
// Make newform 9200.2.a.dc in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9200_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_9200_2_a_dc();".
// 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_9200_2_a_dc();".
// 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 := [-2, 17, -35, 1, 24, -7, -3, 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], [-6, 7, 26, -3, -11, 0, 1], [-2, -21, 16, 15, -9, -2, 1], [-6, 20, 6, -10, -1, 1, 0], [22, -91, 20, 63, -23, -8, 3], [-11, 42, -22, -30, 17, 4, -2]];
    Rf_basisdens := [1, 1, 2, 2, 1, 2, 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_9200_a();"
function MakeCharacter_9200_a()
    N := 9200;
    order := 1;
    char_gens := [1151, 6901, 2577, 1201];
    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_9200_a_Hecke(Kf)
    return MakeCharacter_9200_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+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, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 1, 1], [0, 1, 1, 0, 0, 1, 0], [1, 1, -1, 2, 0, -1, 0], [1, 2, 0, 2, -1, -1, -1], [-1, 0, 0, 0, 0, 0, 0], [-1, 2, -1, 2, -2, -1, 0], [0, 1, 3, -1, 1, 0, 1], [3, 0, 1, 0, 0, 0, 2], [-3, 2, -1, 1, 0, 0, -2], [1, -2, 1, -1, 1, 1, 1], [2, -1, -2, 1, -2, 1, 0], [2, 0, -1, 1, 0, 2, -1], [3, -3, -2, 1, 1, 1, -2], [0, -1, 2, -2, 4, 1, 2], [1, -4, 3, -4, 2, 0, 2], [4, -1, -4, 1, 1, -1, -1], [1, 0, -1, -1, 1, 0, -1], [5, 2, -3, 1, -3, -1, -1], [2, -4, -2, -1, 1, -1, -1], [-3, -2, 1, 1, 1, 1, -1], [-4, 1, 3, -4, -1, 0, 0], [-2, 4, -3, 5, 0, -4, -1], [3, 3, -2, 5, -2, -1, -3], [-3, 2, 2, 0, 1, 3, 0], [-5, 0, 0, 0, -3, 3, -1], [3, 2, -3, 4, -2, -4, -2], [3, 2, 1, 1, 1, -4, 1], [1, 4, 0, -1, -2, -3, 1], [-8, 2, -1, -3, -2, 2, 0], [5, -4, 3, 0, 2, 1, 0], [-8, 0, 3, -1, 0, 0, 2], [-3, -2, 0, -6, 2, 2, -1], [7, -6, 2, -2, -1, 1, 2], [3, 6, -2, 4, -3, -4, -5], [3, -1, 3, -2, 1, 3, 1], [6, -2, 0, -3, 0, -1, 2], [8, -7, -1, -1, -1, -2, 2], [2, -3, -6, 0, 1, -3, 2], [3, 3, 2, -2, -2, 2, -1], [1, -6, 1, -5, 7, 0, -1], [4, -3, 0, 2, 0, 0, -2], [6, -7, 3, -5, 3, 1, 4], [4, 6, -3, 11, -2, -2, -3], [12, 0, 4, 2, 0, 4, 2], [5, 4, -6, 3, -2, -2, -5], [-5, 2, 5, -1, 3, 5, -1], [11, -6, 1, -1, -1, -2, 7], [-2, 7, -2, -2, 3, -2, 1], [5, -1, -2, 6, 0, 