
// Make newform 9300.2.a.ba in Magma, downloaded from the LMFDB on 13 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_9300_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_9300_2_a_ba();".
// 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_9300_2_a_ba();".
// 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 := [-64, -64, 112, 30, -22, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 200, -26, -30, 2, 1], [176, 32, -46, 14, 2, -1], [-32, 16, 22, -14, -2, 1], [-104, -84, 14, 22, 0, -1], [72, -68, -58, 18, 4, -1]];
    Rf_basisdens := [1, 48, 48, 24, 24, 24];
    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_9300_a();"
function MakeCharacter_9300_a()
    N := 9300;
    order := 1;
    char_gens := [4651, 3101, 2977, 1801];
    v := [1, 1, 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_9300_a_Hecke( : Kf := Kf)
    return MakeCharacter_9300_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, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1], [-1, 0, 0, 0, -1, 0], [0, 0, 0, -1, 0, 0], [0, 1, 0, -1, 0, -1], [-1, 1, -1, 0, 0, 0], [2, 1, 1, 1, 0, 1], [-1, 0, -1, 0, -1, 0], [-1, 0, 0, 0, 0, 0], [1, 1, -1, 0, 1, -1], [3, 0, 0, 1, 0, -1], [3, 0, 0, -1, 1, 1], [1, -2, 1, 0, 0, 1], [-1, -1, -1, -1, -2, 0], [-1, 2, 2, 1, 0, -1], [3, -1, -1, -1, -1, -1], [3, 0, 2, 1, 1, 2], [6, -1, 2, 1, 1, 0], [2, 0, 0, 0, 0, 0], [3, 0, 2, 2, 1, 1], [4, -1, -1, 1, 0, 3], [2, -2, 1, 0, 1, 1], [0, -2, 0, 0, 0, 1], [6, 2, 0, 0, 0, -4], [6, 0, 0, 1, 1, -1], [-6, 0, -3, -1, -2, -2], [4, -2, 0, -1, 1, 2], [1, -1, 1, 1, 1, 1], [4, -1, 1, 2, 0, 2], [6, 0, -1, -1, 2, 2], [-1, -2, 1, 0, -1, 0], [1, -4, 0, 0, -1, 1], [-3, 4, 2, 1, 2, -3], [4, -1, -1, 2, -2, 2], [-4, 1, -1, 1, 1, -1], [10, -2, 2, 3, 1, 1], [-3, 3, 1, 3, 1, -1], [4, -3, 3, 0, 1, 2], [-5, 0, -2, 2, 0, -1], [5, 1, 1, 0, -1, 3], [5, 1, -2, 0, 1, -1], [6, -1, 1, 0, 2, 3], [-7, 2, -1, 0, -2, -3], [6, 0, 6, 1, 2, 2], [4, 2, 2, 3, 1, 3], [7, 2, -2, -1, -1, -3], [-7, 0, -1, 0, 2, -1], [-2, 6, -2, -1, -2, -2], [-3, 1, 1, 1, -1, -1], [-5, -1, -1, -1, -2, -2], [11, -1, -1, 0, 3, -1], [5, 1, 1, -1, -2, 2], [-8, 1, 3, 2, -1, 0], [-4, 4, 0, -2, 0, -4], [-2, -1, -2, 1, 1, 0], [12, 1, 1, 1, 2, 4], [0, -4, -2, -2, 2, 0], [7, 1, -1, -3, -1, -1], [10, -4, 0, 1, 1, 1], [-9, 0, -2, -1, 0, -5], [-4, 1, 1, -5, 1, -2], [13, -1, 2, 6, -1, 1], [0, 2, -2, 1, 0, -2], [-7, 0, -4, -5, 2, -3], [4, 