
// Make newform 9300.2.a.bd in Magma, downloaded from the LMFDB on 17 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_bd();".
// 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_bd();".
// 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 := [281, -161, -187, 113, 31, -21, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [-115, 18, 71, -14, -15, 2, 1], [153, 134, -49, -62, -3, 6, 1], [201, -94, -181, 50, 45, -6, -3], [413, 38, -293, 2, 57, -2, -3], [1433, -126, -997, 98, 197, -14, -11]];
    Rf_basisdens := [1, 1, 8, 8, 8, 8, 8];
    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, 0], [-1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, -1, 0, 0], [0, 0, 0, 1, 0, -1, 0], [1, 0, 0, 0, 1, 0, -1], [2, 0, -1, 0, 1, 0, 0], [0, 0, -1, -1, 0, 0, -2], [0, 1, 0, 1, 1, -1, -1], [-1, 2, -1, 0, 2, 0, -1], [1, 0, 0, 0, 0, 0, 0], [1, 3, -1, 0, 1, -2, 0], [4, -1, 0, -3, -2, 2, -1], [2, -3, -1, 0, 0, 0, -1], [2, 0, 1, 0, -1, -2, 0], [4, -3, 0, -1, -3, 1, -1], [-1, 1, -1, 2, 0, -1, 0], [-2, 1, 1, 3, 0, -3, 3], [1, 0, 0, 2, 1, -2, 1], [1, -5, 3, 1, -2, 2, 0], [1, -1, 3, 2, 1, 2, 2], [-4, 2, 1, 1, -1, 1, 2], [2, 1, 0, 1, -1, 3, -1], [1, -2, -1, -3, -2, -1, -1], [2, 2, 0, 4, 2, -2, 3], [4, -1, -2, -3, 0, 4, -3], [1, -1, -1, 0, -1, 0, 1], [2, 4, -1, -1, 1, -3, 1], [0, -3, -1, -1, -1, 0, -1], [6, -2, 0, -3, -4, 3, 1], [8, 0, -2, -6, -2, 4, 0], [-3, 2, 1, 0, 0, -2, 1], [10, -3, 2, -5, -3, 5, -3], [-4, -1, -1, -1, -4, 1, 1], [-6, 2, 0, 1, 4, -1, 0], [-6, -1, 0, 4, 1, 0, 3], [6, 2, -2, -2, 0, 4, -1], [-3, 1, 1, 0, 3, 2, -1], [10, -3, 1, 1, 0, 3, -1], [-4, 5, -1, 7, 4, -5, 3], [0, 4, 2, 0, -2, 2, 2], [2, -7, 1, 1, 0, 3, -1], [-3, 3, 1, 2, 2, 1, 2], [0, 0, 2, -2, 0, -2, 3], [6, -5, 2, 3, -1, 1, 3], [-4, 5, -2, -2, 3, -2, -1], [-6, -1, 4, 0, -2, 1, 1], [-2, -3, 1, 4, 2, 2, -1], [-8, 4, 5, 4, 5, -2, 2], [-2, -7, 3, -1, -4, -1, -1], [8, 4, -3, -1, 1, 1, 1], [2, -6, 0, 1, -4, 3, 0], [-2, 5, 1, 3, 2, -5, 3], [4, -4, 0, 2, 2, 0, 4], [6, 0, -3, -1, -1, -3, -3], [4, 1, -3, -3, 2, -1, -3], [-1, -2, 1, 2, 2, 2, 5], [0, 3, -5, -3, 0, -1, -3], [9, -9, -1, -6, -3, 6, -6], [10, 3, -4, 4, 2, -3, 1], [-1, -7, 3, -2, -7, 2, -2], [8, -5, 0, -3, -5, 5, -1], [-3, 8, -2, 0, 9, -6, 0], [-1, 5, -1, -1, 4, -6, 0], [5, -2, 6, 0, -3, -2, 3], [10, -2, 1, -1, -7, 1, -1], [2, -2, -1, -5, 