4, -1], [5, -1, 7, 1, 0, 6, 1], [8, -3, -4, 7, 0, -3, -2], [-2, 5, -3, -5, 0, 1, -1], [-8, -3, -2, 0, 1, 0, 1], [0, 0, 0, 2, 2, 0, -3], [6, 1, -2, 5, -4, -3, -4], [-8, -1, 1, -1, -5, 1, 0], [4, 0, 2, 1, -3, 1, -1], [2, 2, 5, 3, -2, 6, 1], [8, 2, -2, -5, 0, -1, 0], [8, -5, -1, 1, 5, -2, 2], [6, -2, -6, 4, -2, 0, -1], [5, 3, -2, 3, 3, -5, -3], [-1, 2, -1, 2, -6, 1, -4], [7, -10, 2, -2, 1, -1, -1], [-5, 3, 7, -3, -3, 0, 3], [-10, -1, 0, -4, -5, -4, -3], [1, -8, 5, -5, 1, 6, 3], [-1, 4, -1, -1, 3, -9, -3], [10, 6, -6, 1, -7, -5, -3], [12, -4, -5, 0, -3, -5, -4], [-1, -1, 5, -9, 1, 2, 5], [13, -4, -2, 2, 1, 1, 6], [-1, 3, -1, 5, 1, -6, -3], [7, 2, -3, 4, -1, 3, -1], [-5, 3, 2, 4, -6, 4, -3], [8, 3, 3, -1, 3, 0, 3], [5, 10, -3, 3, -5, -5, -1], [8, 3, -4, 8, -1, -3, -4], [1, 4, 1, 5, -7, 3, -3], [2, 4, 5, 4, -3, 1, 3], [-5, -6, 5, -2, -2, 9, 1], [8, -3, -3, 0, 4, -3, 6], [-17, 1, 3, -8, 7, -3, 0], [-2, -6, 4, -5, -1, 7, 7], [-7, -6, 0, -3, -2, -3, -5], [2, 0, -4, -4, -2, -4, -6], [-8, 0, 1, -1, 0, -4, -1], [3, -1, 2, -2, 6, 0, -1], [-2, -1, -2, 7, 2, 3, -2], [11, 0, -5, 8, -8, -1, -6], [-4, -1, 0, 6, -1, 4, -3], [8, 2, -2, 4, 4, 0, 3], [5, -14, -4, -5, 6, -1, 7], [2, -6, -6, 2, 2, 4, 2], [9, -2, 2, -1, -2, 0, 3], [-6, -7, -1, -9, -1, 0, -2], [6, 3, 1, 4, 4, -3, 4], [3, -2, 2, -10, 3, 5, 0], [2, 2, -3, 1, 2, -4, -9], [7, -11, 0, 4, -2, 0, 3], [-12, 0, -1, 3, 1, 4, -2], [0, 5, -4, 5, -6, 1, -2], [4, 7, -4, 8, -6, 4, 2], [0, 10, -4, 4, -8, 0, -2], [0, 2, -2, 3, -4, -1, -6], [-3, -10, 10, -5, -2, 9, 4], [-3, 7, -3, -4, -5, 3, -3], [2, 4, 3, -8, 5, -1, -6], [-14, 11, 3, 7, -2, 1, -3], [-4, -8, 5, 1, 4, 2, 4], [-9, -4, -5, -3, 7, -5, -1], [-11, -2, 7, -9, 5, -3, 5], [6, -2, -2, -3, 5, 5, -5], [-1, -10, 5, -9, 3, 0, 11], [2, 9, -11, 10, -8, -5, -10], [-9, -13, 2, -4, 2, 2, 5], [5, -3, -13, -3, 3, -8, -3], [-11, 0, 1, -11, 7, -4, 7], [14, -2, 7, -11, 6, 4, -3], [3, -10, 6, 2, -1, 2, 5], [12, -10, 0, -4, 8, -2, 8], [-8, -5, 16, 0, -1, 5, 6], [17, -6, 4, -2, 2, -1, 5], [6, 2, 2, 2, 4, 6, 1], [-3, 11, 5, 6, 0, -7, 0], [-2, 10, -4, -3, 5, 1, -7], [-1, -4, 9, 4, -6, 9, 8], [-2, -10, 8, 1, -1, 7, 2], [1, -8, -1, 7, -5, 3, -1], [21, -14, 1, -2, -4, 2, -2], [-6, -3, -5, -1, -5, -2, -3], [11, 5, -2, 0, 0, -10, 1], [5, 2, 1, -5, 