2, 2, -3, 2, 0], [-3, -1, 1, 0, 1, -1], [-1, -2, 0, 0, 2, -1], [2, 1, 1, 3, -1, -1], [-3, 7, -1, -1, 2, -2], [-1, 3, 1, 1, 1, -1], [4, 3, 3, 0, 4, 2], [1, 1, 3, 3, -3, 0], [2, 1, -1, 3, -1, 4], [4, -2, 0, -2, 1, -5], [3, 1, -3, -1, -1, -5], [-10, -1, 3, 0, 0, -5], [7, 0, 0, 0, -3, -4], [11, 3, 1, -3, -1, -3], [0, 1, 2, 3, 1, 4], [13, 3, 3, 4, 2, -7], [3, -5, -3, -3, -5, 1], [1, -1, -1, 1, 1, 5], [-5, -2, -4, -4, -4, -1], [2, 6, -3, -1, 2, -2], [1, 1, 2, 1, 0, -6], [-4, 6, 4, 3, 0, -4], [11, 2, 2, -2, -3, 2], [7, 0, -4, -3, -3, -3], [-12, -5, 2, 1, -1, 2], [13, 0, 5, 2, 4, 5], [-12, -3, -3, -3, -2, -2], [-3, 4, 0, -2, -1, 0], [-12, -5, -3, -3, -4, 0], [2, -3, 0, -5, -1, 2], [-1, 1, -1, -3, -4, 2], [7, -2, -6, -1, -2, -1], [-4, 1, -3, -2, -8, -2], [-1, -3, -3, -3, -1, 0], [0, -1, 1, 4, -4, -3], [0, -4, 3, 0, -3, -1], [11, 1, 4, 0, 5, 5], [-5, 2, -2, 0, -1, 4], [-12, -2, -2, 2, 2, -4], [-4, 0, -2, 3, 3, 2], [-14, 1, -5, -1, 2, 1], [-6, -1, 3, 6, 5, 8], [-2, 4, -5, 1, 0, 0], [5, 3, 3, 2, 5, -3], [-4, -1, 1, 4, -2, 7], [-1, -7, -3, -3, 1, -1], [12, -3, 1, -4, 5, 6], [5, -2, 2, 3, 1, 1], [6, 3, 3, 5, 2, 4], [11, 3, 3, 3, 2, -8], [0, 1, -3, 1, -6, -4], [-23, 4, 0, -2, -1, -4], [-15, -4, -2, -1, -4, -5], [-1, -5, -1, -3, 3, -1], [9, -3, -3, 1, 3, 4], [11, 5, 1, -4, 3, -1], [-12, 4, -3, 6, 0, 0], [1, 2, 1, 2, 2, 9], [4, -8, -2, -1, 3, -1], [4, -1, 1, 0, -5, -2], [7, -3, 1, 4, 1, 1], [-10, 6, -4, -4, -2, 2], [4, 1, 1, 5, 3, -2], [-3, 2, -2, 2, 4, -6], [10, 1, -1, -1, 4, 2], [-15, -2, -2, -2, 5, -6], [-3, -2, -4, -7, 3, -6], [-7, -7, -5, 0, -3, 1], [13, -1, 2, 1, 5, -1], [20, 0, -4, -2, 4, -1], [-15, 5, 2, 3, -2, -6], [-10, 0, -2, -8, -2, -2], [5, 4, 4, -6, 2, -1], [-5, 2, -3, 0, 3, 0], [-11, -4, 0, 4, 2, 1], [-1, -4, 2, -2, -7, 2], [-2, 6, -4, -2, 0, -6], [-11, 3, 9, -1, -2, -2], [0, 0, -2, -5, -4, -2], [5, 6, -1, 2, 0, 3], [4, -8, -4, -1, 1, 2], [5, -4, -2, 1, -4, 3], [-8, 1, 5, -2, 0, -2], [-11, -1, 1, -1, -5, 3], [0, -3, -1, -4, -4, 0], [-4, -5, 2, -1, -1, 2], [-4, 4, 0, -9, -2, -10], [-11, -7, 4, 2, 3, 5], [14, -8, -6, 3, -3, 9], [0, -4, -2, 2, 3, 5], [-1, -4, 0, 5, 5, 0], [3, -3, 3, 5, -2, 8], [-20, 0, -4, -5, -7, -5], [16, -4, -3, -4, -1, -1], [-12, 1, 1, -3, 4, -3], [1, -5, -3, -3, 3, -7], [14, 4, -4, -1, 2, -12], [-7, 3, 2, 6, -5, -3], [3, -4, 0, -7, 0, 5], [-5, 1, 3, -1, 0, -10], [0, 7, 1, 3, 4, 4], [4, 3, 3, 8, 2, 7], [18, -4, 