, -4, 0], [25, 0, 9, -1, -4, 0, 0, 0], [0, -19, 0, 0, 0, 5, -6, 3], [12, 0, 9, -5, -1, 0, 0, 0], [0, -3, 0, 0, 0, -13, 14, 7], [-2, 0, 11, -10, -3, 0, 0, 0], [0, 3, 0, 0, 0, 9, -8, -3], [-18, 0, 3, 2, -37, 0, 0, 0], [0, 8, 0, 0, 0, -2, -12, -4], [-17, 0, 15, -10, -13, 0, 0, 0], [0, 17, 0, 0, 0, 4, -5, 5], [7, 0, 5, 3, -1, 0, 0, 0], [0, 13, 0, 0, 0, 2, 1, 5], [-29, 0, 16, -3, -24, 0, 0, 0], [0, 7, 0, 0, 0, -3, -20, -9], [0, -14, 0, 0, 0, 2, 0, 4], [0, -19, 0, 0, 0, -4, 1, 19], [2, 0, -14, 1, 19, 0, 0, 0], [0, 4, 0, 0, 0, 21, -5, 8], [0, 2, 0, 0, 0, 4, 0, -8], [0, -4, 0, 0, 0, -16, 4, -6], [-1, 0, -8, -5, -3, 0, 0, 0], [-28, 0, -1, 4, -7, 0, 0, 0], [22, 0, -11, 5, 29, 0, 0, 0], [0, 5, 0, 0, 0, 2, -1, -23], [0, -14, 0, 0, 0, 10, -10, 2], [9, 0, 11, 1, -11, 0, 0, 0], [0, 4, 0, 0, 0, -8, 18, -2], [-5, 0, -3, -7, 28, 0, 0, 0], [11, 0, -1, -1, 15, 0, 0, 0], [0, -6, 0, 0, 0, 6, 18, 4], [-19, 0, 10, -11, -16, 0, 0, 0], [0, -20, 0, 0, 0, 0, 10, 8], [0, 0, 19, -9, -13, 0, 0, 0], [-19, 0, -8, 1, -15, 0, 0, 0], [30, 0, 15, -9, 9, 0, 0, 0], [0, 1, 0, 0, 0, -1, -8, -3], [29, 0, 6, 3, 16, 0, 0, 0], [0, -4, 0, 0, 0, -8, 10, -2], [0, -2, 0, 0, 0, 11, 15, 6], [0, -9, 0, 0, 0, 11, -16, 1], [9, 0, 7, 0, 17, 0, 0, 0], [-4, 0, 2, 4, -9, 0, 0, 0], [-33, 0, 0, -3, -12, 0, 0, 0], [0, 3, 0, 0, 0, 17, -8, 3], [-32, 0, 3, -1, -7, 0, 0, 0], [0, 8, 0, 0, 0, 12, -8, -2], [33, 0, -2, 2, 1, 0, 0, 0], [0, 9, 0, 0, 0, -4, 25, -9], [0, -4, 0, 0, 0, -9, 3, -6], [0, 10, 0, 0, 0, 1, 15, 12], [-23, 0, 14, -7, -19, 0, 0, 0], [14, 0, -7, 9, 29, 0, 0, 0], [0, 0, 0, 0, 0, 1, -3, 14], [0, -2, 0, 0, 0, 8, -14, -10], [0, -11, 0, 0, 0, -8, 13, 7], [25, 0, -2, 9, -12, 0, 0, 0], [13, 0, -4, 15, 20, 0, 0, 0], [34, 0, 12, -7, 17, 0, 0, 0], [0, -11, 0, 0, 0, 2, 17, 17], [0, 4, 0, 0, 0, 0, -4, 14], [-14, 0, -5, 6, -1, 0, 0, 0], [0, -16, 0, 0, 0, -6, -14, 16], [0, -6, 0, 0, 0, -1, 17, 8], [13, 0, 9, 9, -11, 0, 0, 0], [32, 0, 3, 13, -17, 0, 0, 0], [0, 14, 0, 0, 0, 16, -22, 10], [-25, 0, 9, -13, -5, 0, 0, 0], [0, 6, 0, 0, 0, -4, 4, -26], [0, 0, 0, 0, 0, -17, 19, -14], [-12, 0, -9, -4, -11, 0, 0, 0], [-14, 0, -7, -7, 7, 0, 0, 0], [0, 9, 0, 0, 0, 4, 23, -9], [0, -18, 0, 0, 0, 8, 4, 36], [0, -13, 0, 0, 0, -15, 8, 7], [0, 0, 7, 7, -19, 0, 0, 0], [1, 0, -5, 10, 29, 0, 0, 0], [0, 23, 0, 0, 0, -11, 16, -21], [-19, 