5, -3, -1], [-2, -8, 1, -13, 10, 0, 9], [15, 10, 2, 0, -3, -7, 2], [7, 9, 4, 4, -7, 4, -8], [-2, -5, 8, -4, 1, 6, 13], [4, -2, -4, 4, -4, 10, 2], [-9, 8, 1, 7, -3, 6, -9], [8, -8, -9, -5, 8, -4, -3], [-4, -8, 3, -9, 6, 6, -5], [0, 9, -1, 11, -5, 5, 6], [13, 3, 4, 1, -1, -3, 5], [-17, -8, 3, 5, -5, -2, -3], [-6, -3, -4, 8, -3, -6, 1], [-5, 0, -3, -1, -1, -2, -5], [8, -6, -6, -9, -4, -3, -2], [-9, -1, -2, -10, 0, -4, -3], [7, -15, -8, -7, -1, 3, -1], [2, 5, 6, -1, -8, -3, 8], [-1, -8, 8, -8, 5, 9, 10], [7, -8, 1, -3, 3, 11, -5], [-8, -3, 3, -1, -9, 1, -4], [-18, -1, 10, -6, -11, 10, 1], [3, -8, -9, 6, 6, -2, 5], [-14, 10, 3, 1, -5, -2, 4], [-25, 8, 4, 6, -5, 0, -7], [-7, -3, 7, -11, 11, 0, 7], [17, -10, -5, 1, 3, -6, -3], [-3, 13, 7, -5, -1, -2, -5], [-3, 14, -6, 7, -2, -8, 0], [1, 17, -10, 16, -13, -8, -8], [-3, 9, 8, 4, 3, 2, 6], [1, -1, 5, 2, 5, -1, 3], [11, -6, 13, -11, -7, 9, 1], [6, -13, -4, 2, 4, -4, 0], [22, -9, -7, -1, 3, 5, 6], [6, -19, 0, -10, 11, -2, -1], [3, -19, 14, -18, 12, 6, 11], [-5, -13, -6, -10, 3, 8, 0], [-22, 5, 15, -4, 3, 5, 6], [-4, -13, 3, -13, -3, 3, 2], [1, -2, 0, -5, 2, -1, 1], [9, -8, 1, 8, 0, 7, -6], [-14, 1, 5, -7, 4, 3, -1], [15, -11, -1, 11, -2, -4, 5], [23, 12, 3, 11, -9, 2, -7], [16, -1, -10, -8, 0, -3, 0], [-3, 0, 2, -6, -1, 0, 3], [-27, -9, 7, -4, 5, 7, 9], [15, -1, -1, -8, 10, 3, 6], [9, -5, 7, 3, 6, -8, -1], [20, -4, 7, -3, 2, 6, 6], [19, -6, -10, 6, -5, -1, 0], [5, -8, 3, -9, 12, 6, 6], [1, 5, -2, 0, -1, -4, -12], [-20, -19, -2, -1, 0, 9, 0], [-3, -1, 3, 8, 3, 1, 1], [-27, 1, -10, 0, -2, -4, -1], [-22, 0, -11, 1, -4, 2, -11], [-16, -4, -15, 4, -3, 1, -6], [-25, 16, 11, -1, 5, 1, 0], [10, -7, -3, 1, 5, 5, -2], [6, 0, -4, 0, -12, 2, -4], [-12, -12, -10, -6, -2, 0, -2], [-14, -7, 10, -3, 2, 9, 14], [11, -19, -11, -9, 9, -4, -1], [3, -21, -6, -7, 1, -3, 3], [17, -4, 1, -10, 17, 3, 3], [14, -3, -19, 11, -3, -1, -10], [-16, 0, 6, -10, 12, -8, 2], [-11, 10, 3, -7, -3, 2, -13], [-8, -19, -5, -21, 5, -1, -6], [9, 6, 1, 4, -7, -7, -3], [18, 3, -7, 15, -5, -10, -7], [0, 15, -9, 3, -5, -8, 2], [10, 1, -12, 7, -2, -9, -2], [-18, 17, 3, -1, -9, 4, -6], [-19, 18, 6, 4, 1, -5, 3], [6, -2, 12, 2, 4, -4, 2], [23, -17, 0, -9, 1, -3, 6], [3, 2, -7, -3, -7, -9, -9], [38, -9, 2, 5, 8, -11, 4], [1, 