2, -4, -2, -4], [0, -5, 4, 5, 5, 14], [-17, 1, 7, 1, -5, 1], [0, -6, 4, 0, -2, 0], [-15, 1, -1, -5, 1, -1], [-16, -3, -3, -3, -7, -1], [-11, 3, -3, 3, -3, -9], [-5, -7, 1, 5, -5, -1], [2, -1, 3, -5, -1, -4], [23, 4, -2, -3, 0, -3], [33, -7, 5, 6, 10, 8], [1, -3, -1, 5, -3, -3], [-12, -1, -5, -8, 6, -5], [-16, 0, -6, -2, -3, -1], [-8, 4, 0, 2, 2, -5], [-5, -1, -5, -5, -8, 2], [3, 1, -10, 2, -3, 3], [-20, 1, 0, -1, 2, -5], [18, 12, 6, 4, 4, -1], [14, 1, -3, -5, -3, -4], [9, -3, -3, -2, 3, 5], [24, -2, 1, -1, 4, 6], [-13, 3, -3, 2, -1, 1], [-19, -6, -4, -1, -4, 3], [-28, 4, -6, -9, 2, -4], [-1, 13, -3, 1, -1, -5], [-25, -1, 1, 3, -7, -5], [-23, -5, 2, 1, -4, 0], [24, 2, 0, 7, -2, 2], [9, 10, -2, -1, 5, -5], [21, 1, -5, 1, 3, -5], [-17, 2, -2, 9, -2, -3], [-17, 1, 3, -5, 1, -7], [8, 2, 6, -1, -4, 6], [26, -11, 1, -5, 1, 2], [2, 5, -1, -3, 3, -7], [-22, 3, -2, -3, -3, 4], [43, -2, 5, 10, -2, 3], [8, 0, -2, 5, 1, 7], [-8, -8, 0, -2, 1, 5], [22, 0, 2, -4, 4, 6], [-4, -1, 9, 0, 4, 1], [-25, -3, -3, 0, -1, -1], [11, -2, 1, -8, 4, -1], [16, -12, -4, -1, 3, 9], [-13, 6, 2, 3, -4, -9], [9, -5, -1, 1, 0, 10], [-24, 3, -3, -5, -11, -9], [-1, -2, -2, 1, 5, 0], [0, 7, -5, 4, -5, -8], [-7, 4, -2, 6, -3, -8], [-16, 6, 2, -10, 0, -6], [-7, -1, 3, -1, -1, 3], [-10, -5, -11, -4, -2, -5], [-12, -4, -4, -2, -12, -4], [25, 4, 14, 5, 3, 4], [3, -3, -5, 1, 7, -5], [-13, -1, 3, -1, -1, 9], [-8, -1, 5, 2, 3, 8], [2, 1, -1, -7, 4, 1], [2, -9, -5, -2, -8, 4], [-2, -4, 5, 1, 4, 8], [-4, -2, 6, -3, -6, -2], [8, -3, 1, 1, 1, -2], [-16, -5, -3, -4, 6, -7], [1, 0, 2, 4, 8, 11], [-1, -6, -4, -6, 5, -5], [8, -5, 5, 5, -2, 11], [21, 9, 1, -5, 1, 4], [-8, 3, -5, -2, -1, 4], [-36, -5, -1, -5, 0, -1], [34, 0, -4, 0, 1, -1], [-24, -6, -9, -1, -4, 2], [-15, -2, 4, 2, 7, 3], [-10, -4, -2, 1, 4, -4], [7, 1, 3, -1, -1, -7], [-5, 3, 0, 3, -6, 4], [-1, -2, 10, 7, -6, 5], [-1, 0, 4, -4, -3, -5], [-22, -6, -2, 0, 3, -13], [-30, 1, -7, -7, -2, -10], [1, -3, 7, 1, 3, -3], [20, 10, -2, -5, 3, -5], [-2, -3, -7, 6, -1, 6], [-33, 3, -1, -5, 0, -8], [-3, 9, -9, 2, -8, -11], [-6, 0, -2, 2, 2, 12], [-1, 2, 1, 4, 4, 9], [6, 10, -2, 2, 8, -7], [19, 4, 0, 6, 1, 5], [18, -8, 4, 8, 0, 2], [7, -3, 1, 2, -8, 7], [7, -4, 4, 0, 0, -9], [5, -1, 3, -3, -2, -14], [25, 11, 1, -1, 3, -1], [0, 2, -4, 3, 1, 11], [-21, -8, -8, -4, -7, 0], [-17, 0, 8, 4, 2, 3], [6, -1, 3, -6, 5, 2], [-9, 6, 8, 1, 4, -7], [25, 4, 6, 3, 