0, 11, -4, -5, 0, 0, 0], [0, 7, 0, 0, 0, -24, 25, -15], [-3, 0, -6, 15, -16, 0, 0, 0], [0, 26, 0, 0, 0, -11, 5, -20], [0, 18, 0, 0, 0, 5, -15, 2], [16, 0, -18, 12, 11, 0, 0, 0], [0, -2, 0, 0, 0, -2, 6, 10], [-28, 0, -2, 11, 1, 0, 0, 0], [-45, 0, -7, 3, -21, 0, 0, 0], [31, 0, 8, 15, -28, 0, 0, 0], [0, 16, 0, 0, 0, -7, 3, 2], [-33, 0, 7, -5, -11, 0, 0, 0], [0, -9, 0, 0, 0, -15, 24, -11], [51, 0, 5, -5, 15, 0, 0, 0], [0, 19, 0, 0, 0, -3, 14, 5], [0, -22, 0, 0, 0, -9, -13, 28], [0, 0, 0, 0, 0, 19, -25, 4], [0, 6, 0, 0, 0, -11, -3, -10], [-10, 0, 7, -2, 19, 0, 0, 0], [10, 0, -2, 7, -11, 0, 0, 0], [-13, 0, 0, -8, -5, 0, 0, 0], [0, 10, 0, 0, 0, 1, -17, -14], [0, -10, 0, 0, 0, 2, -12, -16], [24, 0, -22, 5, 37, 0, 0, 0], [0, -2, 0, 0, 0, -26, -8, -10], [-32, 0, -11, 8, -31, 0, 0, 0], [0, 6, 0, 0, 0, 5, 37, -12], [10, 0, -5, 8, 40, 0, 0, 0], [0, 22, 0, 0, 0, 16, -12, -10], [0, -15, 0, 0, 0, -16, 13, -7], [0, 0, -15, 22, 0, 0, 0, 0], [0, 18, 0, 0, 0, 11, -7, -14], [0, -3, 0, 0, 0, -26, 21, -25], [-8, 0, -15, -9, 29, 0, 0, 0], [-12, 0, 5, -19, 13, 0, 0, 0], [0, -17, 0, 0, 0, 2, -1, 3], [0, 0, -3, 13, -17, 0, 0, 0], [20, 0, 2, 0, 31, 0, 0, 0], [0, -12, 0, 0, 0, -7, -11, 10], [36, 0, 7, 4, 7, 0, 0, 0], [1, 0, 12, -2, -5, 0, 0, 0], [0, -7, 0, 0, 0, 0, 5, 5], [1, 0, 8, 6, 21, 0, 0, 0], [0, 11, 0, 0, 0, 5, -8, 9], [0, -16, 0, 0, 0, 5, -15, -14], [-7, 0, -12, 6, 1, 0, 0, 0], [0, -8, 0, 0, 0, 19, -33, 2], [11, 0, 0, -13, -15, 0, 0, 0], [27, 0, -12, 18, 23, 0, 0, 0], [-15, 0, -17, -1, 12, 0, 0, 0], [7, 0, -7, 7, -7, 0, 0, 0], [-3, 0, 3, -3, -25, 0, 0, 0], [47, 0, -9, 2, 39, 0, 0, 0], [0, 3, 0, 0, 0, 4, 5, 17], [9, 0, 7, 9, 33, 0, 0, 0], [-30, 0, -1, -10, 9, 0, 0, 0], [0, -4, 0, 0, 0, -19, 9, -22], [0, 20, 0, 0, 0, -22, -14, -40], [12, 0, -7, 0, -15, 0, 0, 0], [0, 2, 0, 0, 0, -24, 4, -14], [-36, 0, 0, -11, -9, 0, 0, 0], [6, 0, 5, -10, -12, 0, 0, 0], [0, -4, 0, 0, 0, -15, 1, -14], [-37, 0, 10, 3, -1, 0, 0, 0], [0, 2, 0, 0, 0, 9, 17, 2], [-51, 0, -1, 2, -19, 0, 0, 0], [0, -24, 0, 0, 0, 4, -10, -2], [0, 12, 0, 0, 0, -15, 1, -14], [0, -7, 0, 0, 0, 8, 9, 43], [0, 18, 0, 0, 0, 4, -12, 18], [0, 11, 0, 0, 0, 14, -9, 25], [-20, 0, -2, 8, 9, 0, 0, 0], [0, 11, 0, 0, 0, -29, 0, -31], [-3, 0, -7, 8, -9, 0, 0, 0], [0, 10, 0, 0, 0, -21, 27, -6], [0, 30, 0, 0, 0, 19, 9, -30], [-32, 