0, -5, -2, 6, -13, 8], [-24, 4, -9, 14, -9, -9, -3], [0, -1, 2, 11, -6, -1, 4], [13, -2, -5, 0, -6, -5, 0], [6, -3, -7, 5, -3, 1, -2], [1, 8, 5, -9, -4, 2, 0], [15, 4, -10, 3, 6, -14, 4], [24, 7, 2, 23, -8, 1, -4], [-23, 0, -3, -16, 4, -3, 2], [-17, -2, 13, -6, -10, 18, 5], [12, -11, -5, 15, -7, -6, 5], [-12, -12, 11, -7, -4, 18, 6], [13, 3, 0, -1, -9, -1, 10], [24, -6, -3, 19, 4, -10, -4], [-1, 20, -8, 8, -10, 1, -11], [1, -13, 8, -4, -2, 0, 11], [-4, -24, 5, -12, 16, 1, 3], [-10, 8, -3, 9, -4, 10, 9], [-21, 12, 4, 7, 4, 2, -3], [0, -16, 0, 0, -2, -8, 2], [-27, -8, 6, -5, 2, -8, 1], [8, 9, -20, -4, 5, -14, -13], [0, 3, 4, 3, 5, 13, -1], [-10, -4, 3, 1, -2, -4, -7], [-4, -9, -2, 0, 1, -4, -5], [-14, -7, 4, -9, 6, 7, 2], [26, -9, 1, -1, 1, -8, 10], [-3, -17, 1, 0, 5, -1, 3], [10, -9, 0, 6, -3, 10, 6], [-12, -5, 10, -9, -10, 17, -2], [2, 8, 3, -3, 0, 0, -2], [-3, 1, -9, 21, 1, -4, -3], [-11, -3, 17, -8, -2, 13, 6], [28, 16, -11, 20, -7, -5, -12], [-9, 3, -18, 6, 2, -6, 1], [20, 0, -4, 4, -12, 8, -1], [4, 11, -5, 7, 1, -17, -2], [3, -27, 13, -16, 3, -1, 15], [9, 14, 5, 2, 10, -7, 6], [-3, 10, -4, 0, 0, -1, -13], [20, 18, -1, 2, -5, -1, -2], [20, -5, 1, 4, -14, 5, -2], [-12, 13, 16, 13, -4, 7, 0], [5, -21, -6, -2, -3, -2, 2], [-7, -20, 6, -10, 15, 1, 10], [15, 17, -6, 3, -9, -9, -2], [9, 0, -5, 10, -4, 0, -2], [-4, -6, -2, -12, 4, -2, 10], [36, 7, 0, -6, 6, -1, 0], [1, 6, 10, -6, 10, 7, 3], [1, 0, 3, 9, -3, -11, -3], [-13, -6, 12, -4, -5, 3, 4], [-12, 17, -7, 2, -10, -1, 4], [-28, -1, -2, -4, -1, 8, -3], [18, -2, 5, -5, 2, -8, 1], [-17, -6, 17, -3, -5, 3, 6], [-22, 7, -5, -7, 1, -9, 2], [-25, -18, 11, -3, 5, 22, 5], [-18, -9, 2, -13, 8, 1, 0], [13, -19, 5, -11, 9, 0, -9], [4, -12, 3, -17, 8, 6, 15], [1, -15, 2, -4, 10, 8, -9], [22, -2, -17, 3, 0, -10, -14], [20, -7, -16, -1, 4, 3, -4], [-17, -7, 4, -10, -2, -6, -7], [-12, -1, -21, 12, -1, -8, -2], [10, 6, 10, 4, 4, 20, 2], [-18, -8, 7, 9, -16, 2, 0], [-11, 9, 6, 0, -9, 10, -12], [0, 30, -7, 9, -6, -4, -3], [9, 0, 7, -9, -3, -5, -9], [7, -3, 9, -5, -7, -6, 15], [15, 15, 1, 21, -7, -6, 1], [0, 8, -10, 18, -14, -10, -2], [-22, -2, 6, -21, 11, 1, 9], [-6, -6, -2, -12, -6, -10, -4], [5, 12, -10, 14, -5, 7, -9], [26, -9, -4, 10, 10, 0, 6], [-35, 7, 5, 1, -5, -4, 7], [11, 