
// Make newform 4114.2.a.bg in Magma, downloaded from the LMFDB on 29 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4114_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_4114_2_a_bg();".
// 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_4114_2_a_bg();".
// 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 := [59, 67, -92, -80, 51, 28, -12, -3, 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], [505, 470, -500, -378, 87, 90, 1, -6], [694, -174, -934, 308, 374, -134, -48, 15], [-354, -998, 82, 980, 72, -255, -18, 17], [-600, -544, 1177, 77, -589, 103, 79, -19], [-1215, -46, 2049, -42, -831, 88, 94, -18], [586, 1377, -491, -1358, 15, 345, 19, -23]];
    Rf_basisdens := [1, 1, 91, 91, 91, 91, 91, 91];
    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_4114_a();"
function MakeCharacter_4114_a()
    N := 4114;
    order := 1;
    char_gens := [3027, 2179];
    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_4114_a_Hecke(Kf)
    return MakeCharacter_4114_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 := [[-1, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0], [-1, 0, 1, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 1, -1, 0, -1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2, 1, -1, 0, 1, -1], [-3, 1, -2, -1, -1, 0, 0, -1], [-4, 1, 0, -1, 2, 0, -1, 0], [0, 1, 2, 0, -1, -2, 0, -3], [2, 1, 0, 1, 0, -1, 0, -2], [-2, -1, 0, -1, 0, 1, 1, 2], [-2, -3, -4, -1, 0, 0, 1, 0], [2, 0, 0, -1, -1, 1, -1, -1], [4, 0, 1, 2, 0, 2, 1, 2], [0, 0, -2, 0, -2, -2, 1, 0], [2, -1, 2, 0, 1, 2, 1, 3], [2, -1, 0, -1, 0, 1, 2, 2], [-2, 1, 0, 0, -1, -1, 1, -3], [0, 1, -4, 3, -2, 0, 2, 0], [-3, 1, 1, 0, 2, -1, 0, 1], [-4, -3, 0, 0, 1, -1, 0, 1], [1, -1, -3, 2, 1, 1, 0, 1], [-10, -2, -4, -1, 1, 1, 0, 1], [-4, 2, -1, -2, 2, 1, -1, 2], [-2, 0, -2, 5, -1, 3, 0, 1], [1, -1, 3, 2, -4, -2, 1, 1], [0, 4, 0, 1, 1, 1, -1, -3], [6, 0, 2, 3, -1, 0, 1, 3], [-2, 0, 4, 0, 0, -2, 1, 0], [-7, 0, -6, 0, -1, -1, -1, 1], [2, 0, 0, 3, -1, -1, -2, 3], [0, 1, -4, 1, 0, 2, 2, 2], [-6, 0, 3, -1, 1, 2, -2, -1], [-4, -3, -4, 0, -3, 0, -2, 3], [5, -4, -5, -1, 1, 1, 0, 1], [-1, -3, -5, 1, 1, -1, 2, 2], [3, -3, 0, 0, 2, 3, -2, 2], [4, -2, 6, 0, -2, -1, -3, -2], [-2, 1, -4, 0, -3, 0, 1, -1], [-8, -2, -8, 0, -2, 1, 0, -2], [-2, 0, 2, -1, -1, -1, 1, 1], [4, -2, -4, 1, 3, 1, -1, -1], [-8, 4, -2, -1, -1, 1, 0, -1], [-3, 