0, -5, 4, -55, 0, 0, 0], [25, 0, 3, 10, -1, 0, 0, 0], [0, -6, 0, 0, 0, 19, -13, 14], [13, 0, 11, -14, -15, 0, 0, 0], [0, -22, 0, 0, 0, -12, -2, -8], [1, 0, -8, 1, 8, 0, 0, 0], [-32, 0, 10, -25, -3, 0, 0, 0], [0, 4, 0, 0, 0, 9, -17, 0], [-13, 0, -2, 7, 3, 0, 0, 0], [0, 14, 0, 0, 0, 28, -12, -6], [29, 0, -7, -2, 39, 0, 0, 0], [0, 28, 0, 0, 0, -15, -3, -14], [0, 15, 0, 0, 0, 20, 1, -21], [0, 4, 0, 0, 0, -5, 27, -4], [-38, 0, 3, 2, 9, 0, 0, 0], [0, -10, 0, 0, 0, 7, -19, 10], [-17, 0, 5, -21, 3, 0, 0, 0], [0, 8, 0, 0, 0, -4, 36, 0], [0, 18, 0, 0, 0, 35, -23, 16], [-19, 0, -16, 3, -8, 0, 0, 0], [0, -22, 0, 0, 0, 6, 10, 8], [20, 0, -3, 3, 15, 0, 0, 0], [-25, 0, -17, 4, -5, 0, 0, 0], [0, 26, 0, 0, 0, 2, 4, -28], [29, 0, -8, 0, -13, 0, 0, 0], [0, -9, 0, 0, 0, 3, 6, 29], [0, 0, 0, -14, 28, 0, 0, 0], [18, 0, 1, -11, -13, 0, 0, 0], [11, 0, 6, -17, 19, 0, 0, 0], [0, 2, 0, 0, 0, 6, 22, 12], [0, -5, 0, 0, 0, -21, -12, -1], [-10, 0, 7, 2, -25, 0, 0, 0], [0, -11, 0, 0, 0, 16, -21, 3], [-12, 0, 30, -3, -19, 0, 0, 0], [0, -12, 0, 0, 0, -8, 24, -24], [6, 0, -9, 9, -13, 0, 0, 0], [0, -4, 0, 0, 0, 5, 11, 26], [10, 0, 14, -14, 5, 0, 0, 0], [0, 4, 0, 0, 0, -13, 3, -12], [-22, 0, 29, -11, -15, 0, 0, 0], [14, 0, 1, -14, 15, 0, 0, 0], [7, 0, 14, 5, 20, 0, 0, 0], [0, 5, 0, 0, 0, 10, -7, 9], [0, -5, 0, 0, 0, 3, -26, 9], [47, 0, 10, 0, 37, 0, 0, 0], [0, -7, 0, 0, 0, 24, -31, 9], [0, 4, 0, 0, 0, -6, -4, 18], [-1, 0, -28, 2, -7, 0, 0, 0], [-27, 0, -6, 10, 13, 0, 0, 0], [0, 1, 0, 0, 0, -4, 3, 27], [0, 0, 0, 0, 0, -8, 32, 8], [4, 0, -10, 12, 31, 0, 0, 0], [-4, 0, 19, 0, -37, 0, 0, 0], [0, -2, 0, 0, 0, 5, -19, -28], [-35, 0, -5, 4, -17, 0, 0, 0], [7, 0, -8, 23, -11, 0, 0, 0], [0, 7, 0, 0, 0, -21, 6, -1], [0, 3, 0, 0, 0, 17, 8, -9], [0, -16, 0, 0, 0, -31, 31, -6], [0, -4, 0, 0, 0, 1, -21, 22], [0, 16, 0, 0, 0, 7, 9, -34], [-15, 0, -16, -1, -28, 0, 0, 0], [0, 28, 0, 0, 0, 6, -6, 14], [0, -3, 0, 0, 0, 6, -7, -19], [0, -23, 0, 0, 0, 13, 10, 21], [-6, 0, -14, 6, -15, 0, 0, 0], [-14, 0, 7, 2, 1, 0, 0, 0], [0, -15, 0, 0, 0, 5, -30, 21], [-10, 0, 7, -5, 5, 0, 0, 0], [42, 0, -16, 2, 13, 0, 0, 0], [0, 3, 0, 0, 0, 3, 10, -15], [-39, 0, -12, 5, 0, 0, 0, 0], [0, 19, 0, 0, 0, -2, 11, -37], [0, -2, 0, 0, 0, 9, -13, -24], [3, 0, -16, -6, 15, 0, 0, 0], [8, 0, -8, 0, -13, 0, 0, 0], [0, 10, 0, 0, 0, 2, 