3, -4, 4, -14, -4, -7], [17, 2, 1, 8, 7, 17, 5], [3, 22, -11, 10, -8, -5, -10], [3, -12, 5, -8, 15, 7, -3], [-15, 22, 4, 11, 0, -4, 4], [-21, 18, 8, 0, -9, 7, -12], [-25, -9, 0, -14, -8, -2, -5], [-7, 10, -13, 13, -5, -10, -1], [-12, -2, 15, 1, 4, -6, -2], [-41, -4, -5, 5, 3, 2, 5], [-35, -7, 7, -19, 7, -10, 7], [12, -30, 3, -15, 0, 6, 6], [3, -15, 2, 2, -6, 0, -7], [-7, 1, 15, -1, 15, 12, 13], [19, -30, 5, -21, -5, 6, 3], [-1, 4, -2, -14, 5, 9, 1], [-13, -9, 8, -16, 2, 12, 3], [-1, -24, 5, -14, 5, 1, 7], [-5, 4, -3, 1, 17, -13, -2], [-8, 5, 13, -9, -15, 11, 2], [25, 12, -11, 10, -6, -17, -14], [4, -13, 1, -21, -3, 2, 1], [-2, 6, -3, -5, -6, -2, 7], [-26, 27, 7, 6, -5, 13, -8], [-12, 19, 8, 10, -12, -3, 10], [-6, 13, -4, 5, 3, -22, -13], [-28, -7, 15, 1, -1, 14, 12], [-21, -1, 13, 10, -3, -3, -3], [-15, 5, -7, 2, 2, -5, -24], [-8, 8, 6, -8, 4, 6, 4], [-7, 11, -21, 13, 0, 0, -17], [-13, 6, -12, 6, -7, 6, -11], [35, -2, 1, 0, -4, 0, -9], [0, 12, -2, 13, -7, 11, 9], [-10, -1, 14, -16, 5, -8, -3], [-28, 27, 4, 14, -8, -6, -2], [32, -25, 2, -14, 1, -4, 9], [-16, 21, 7, -4, 2, 1, -4], [27, -8, -15, 13, 1, -9, -5], [-18, 7, -17, 10, -16, -1, -8], [-42, 12, 3, -15, 5, 6, -6], [23, 0, 2, 13, -2, 2, -5], [-11, 0, -12, 2, -14, 2, -7], [-24, -3, 7, -21, 0, -1, 7], [-13, 10, -5, 15, -3, 8, 1], [-40, 9, 4, 8, -16, -1, -2], [-24, -4, -6, 2, -8, -10, -6], [-34, 16, 6, 6, -4, 4, -18], [8, 26, -2, -1, 5, -11, -3], [-2, -14, -11, 1, 6, 4, 9], [8, -9, -2, -8, 5, 2, 1], [17, 1, 16, 9, -11, 15, -3], [18, -4, 12, 0, -2, -4, 10], [14, -14, 2, -21, 13, 3, 7], [-29, -6, 7, -18, 8, 15, 2], [-29, 32, -13, 27, -1, -8, -7], [-17, -1, 16, -1, -3, 5, 6], [25, 3, 4, 10, -1, 6, -2], [37, -15, -6, 4, -4, 2, -1], [-41, 9, 9, -18, 7, -1, -1], [37, -13, -2, -2, 10, -10, -5], [23, 3, -15, 19, 6, -2, -7], [-3, 0, 23, -19, 13, 6, 11], [-3, 11, -17, 20, -9, -11, -17], [-26, -10, 2, -11, -5, 9, -3], [-27, 6, -9, 16, 2, -5, -8], [-3, -8, 11, 11, -7, 12, 9], [-16, -12, 13, -7, 11, 14, 14], [59, -18, -7, -8, -9, 5, 9], [23, -8, 15, -1, 7, 10, 15], [-26, 3, 6, 2, 10, 4, 8], [5, -4, 7, 2, -12, 6, 6], [-4, -20, -2, -16, 0, 0, -8], [9, 14, 1, 8, -10, -3, 8], [12, -12, 5, -7, -10, 12, 13], [9, -10, 3, 3, 9, 6, 3], [-4, -18, 1, -21, 11, 4, 12], [0, -26, -6, -10, 