9, 6], [-9, 0, -8, -1, 0, 3], [-19, -5, 1, 1, 3, -6], [14, 2, -4, -2, 6, -12], [-7, -3, 11, 6, -1, 11], [-3, 6, -2, 1, 3, -9], [-11, -4, -2, 1, -9, 3], [17, 5, 1, 2, 12, 0], [-7, 7, -2, -6, -5, -15], [16, -8, 2, 4, 8, 0], [-28, -4, -10, -6, -6, -8], [-29, -4, 2, 0, -3, -4], [-14, -5, -4, 1, -3, 6], [10, -9, 9, 2, 4, 5], [9, 2, 12, 11, -2, 9], [10, -12, 0, -5, 7, 0], [19, 6, 2, 3, -7, 8], [-6, -3, 5, -7, 1, 4], [-7, -11, -7, -3, -5, 13], [5, -2, -1, 0, -9, -2], [-7, -7, 3, 0, 4, 16], [22, -1, 2, -1, 2, 13], [12, -2, -2, 4, 4, 10], [17, -10, 1, -2, 0, 1], [1, 12, -2, 6, -3, 0], [3, 1, -3, -9, 3, 1], [-11, 7, -7, -3, -9, -1], [-17, -4, 3, -6, 3, -10], [19, 18, -4, -2, 1, -9], [-9, -7, -3, 3, 3, 5], [-8, -1, 7, -3, 2, -13], [44, 1, 1, 1, -2, 4], [-2, -10, -6, -6, 2, 3], [18, 4, 11, 6, 5, 7], [0, -4, 6, 9, 6, 8], [7, 6, 8, 3, 0, 9], [-22, 1, -5, 8, 4, -2], [-17, -1, 5, 4, 6, 3], [1, 0, -6, 2, 7, 10], [10, -14, 0, 6, -2, 9], [0, 1, 3, 11, 8, 5], [27, -3, -9, 3, -7, 8], [14, 0, 10, 6, 7, 9], [-23, 2, -3, 2, 2, -7], [-13, 10, 4, 13, 2, 7], [6, -7, 11, 3, 10, 8], [-39, -3, 1, -1, -5, -9], [34, 6, 10, 3, -2, 10], [-32, -2, -2, -2, -5, -11], [1, -7, 1, -5, -5, -5], [-26, -1, 3, -2, -6, 5], [-25, 10, 1, -6, 0, -13], [20, -1, -1, -6, 8, 4], [-22, 6, 2, 4, -8, 2], [26, -1, 1, -6, -1, 0], [12, -5, 3, -6, 0, 3], [1, 3, 7, 7, 7, 3], [4, 0, 2, -8, -2, 4], [9, -2, -10, -2, 2, 1], [4, 9, -5, -4, -2, -5], [20, -1, 3, -5, 3, 6], [-17, -1, -9, -7, -2, 8], [-17, -1, -13, -3, -3, 0], [12, -4, -2, 2, 6, -2], [13, -7, 7, 6, -8, -2], [-27, -7, -15, -3, -4, -4], [-5, -3, -3, 8, -2, 6], [-21, 1, 1, -3, -5, 1], [6, -2, -6, 0, 4, 0], [20, -4, 4, 6, 2, -6], [6, 2, 0, 12, 0, -1], [-15, 8, 2, 3, 1, 1], [0, 4, 10, 6, 0, -2], [14, -9, -6, -13, -1, -12], [-12, 7, -8, -3, -5, -6], [2, 1, -3, 1, 4, -2], [8, -10, -6, -10, 4, 10], [8, 1, 2, -1, 9, 4], [-40, -2, 0, -1, 1, -1], [4, 1, 3, 8, 2, -4], [-15, -8, 0, -6, -1, 2], [-20, -3, -9, -14, -1, -12], [-22, 0, 4, 5, 6, 8], [-21, 11, -7, -6, -1, -3], [-20, 10, -10, -6, 2, -6], [-3, 2, -2, -4, -7, -15], [28, -1, 1, 0, -4, -9], [-38, 1, 2, 3, -1, -12], [-17, -3, 3, 3, 8, 2], [-19, -3, -3, -11, -7, 3], [15, 6, -3, 4, -1, -10], [2, -4, -5, 3, 6, 6], [24, 2, 6, 5, 11, 7], [35, -4, 0, 2, -5, -2], [-7, 3, -1, -1, 1, 15], [9, 2, -1, 4, -2, -9], [1, -18, -14, -6, -6, 13], [-22, 8, 4, -2, 2, -12], [35, 3, -5, 3, -9, -3], [4, -4, 0, 