3, 3, 0], [11, -1, -1, -4, -5, 4, -4], [6, 1, -3, 1, 0, -1, 3], [0, -6, 1, -7, -1, 7, 0], [6, 6, 1, 2, -3, -6, 6], [13, -3, -3, -6, 2, 1, -2], [16, -4, 4, -6, -8, 6, 2], [4, -1, 1, -2, -4, -2, 4], [-6, 3, 3, 3, 4, -1, -1], [4, -5, 4, 1, -5, -1, 7], [-3, -2, 7, -5, -4, -1, 3], [0, 5, 1, 12, 6, -6, 2], [3, 4, 1, 7, 4, -1, 1], [6, 7, -1, -3, 4, -1, -7], [3, 3, 1, 7, 2, -6, 8], [-6, 8, -4, 0, 7, -1, 2], [13, -4, 1, -7, -4, 7, -1], [7, 1, 1, 0, 3, -6, -4], [-6, -5, 6, -2, -4, 3, 1], [-10, -1, -2, 6, 7, -2, -6], [3, -2, -1, -3, -4, 5, 3], [11, -3, -3, 2, -1, 8, -6], [9, -10, -1, -6, -8, 8, -9], [2, 3, -3, 0, 0, 0, 5], [2, -3, -2, -2, 1, 2, 2], [5, 3, -3, 1, 2, -6, -6], [2, 6, -2, 2, 6, -2, -4], [2, 2, 4, 2, 0, -4, 2], [12, -2, -4, -8, 0, 6, -2], [0, 7, 4, 8, 3, -4, 8], [-5, 8, 1, 9, 8, -7, 1], [10, 2, -2, -1, -6, -1, -2], [-6, 2, 2, 2, -2, -2, 0], [-8, 0, 4, -2, 1, 1, 0], [3, 1, 3, -2, -1, -2, -3], [16, -8, -1, -2, -5, 6, -10], [0, -6, 5, 1, 3, 7, 5], [9, 0, 3, 0, -8, -2, 7], [-18, -1, 5, 7, 10, -1, -5], [8, 3, -5, -10, 4, 6, -3], [-10, 5, 7, -3, -2, 3, 3], [12, -4, -6, -5, -6, 1, -7], [11, -3, 1, 1, 0, -4, -2], [-10, -2, 2, 6, -4, -2, 6], [1, -12, 6, 2, -9, 0, -1], [9, 4, -4, 2, 5, 0, 3], [0, 4, 0, -2, 4, 4, 6], [-4, 8, -5, 1, 3, -3, 0], [10, -4, -2, -4, -6, 2, -6], [9, -4, 1, 0, 2, 0, 5], [-8, 8, -8, -8, 0, 2, -4], [8, -6, 4, 2, 2, 6, 6], [-2, -7, 6, 3, -3, -1, 3], [3, -3, -3, -1, 6, -4, -2], [-2, 2, -3, 9, 4, -6, 0], [13, -12, 6, -4, -13, 4, -3], [4, 3, -2, -7, 1, -1, -1], [-14, -11, 4, 0, -1, 8, 0], [-2, -6, 5, -9, -4, 6, -2], [14, 3, -4, -5, 8, 0, -3], [-4, 4, 2, 4, -2, -2, 6], [12, 2, 2, 3, 0, 3, 0], [3, 5, -1, 10, 7, -6, 9], [12, 1, -7, 2, -2, -4, -4], [-12, 9, -2, -2, 3, 2, 7], [-8, -9, 3, -3, -2, -1, -3], [-6, 8, -5, 1, 6, 2, 0], [-1, 4, 2, 0, 5, -6, 5], [5, 6, -7, 0, -2, -6, 3], [10, -5, -3, 3, -7, 4, -1], [4, 6, -1, -6, 1, 2, -6], [6, 5, -3, 2, -2, -8, -3], [-18, 9, -2, 3, 1, -11, 9], [23, -2, -1, -9, -2, 9, -5], [-14, -4, 4, -2, -6, -2, -2], [5, 0, 7, -3, -8, -3, 7], [-2, 5, -3, -4, -6, -4, 1], [-6, 15, -2, 1, 3, -5, 11], [26, -4, 2, -4, -10, 4, 0], [-5, 6, 3, 10, 8, 2, 9], [30, -3, -5, -4, 0, 2, -1], [6, -5, 3, 9, 0, 1, -1], [0, 0, 0, -2, 0, 6, 0], [-10, 0, 2, -4, 8, 4, 0], [-10, 6, 0, 10, 2, -6, 9], [9, -4, 5, -1, 0, 1, -1], [-10, -7, 1, 2, -2, 