30, 8], [0, -5, 0, 0, 0, 20, 19, 37], [49, 0, 3, -2, 3, 0, 0, 0], [0, -22, 0, 0, 0, -2, 32, 6], [23, 0, -17, 7, 31, 0, 0, 0], [0, 10, 0, 0, 0, 1, 5, -30], [0, 22, 0, 0, 0, -19, 1, -10], [0, 0, 0, -20, -16, 0, 0, 0], [0, 41, 0, 0, 0, 5, 4, -25], [-16, 0, 2, -15, -23, 0, 0, 0], [0, 26, 0, 0, 0, -6, -2, -4], [18, 0, 21, -18, 19, 0, 0, 0], [-1, 0, 13, -5, -37, 0, 0, 0], [-58, 0, 7, -13, -19, 0, 0, 0], [0, -33, 0, 0, 0, -8, 3, -1], [-15, 0, 1, 4, 21, 0, 0, 0], [0, 21, 0, 0, 0, -31, 34, -17], [0, -20, 0, 0, 0, 0, 10, 10], [-30, 0, -13, -3, -17, 0, 0, 0], [0, -6, 0, 0, 0, -4, -12, -12], [70, 0, -27, 22, 47, 0, 0, 0], [-15, 0, 6, -11, 8, 0, 0, 0], [0, 20, 0, 0, 0, -22, -14, -24], [20, 0, 14, -12, 39, 0, 0, 0], [0, 37, 0, 0, 0, -16, 9, -15], [33, 0, -11, 18, 19, 0, 0, 0], [0, -20, 0, 0, 0, -28, 18, -12], [-14, 0, 6, -1, -27, 0, 0, 0], [0, 10, 0, 0, 0, 1, 15, 32], [0, 10, 0, 0, 0, 13, 3, 4], [-11, 0, 27, -15, -11, 0, 0, 0], [40, 0, 0, 3, 49, 0, 0, 0], [0, -2, 0, 0, 0, -6, 6, -8], [9, 0, 14, -12, -9, 0, 0, 0], [0, 22, 0, 0, 0, 8, -4, 26], [-14, 0, 5, -5, 31, 0, 0, 0], [0, 20, 0, 0, 0, -14, 22, -12], [0, 13, 0, 0, 0, 0, -21, 1], [0, 0, -5, -12, 49, 0, 0, 0], [26, 0, -6, -3, 9, 0, 0, 0], [0, -4, 0, 0, 0, -48, 42, -14], [0, 13, 0, 0, 0, 32, -21, 7], [-11, 0, 26, 5, -36, 0, 0, 0], [0, -18, 0, 0, 0, -10, 20, -8], [0, 33, 0, 0, 0, -6, 5, -43], [0, 12, 0, 0, 0, 47, -11, 14], [-27, 0, -1, -14, -41, 0, 0, 0], [0, 16, 0, 0, 0, -7, 19, -6], [0, 5, 0, 0, 0, 25, -24, 33], [-9, 0, 31, -16, -17, 0, 0, 0], [0, 9, 0, 0, 0, 2, 31, -3], [0, -6, 0, 0, 0, 6, -34, 0], [0, -41, 0, 0, 0, -25, 8, 9], [-27, 0, 4, 2, -7, 0, 0, 0], [0, -14, 0, 0, 0, -5, 11, -20]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7569_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7569_2_a_bf();".
// 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_7569_2_a_bf(:prec:=8)
    chi := MakeCharacter_7569_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_7569_2_a_bf();".
// 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_7569_2_a_bf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7569_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 0, -31, 0, 36, 0, -11, 0, 1]>,<5,R![16, 0, -116, 0, 121, 0, -21, 0, 1]>,<7,R![5, 20, -15, 0, 1]>,<19,R![5, -5, 1]>],Snew);
    return Vf;
end function;