3, -10, -6], [-14, -1, 5, -1, 1, 10], [-15, 6, 8, 7, 12, 3], [-33, 12, -4, -6, -12, -5], [0, 9, -6, -9, 3, -8], [0, -8, 8, -10, 4, 12], [34, -7, 1, 1, -9, 4], [-25, -13, -17, -7, -9, 1], [-5, -12, 2, -2, 3, 8], [21, 13, -1, 1, 1, -1], [-30, 9, 5, -5, 5, -10], [2, -14, 4, -2, 0, -4], [8, 6, 2, 9, -2, -10], [-3, -4, -7, 8, -3, 6], [23, 13, 2, 7, -1, -11], [-18, -5, -3, 5, 2, -6], [-17, 9, -9, -7, -2, -12], [4, -10, 8, 6, 4, 0], [-2, 2, -7, -7, 0, -2], [11, 1, -1, -6, 10, 8], [-8, -4, 6, -1, -10, 6], [-12, -15, -7, -1, -6, 6], [-31, -3, 1, -7, 3, -5], [-6, -4, 4, 0, -14, 4], [-26, -14, 3, 4, 2, -2], [12, 1, 4, -7, 1, -10], [-7, 14, 10, 5, 3, -6], [-18, -9, -8, 1, -3, 8], [-42, 3, 1, -3, 5, -10], [22, 14, 0, -2, 12, 6], [47, 1, 7, 9, 7, -1], [-34, 6, -10, -1, -4, 0], [-10, 6, 1, -8, -1, -5], [16, 4, -6, 1, -8, -2], [-6, -4, -2, -4, -8, -12], [17, -3, 9, -2, 8, 4], [22, 13, -1, 0, 1, -22], [-11, 4, -7, -6, 5, 6], [22, -9, 5, -1, 7, 8], [8, -9, 11, 6, 2, 4], [10, 3, 12, 9, 3, 16], [-38, 0, -2, -5, -8, 0], [20, -10, -6, -2, 1, 7], [-5, 7, -5, 9, -3, -5], [1, 1, 1, 7, -7, -13], [1, -21, 1, 13, 1, 11], [8, 7, 9, 8, 2, 14], [-9, -11, 3, 6, -6, 11], [35, -5, -1, 1, 3, 1], [-15, -7, -13, -11, -4, -12], [-29, 0, 0, -3, -1, 0], [20, 7, -5, 4, -3, 4], [-5, 7, 5, -3, 3, -15], [29, 1, 13, 7, 2, -4], [21, 3, 5, 6, 1, -3], [-10, 3, 7, 3, -5, 8], [13, -4, 11, 4, 0, -9], [-48, 12, 4, 3, 1, -10], [7, 3, -3, -1, -9, -5], [32, 6, 9, 1, 0, 8], [7, -6, -3, -4, -18, -5], [7, -2, 4, 11, 11, 19], [-21, 2, -8, 8, 2, -5], [-22, -1, 7, 2, -16, -1], [3, 11, 2, 10, 5, -17], [-20, 5, -7, -9, 2, -10], [-29, 23, 1, 7, -5, -7], [-2, 8, -10, -10, -6, -20], [19, -10, 13, 4, 7, 4], [-27, 1, 9, 1, -3, -15], [21, -1, -1, -15, 1, -5], [24, -13, 1, -15, -4, 4], [28, 5, 5, -6, -2, -15], [-44, 2, -5, 7, -6, -4], [21, 6, 14, 13, 6, 7], [16, -3, -5, -4, -4, 17], [47, 5, -2, -6, 11, 5], [15, -7, -5, 5, 5, 7], [24, 7, -2, -7, -2, 1], [-28, -3, -9, -4, -8, -9], [5, -15, 2, -6, 3, 3], [3, -18, -8, -12, -4, -2], [5, -5, 1, -5, 3, 8], [-12, -12, 4, 1, 5, 9], [47, 7, 3, -5, 6, 14], [8, 5, -12, -7, 13, -8], [21, -17, -3, 7, 3, 9], [33, -1, 13, 9, 5, 5], [-6, 5, -7, -2, 3, 10], [0, 2, 0, -10, 6, 1], [-5, -5, -13, 1, -6, -2], [31, -5, -3, 1, 11, 13], [-17, -4, 6, 1, 8, 1], [47, -7, 7, 10, 10, 6], [-33, -5, 3, 0, -2, -8], [0, -6, -2, 2, -3, 17], [-2, -2, -14, -4, 4, -12], [-4, 11, -1, 4, 0, -14], [-30, 