8, -9], [2, 0, -11, -5, 3, 9, -7], [-1, -2, -2, 0, 1, 8, -6], [-16, 12, -4, 1, 12, -3, -2], [2, 1, 7, 1, 2, -11, 1], [11, -12, 1, 0, -16, 0, 3], [-4, -11, 3, -2, -6, 8, 5], [9, -2, 11, 7, 6, -1, 5], [-4, -15, 0, -7, -3, 11, -7], [-6, 5, 0, 5, 11, -9, 1], [18, -5, 11, -6, -10, 4, 5], [25, 0, 1, -13, -4, 9, -5], [6, 9, 1, -2, 8, 2, -8], [-2, 10, -6, -6, 2, -2, 2], [-6, -1, -3, 7, 2, -5, -5], [8, -3, 5, -4, -10, -4, 0], [-4, -3, 1, 7, 6, -7, 5], [-4, -13, 4, -3, -1, -1, -3], [2, 8, -2, -3, 2, -1, 0], [0, -15, 3, -1, 0, 9, -7], [19, 6, -7, 1, 0, -1, 3], [10, -4, 4, 0, 0, 2, -1], [2, -4, 5, -11, -8, 2, 0], [5, -4, -11, 0, 0, 6, -3], [-14, -12, 5, 3, 0, 0, 4], [-10, -1, -2, 6, 4, -5, 5], [-15, 5, -9, 8, 7, -8, 0], [-28, 7, -3, 5, 0, -5, 3], [-27, 2, 6, -2, -1, 0, -4], [-8, -6, 10, 8, 8, 2, 0], [-6, 2, 0, 6, -2, -8, 11], [-2, -8, 5, 12, -1, -4, 0], [0, 5, -4, -14, 5, 2, -4], [7, -2, -5, 2, 2, 4, 1], [14, -9, -7, -12, -4, 14, -3], [-19, 3, -3, 12, 1, -6, 11], [12, 0, -8, -8, 2, 4, -6], [-5, 13, 3, 2, 14, -9, -2], [13, -7, -7, -4, 1, 8, -4], [-18, 9, 2, 12, 1, -12, 10], [-14, -9, 0, 4, 1, 8, -5], [25, 4, 1, 0, 4, -4, -5], [-28, -2, 1, 2, -1, -8, -4], [2, 4, 3, 0, 13, 4, -8], [-2, 4, 0, 14, -2, -12, 0], [-2, 6, 0, 2, 0, 4, -6], [18, -7, 5, -4, 0, 8, -2], [-16, -1, 4, 22, 7, -10, 4], [24, -15, -6, -10, -4, 9, -13], [-4, 1, -1, -5, -8, 7, -1], [-4, 3, 7, 0, 0, 0, 5], [-8, 7, 2, -4, 0, 5, -7], [7, -6, 5, -6, 6, 2, 1], [12, 4, 1, -6, -3, -2, -2], [0, -5, 12, -2, -2, 5, -1], [-2, 3, 2, -9, -5, -7, -1], [6, 7, 8, 5, -2, 0, 9], [24, -8, -5, 5, -4, 10, -10], [-25, 12, 2, 6, 13, -18, 9], [1, 14, -3, -1, -2, -5, 15], [5, -7, -3, 2, 5, 8, -10], [-28, 6, 1, 11, 7, -11, 7], [0, -18, 8, 5, -10, 5, 4], [-18, 13, -1, 7, 19, -2, 5], [5, -13, -5, -10, -3, 14, -18], [12, -13, 8, 1, 0, 10, -7], [6, 11, -6, -3, 15, -5, -9], [10, -18, 6, -5, -14, 13, -7], [-16, 8, -12, -4, 4, 2, -10], [-18, 7, 0, 16, 12, -9, 9], [-10, 7, -4, -3, 6, -4, -7], [-1, -11, -5, -16, -9, 6, -9], [-24, 6, -1, 7, 9, -3, 7], [4, -2, 0, 18, 0, -10, 4], [14, -4, -3, -11, -9, -5, -7], [-4, -12, 4, -5, -6, 9, 6], [0, -6, 6, 8, 4, -6, -6], [15, -8, -7, -7, -14, 9, -9], [7, 12, -2, -12, -3, -6, -1], [-20, -11, -2, -8, -8, 1, -3], [-8, 2, 9, 7, -2, -10, 8], [29, -4, 5, 1, -12, -3, 5], [0, 5, -13, -2, 2, 2, -5], [-2, 4, 2, 