14, -6, 8], [60, 3, -13, 8, 4, -1, -8], [14, 21, -10, 6, -21, -6, -3], [30, -1, -17, 1, 9, -15, -10], [-9, 17, -18, 4, -9, -10, -8], [-4, -10, 10, -12, -8, 8, 4], [-19, -8, -21, -7, 7, -6, -9], [-26, 10, -9, -17, -1, 2, -8], [4, 7, 0, 2, -4, 13, 2], [-12, 14, 22, 0, 0, 10, 10], [-5, -26, 17, -27, 5, 10, 15], [-30, 21, -1, 9, -7, 9, -6], [-10, -15, 20, -21, 12, 5, 12], [-12, -9, 7, -3, 1, 8, 9], [31, -9, -14, 5, 3, -5, -17], [-16, -18, -2, -19, 25, -5, 1], [10, -2, 3, -2, -13, 9, 9], [-35, 9, 12, -4, 16, 4, 7], [-29, 19, 0, -9, -9, 3, -1], [32, -1, 2, -6, -1, -4, -11], [43, 3, -17, 22, -9, -5, -17], [0, -1, -13, -1, 3, -29, 2], [-14, 10, 9, -26, 5, -3, 15], [15, 22, -3, 11, -9, 1, 5], [2, 17, -15, 21, 3, -29, -6], [-7, 15, 1, -3, -2, 0, 1], [-14, 2, -11, -9, 8, 0, 1], [-17, -4, 27, 0, -10, 17, 8], [-25, -8, 11, -9, 3, 16, 7], [-9, -32, 17, -26, 8, 12, 14], [-6, 20, 7, -8, 19, -3, 13], [-17, 18, -1, 4, -6, -1, -16], [18, -11, -6, -21, -1, 5, 1], [11, 0, -8, -6, 13, 3, -10], [-3, -16, -15, 8, 12, -9, 5], [-7, 0, 2, -7, 2, -4, -17], [17, 19, 5, 8, -11, 7, -17], [3, -24, 1, -15, 21, -2, 13], [6, -9, -8, 26, 12, -2, 0], [-63, 6, 2, -9, -4, 14, -5], [7, -1, 11, 15, 1, -2, -1], [25, -14, 9, -5, 3, 13, 10], [-20, -5, 9, -3, 5, 16, 8], [-3, -19, 20, -17, 1, 23, 21], [17, 23, -17, 22, -9, -11, -17], [-52, 9, 17, -7, 10, 15, 9], [-39, 12, -10, -6, 5, -5, -9], [-7, 0, 2, -2, -10, -5, 3], [25, -16, -1, 7, -3, -3, 15], [12, 8, -10, 2, -4, -2, -14], [-36, -16, 10, -30, 14, 6, 12], [-21, 8, -17, 11, -19, -2, -9], [3, -4, -13, 7, -13, -6, -13], [3, -2, 3, -9, 15, 11, -5], [-31, 11, -9, -2, -15, -3, -6], [-18, 19, -13, 1, -7, 0, 8], [-29, 8, 3, 1, -18, -2, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9200_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9200_2_a_dc();".
// 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_9200_2_a_dc(:prec:=7)
    chi := MakeCharacter_9200_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_9200_2_a_dc();".
// 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_9200_2_a_dc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 17, -35, 1, 24, -7, -3, 1]>,<7,R![92, 12, -116, 10, 41, -8, -4, 1]>,<11,R![-128, -1780, -1224, 248, 198, -27, -7, 1]>],Snew);
    return Vf;
end function;