10, 4, 6, -8, -16], [-19, -4, -4, -15, -1, -4], [-37, -3, -1, 5, -5, -3], [-12, -1, -14, -1, -10, 1], [-60, 1, -7, -8, -4, -3], [-10, 0, -8, -7, 10, 4], [33, 4, 2, 2, 9, 18], [41, 6, -10, 0, 9, -12], [-34, -2, 2, -8, 8, -6], [14, -10, -4, -10, -12, -2], [4, -14, -10, -1, 3, 7], [26, 3, 5, -2, 9, 10], [-16, -5, -3, -11, -6, -10], [19, -7, 7, 7, 9, 9], [-3, 11, -2, 5, 6, 16], [29, 1, -3, -9, -5, -19], [-28, -17, -7, -4, -9, 4], [-8, 14, 2, -2, 6, -7], [36, 11, 9, 2, 5, 12], [24, -16, 6, 0, 8, 2], [-49, 3, -5, -9, -7, -7], [-9, 7, 3, -2, 6, 3], [19, -1, 2, 12, 9, -1], [5, -6, 8, 7, 5, 25], [13, -7, 9, 6, 0, 4], [27, -6, -2, -10, -4, 6], [7, -2, -12, -4, -2, 3], [15, 9, 5, -4, -1, -15], [-7, -2, -3, -14, -7, -12], [56, -8, 11, 13, 0, 6], [12, 6, -6, 2, -4, -23], [-29, 0, -6, -9, -3, 3], [-37, 9, -3, -17, -1, -17], [12, 7, -7, 6, 4, -15], [-20, 9, -9, -8, -2, -10], [2, 3, 5, 2, 15, 2], [37, -5, -7, 4, -2, 14], [58, 6, 14, 16, 8, -10], [48, -8, 2, 8, 4, 10], [-24, 6, -12, -5, 4, 8], [37, -8, 9, 0, 6, 1], [-11, -4, -16, -6, -5, 2], [37, -2, 2, 11, -3, -2], [-39, 10, -6, -8, 8, -2], [-46, -6, -10, 3, -1, 7], [-7, 1, -9, 3, 9, -9], [-8, 13, 3, -3, 16, -7], [-23, -13, 3, 10, 5, 11], [14, 7, 5, -4, -9, 10], [-1, -3, 3, 7, 9, 9], [-29, -2, 4, 2, 0, -6], [-13, -13, -7, -5, -1, 7], [3, 15, -3, 5, -1, -2], [14, 20, 0, -2, -3, 3], [-1, 2, 10, -3, 15, 4], [51, -1, 7, 4, 1, 9], [-9, 1, 3, 2, 6, -12], [3, 14, 2, -5, -6, 1], [16, 8, 5, 9, -6, 10], [-14, 7, -3, 4, 8, -3], [55, -2, 2, 3, 2, 17], [-12, 10, 6, 8, 4, 4], [-21, 11, 7, 5, -3, -9], [35, 0, 6, 2, 3, 12], [-24, 12, 6, 2, 6, -10], [20, 3, -1, -1, -3, -22], [1, 8, 4, -5, -14, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9300_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9300_2_a_ba();".
// 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_9300_2_a_ba(:prec:=6)
    chi := MakeCharacter_9300_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(3833) - 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_9300_2_a_ba();".
// 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_9300_2_a_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9300_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-192, -256, 112, 68, -20, -4, 1]>,<11,R![4, -84, 210, -67, -39, 3, 1]>,<13,R![192, -704, 400, 76, -44, -2, 1]>,<17,R![-13127, -2374, 1577, 194, -67, -4, 1]>],Snew);
    return Vf;
end function;