12, 8, -8, 2], [-18, 12, -1, -7, 4, -2, -6], [10, -2, -2, 2, 2, -8, 6], [42, -12, 0, -7, -10, 15, -2], [26, -14, -6, -19, -24, 21, -4], [-6, 11, -12, 1, 5, -3, 3], [-8, 18, -12, 10, 16, -4, 6], [10, 3, -9, 0, -2, 4, 3], [12, -14, 2, -4, -2, 16, 0], [12, 7, 0, 9, 3, 3, -1], [-11, 14, -5, 4, 18, -6, 5], [-2, -7, 7, 2, 4, 8, -9], [2, -2, 0, -3, 14, -5, 2], [2, -11, 3, 3, -6, -1, 5], [-4, 9, -4, -1, -3, 7, 7], [4, 1, -5, 2, 0, 2, 0], [4, 15, -4, 7, 0, 0, 1], [-16, -5, 5, 9, 4, 3, 11], [6, -6, -4, -10, -13, 7, 0], [-16, 9, -3, -2, 14, -6, -6], [-5, -20, 5, 8, 4, 2, 5], [-16, 13, -4, 2, 8, -1, 7], [-4, 3, 3, 4, 0, -2, 15], [-4, 4, 1, -5, -3, 1, 7], [4, -3, -5, -8, -4, 0, -10], [-13, 22, -2, 6, 11, -10, 9], [10, -11, -2, -5, -7, 9, -9], [-28, 14, -8, 4, 0, -16, 8], [18, 2, 2, -4, 0, 8, 5], [-28, -6, -7, -10, -1, 6, -8], [-10, -9, 8, -8, -10, 5, 9], [-24, -11, 0, 0, -6, 5, 1], [-4, -6, -2, -5, -12, 13, -6], [-31, -1, 11, 14, -11, -2, 6], [-10, -7, 9, 6, -4, -2, 6], [-20, -5, 4, 18, 7, 0, 3], [-30, 3, -1, 10, 10, -6, -3], [7, 25, -11, -6, 7, -16, 10], [4, -10, 8, 12, 0, -6, 6], [14, 1, -3, -10, -12, 8, -2], [-11, 10, 8, 4, 5, -16, 20], [-4, -3, -12, -4, -7, 12, -6], [-18, 9, 1, 3, 12, 5, -5], [0, 3, 7, -5, -6, -11, 7], [-14, -10, 4, 8, 8, 6, 2], [4, -6, 9, 1, 7, -1, 3], [-6, 13, -4, 0, -8, 5, 3], [-24, 6, 3, -3, 5, -5, -3], [-28, -2, 1, 7, 2, -8, 2], [-37, -3, 5, 0, -3, -4, -5], [20, 16, -10, 4, 8, -10, 0], [-1, 3, -1, -10, -5, -2, 0], [-24, -2, -3, 10, -1, 2, -2], [-10, -2, 0, -4, 8, 4, -8], [11, 0, -13, -12, -4, 4, -11], [10, 1, -2, 13, 0, -2, 3], [12, -28, 15, 11, -13, 17, 3], [-16, 7, -8, 5, 15, -5, 3], [25, 2, 7, 5, 0, -3, 3], [3, -8, 8, 2, 7, 6, -1], [-19, 22, -7, -3, 6, -19, -5], [-8, -2, 1, 7, -13, 7, 2], [-2, 9, 8, 7, 7, -9, 11], [-6, 6, -6, -4, -6, -6, -18], [-20, -2, 2, -4, 4, 6, 4], [6, -23, 5, 6, -6, 12, 3], [-18, 14, 3, 4, 7, -8, 14], [2, 0, 0, -4, -10, -4, -10], [-22, 18, 5, 16, 1, -22, 22], [16, 9, -8, -8, -5, -10, 7], [-36, 4, -4, 10, 12, 0, -2], [-6, -7, -2, 7, 1, -3, -5], [4, 6, -1, -7, -2, 4, 10], [6, -18, 10, 10, 0, -6, 2], [-20, 9, 1, 6, 14, 4, 8], [-16, 15, -4, 9, 7, -13, 17], [-1, -4, 11, 3, 6, -7, -5], [-18, -21, 10, 6, -10, -3, 5], [-8, 12, -10, -2, 18, 4, -8], [-24, -12, -1, -7, -17, 11, 1], [14, -8, 14, 14, 0, 8, 12], [36, -4, -4, -2, -6, -2, 2], [2, 10, 0, -9, -6, -3, -2], [20, -1, -7, 12, 4, 4, -14], [-21, 10, 3, 15, 10, -1, 9], [14, 5, 4, 2, -9, -2, 1], [-37, 10, 6, 2, -1, -2, 9], [-16, -1, 4, -5, -8, 0, -11], [-41, -1, 3, 8, 13, 2, 1], [16, 7, -10, -9, -15, -1, 3], [22, -3, 11, -5, -10, 13, 7], [14, -4, -7, -1, 6, 10, -4], [-9, -3, -5, -6, 7, 0, -11], [12, 2, 0, -6, -2, 6, 8], [-12, 17, -8, -7, 9, -1, -3], [2, -19, -4, -10, -22, 11, 5], [20, -18, -5, -9, -15, 13, -17], [-14, -8, 11, -7, -2, 2, 0], [4, 5, -10, -11, -9, 15, -11], [-12, 16, -7, -15, 0, 8, 2], [-34, 0, 9, 11, -9, -7, 11], [34, -5, -9, 1, 2, -3, -3], [-8, 7, -2, 3, 1, -17, -9], [2, 1, 5, -7, -5, 10, 17], [2, -13, 7, -6, -6, -6, -3], [13, -8, -4, 4, 1, -2, 7], [-22, -1, 6, 9, 19, -9, 3], [-13, 0, -7, -2, -6, 14, 1], [-14, 10, -6, -6, -2, 2, -8], [22, 6, -6, -14, 2, 16, 2], [7, 13, -15, -6, 0, 3, -6], [-12, 16, -1, -7, 7, -9, 10], [-22, -4, -12, -18, -10, 10, -16], [-22, 8, 4, 12, 8, -22, -2], [-6, 0, -3, -5, -1, 1, -13], [4, -1, -6, -2, 11, -4, -7], [1, 8, -4, -2, 1, 8, -7], [-14, 5, 8, 16, 2, -7, 13], [4, -25, 11, 4, -4, 18, 1], [-16, -12, 1, -11, -10, 8, -8], [20, 3, 2, 3, -7, 5, -5], [34, -24, -4, -8, -4, 4, -24], [10, 12, 8, 5, -2, -9, 14], [18, 6, 5, 4, 5, -4, -2], [-22, 7, -10, 0, 19, -8, -12], [-6, 2, 4, 10, -2, -2, 5], [1, 6, -13, -9, -8, 3, -13], [-22, -5, -2, 7, 3, -7, -9], [-7, 1, 11, 14, -6, -5, 18], [-35, 24, -6, 6, 5, -12, 14], [9, -13, -9, -12, -11, 12, -8], [12, 2, -1, -1, -11, -11, 21], [2, -7, 7, 18, 0, -10, 21], [10, -4, 11, 15, -3, -3, 12], [27, 8, -7, -6, -8, -8, -3], [18, -2, -12, 6, 0, -2, 8], [-42, 15, -2, -3, 7, 1, -1], [0, -11, 4, -17, -8, 16, -7], [2, -5, 6, -6, 14, -3, 1], [-5, 2, 10, 0, -9, -12, 17], [24, 6, -16, -15, -14, 5, 4], [15, 11, -7, -10, 20, -7, -4], [-22, 20, 4, 14, 0, -18, 21], [-2, 16, -8, -11, 6, 5, 4], [17, 7, 7, 6, -7, 4, 8], [15, 2, -3, 1, 10, 5, -5], [6, -13, 4, 9, -11, -1, -1], [24, 2, -4, 2, 6, 10, -6], [-10, -17, 18, 3, -21, 9, 13], [-10, 20, 4, -6, 2, 6, -6], [-14, -13, 6, 8, -9, 8, -4], [2, -6, -15, -11, 12, 4, -18], [29, 2, -10, -16, -15, -2, -1], [10, 8, 8, 8, 10, 4, 10], [10, -17, 2, -5, -17, -7, 5], [20, -5, -3, -13, 2, 3, -21], [-14, 16, -3, 18, 19, -18, 0], [5, 9, 5, 1, -8, -8, 22], [-25, -1, -5, -8, 7, 4, -4], [27, -24, 5, -1, -2, 13, -19], [20, -8, 6, -12, 2, -4, -4], [32, 3, -13, 3, 4, -3, -15], [-30, 1, -2, 9, 6, -20, 3], [14, -14, 13, 3, -13, 5, 8], [7, -6, -3, -18, 2, 14, 1], [-10, 27, -13, 1, 30, -3, -3], [12, -24, -1, 5, -13, 15, 3], [31, 0, -12, 14, 3, -2, -13], [-8, -12, -1, 11, 11, -3, -7], [-11, -6, 17, 16, -4, 8, 11], [8, 7, 10, 14, 10, -13, 15], [-52, -11, 5, 4, 4, 8, 7], [-11, -10, 9, -13, -8, 5, 7], [14, -29, 13, 0, -8, 14, -3], [6, -8, 8, -1, 0, -3, 4], [1, 14, 6, 12, 13, -20, 9], [32, 2, -14, -17, 12, -3, -11], [-40, 16, 8, -2, 0, 2, 10], [-24, 9, -4, -16, 7, 6, -5], [-2, 9, 7, 16, 6, -12, 0], [22, -25, -6, -18, -16, 1, -17], [8, -22, 2, 11, -4, 17, 2], [-27, 2, -6, 2, -5, 0, 8], [12, -15, 2, -9, -9, 15, 1], [30, -13, 0, -8, 9, 8, -2], [-11, 8, -5, 9, -2, 3, -9], [45, -3, -5, -10, -17, 0, 6], [54, -3, -2, -19, 5, 3, -15], [-19, 17, -13, 12, 16, -11, 4], [52, -8, 14, -4, -12, 10, 11], [36, -2, 1, -26, -5, 22, -12], [-16, 8, -3, 12, 9, 6, 6], [3, 22, -19, 0, 14, -18, -5], [12, -8, 18, -14, -16, 8, 12], [36, 2, 4, -2, 0, 14, 0], [12, 1, -11, -19, 9, 18, -9], [-41, 15, 7, 9, 14, -10, 0], [24, -23, 3, -19, -16, 17, -25], [36, -3, -8, -20, -3, 8, -6], [23, 0, 0, -14, 7, 4, -7], [-21, -8, 5, 3, 8, 9, -9], [12, -23, -3, 1, 0, 17, -23], [16, -4, -2, 14, -8, -6, 2], [-44, 8, -2, 6, 18, -10, 8], [28, -16, -12, -14, -20, 14, -8], [34, -18, -1, -1, -9, 21, -21], [-10, -3, 8, 12, 4, -7, 7], [-22, 2, -13, -1, -13, -7, 2], [7, -2, 15, 20, 8, -10, 15], [-36, 1, 9, 13, 6, -17, 1], [26, 12, 5, 10, -5, -14, 12], [21, -19, 1, 0, -1, 0, -9], [46, 17, -14, -13, 5, -1, 5], [44, 8, 1, 1, 2, -12, 12], [26, -25, 0, -24, -11, 18, -15], [-33, 19, -13, -6, 7, 2, -4], [14, 4, 6, -1, -2, 13, 16], [18, -2, -13, -13, -17, 5, -7], [-24, -14, 4, 7, 6, -11, -2], [31, -3, 7, 0, 5, 4, 10], [18, 2, -2, 5, 10, -11, -6], [-20, 24, -7, 5, 22, -16, 0], [-24, -8, -2, 1, -2, 5, 6], [-9, 27, -5, 6, 7, 0, 12], [-26, -8, -4, 5, 14, 7, -13], [-10, 1, 0, -4, -12, -7, -13], [6, 21, -4, -6, -4, -9, 11], [-12, -4, 8, -11, -10, 15, -2], [0, 3, -7, -9, -12, 13, -13], [4, 8, 8, -2, 4, -2, 12], [-27, -20, 20, 2, -19, -6, 3], [-34, 7, -10, 6, 6, 1, -7], [10, -25, 6, -1, -5, 25, -5], [-23, -8, 5, 3, 8, -11, -1], [4, -3, 12, -8, 2, 13, 9], [0, -16, 14, 8, -10, 16, -4], [10, -4, 2, -14, 4, 2, 0], [22, 12, 10, 6, -2, -14, 16], [18, 29, -13, -1, 26, -11, -1], [-18, 32, -2, 12, 10, -28, 8], [-2, 4, 2, 20, 4, 4, -2], [-20, -16, 10, -13, -10, 15, 2], [26, 13, -12, -14, -11, -16, -3], [4, 5, -4, 4, -7, 4, 10], [-10, -6, -9, -16, -1, -8, -2], [19, 12, 2, -18, 1, -4, -7], [4, -15, 18, -12, 0, 17, -7], [30, 24, -14, 6, 14, -12, -7], [0, 8, -2, -6, -4, 8, 4], [-36, 10, 2, 22, 8, -22, 20], [18, -3, 4, -7, 3, -5, -7], [14, 10, 6, -2, 19, -7, -4], [-21, 3, -7, 11, -14, -2, 8], [0, -7, 3, 11, 4, -1, -13], [-32, 24, -7, 11, 24, -26, 2], [52, 0, 4, -12, 0, 0, -12], [32, 15, -12, 7, 7, -7, -1], [42, 11, 1, 5, 2, 7, 5], [42, 0, 7, -2, -7, -6, 4], [-13, -7, 5, -10, -4, -1, 2], [-18, -5, 15, 25, 2, 9, 5], [32, -11, -5, -26, -18, 20, -7], [-36, -6, -7, -4, 19, 16, -8], [-10, -14, 3, -13, 6, 18, -18], [26, -27, 6, 4, -9, 22, -7], [-2, 17, 2, 11, -2, -2, 3], [33, -21, 3, -20, -13, 24, -17], [12, 7, -1, 14, 6, -10, 0], [26, 6, -12, 0, -8, -6, -10], [-19, 13, -15, 16, 1, -14, 12], [4, -3, -14, -15, 3, 3, -9], [14, 23, -9, -3, 12, -5, 11], [36, -16, -12, -2, -12, -2, -4], [-32, -19, 15, 14, -12, 8, 0], [3, -10, 0, 0, 5, -2, 11], [11, 11, -15, -1, -4, -6, 6], [-16, 10, 8, -11, 4, -3, 10], [-9, 21, -1, 8, 1, -8, 4], [0, 10, 1, 37, 15, -17, 9], [-24, 8, 4, 14, 0, -22, 12], [42, -19, 3, -10, -18, 12, -16], [-15, 16, -13, 11, 8, -3, 5], [28, 7, -8, -2, -8, -7, 21], [6, 12, 2, -7, 18, -7, 2], [21, -7, 5, -24, -1, 12, -13], [-11, 2, 10, 8, -11, 6, 13], [14, -11, 12, -8, -22, 3, 19], [-20, -1, 2, -10, -4, -3, 7], [70, -15, 6, -3, -19, 1, -3], [8, -27, 11, -1, -21, 8, -9], [3, 21, -7, 1, 10, -8, -8], [17, -34, 6, -2, -7, 10, -13], [14, -8, 7, -16, -3, 8, -10], [-12, -2, 6, -6, 0, -10, 22], [-4, 11, -20, -12, -10, -7, -15], [-21, 25, -5, -8, 17, 0, -6], [18, 21, -2, 5, 1, -19, 35]];
    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_bd();".
// 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_bd(:prec:=7)
    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_bd();".
// 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_bd( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9300_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 148, -216, -10, 82, -17, -4, 1]>,<11,R![32, -96, -104, 96, 60, -24, -4, 1]>,<13,R![-224, 160, 482, -604, 192, 6, -10, 1]>,<17,R![2688, -7008, 6556, -2716, 424, 26, -14, 1]>],Snew);
    return Vf;
end function;