2, 5, 1, 2, 2, -1, -1], [-4, 3, -4, -4, 5, 1, -2, 3], [2, 4, 8, -1, 1, -2, 2, -1], [-3, 1, 5, -4, 6, -2, -2, -3], [-2, -2, -3, 2, -2, 2, 2, -2], [-6, -1, -2, -4, 1, -1, -1, -1], [-6, 5, 6, 4, 1, -2, 3, 1], [-2, 4, -6, -3, 1, 1, 0, 1], [-4, 2, 8, -4, 0, 0, 1, 0], [14, -1, 9, 2, -1, 3, 2, 3], [-2, 0, 2, -5, 1, 0, -2, -3], [2, 4, 2, 0, 8, 3, -1, 0], [-2, 0, 2, -5, 3, -3, -1, -3], [2, 2, 8, 5, 3, -1, 0, -1], [1, 3, 3, 2, -1, -2, 4, -3], [-7, 5, -3, 0, 6, 0, 0, -3], [-1, -1, 7, -1, -2, 0, 3, 4], [-6, -2, -10, 1, 1, 0, 4, 5], [1, 4, 1, 3, -4, -3, -1, -1], [6, -1, -4, -2, 7, 3, -1, 1], [-8, -2, -4, -7, -1, 4, -1, -1], [4, -2, 4, 3, -5, -3, 1, -1], [2, 8, 8, 1, 1, 0, 2, 1], [3, -2, 0, 1, -2, 4, 2, 4], [-4, -4, 1, -1, -3, 4, -3, -1], [-18, 3, -15, 2, -1, -2, 2, 1], [-4, -1, 6, 4, -3, -1, -3, -1], [-15, 2, -7, -2, 3, -1, 0, -4], [-6, 3, -3, 3, -2, 1, -1, 0], [1, -4, 7, -2, 1, 1, -5, -2], [4, 0, 2, 1, -5, 6, -1, -1], [-4, -5, 3, 0, -5, -1, 4, 1], [6, 4, 14, -1, -1, -1, 2, 3], [-16, 2, 0, -2, 0, 3, -2, 0], [-3, -3, -13, -4, -1, -3, 1, 1], [-2, 1, 2, 0, 3, 0, 5, 5], [5, -1, 1, 2, -5, 5, 1, -1], [-3, 2, 2, 1, 2, -2, 0, 0], [7, -2, -9, -2, -10, -1, 1, -4], [-7, -5, -2, 0, -4, 2, 3, 0], [-6, 6, -2, 2, -4, -4, 1, -6], [-12, -6, -12, 0, 2, -1, -1, 2], [16, 0, 9, -2, 2, 0, -2, -8], [-4, 2, -7, -3, 5, 0, -1, 3], [-2, -4, 4, 0, -2, 1, -8, 0], [0, 2, 10, -1, 1, -3, 2, -1], [-3, -4, -1, -2, -3, 0, 2, 6], [-27, 3, -13, -2, 0, -3, -1, -3], [6, -4, 2, 5, 1, 6, 5, 5], [-7, -1, -8, 4, -2, 5, 3, 6], [1, -6, -12, 1, -6, 3, -4, -4], [-8, -8, -1, -5, 5, 3, -1, -1], [-10, 4, 2, 1, 1, 0, -1, 1], [4, 3, 14, 4, -3, -2, 1, -11], [4, 0, 4, 1, 1, 0, -4, -7], [11, -8, 0, -4, -3, 3, -1, -1], [-1, 6, 13, 0, 6, 0, 0, -4], [10, 8, 6, -3, 3, 0, 0, 3], [-19, 3, -3, -2, -5, -1, -2, 5], [5, 0, 8, 4, -9, -1, -3, -3], [2, -3, -1, -2, -3, 5, 2, 9], [-6, -4, -10, 0, 0, -3, 3, 8], [-11, 1, 13, 1, 8, -2, 2, 2], [-12, -5, -6, 2, -7, 5, 2, 3], [6, -3, -4, 6, -11, -1, 2, -1], [23, -1, 8, -1, 3, 2, 0, -1], [-7, -1, -1, -7, 4, -4, 0, -2], [8, 2, 6, 2, 0, 5, -4, 0], [-12, 4, -2, 8, -4, -6, -1, 0], [-8, 0, -4, -4, -6, -5, -5, -8], [4, 2, 14, -2, -4, -5, 4, -8], [-1, 10, 2, -2, -1, -2, -2, -3], [-6, 3, 4, -3, 6, 2, 0, 6], [0, -3, 8, -3, 4, -3, -2, 2], [-12, -8, -14, -4, -6, -1, -1, 2], [7, -11, 1, 5, -6, 7, 1, 4], [2, -12, -14, -5, -1, 4, -4, 3], [14, -9, 0, -7, 6, 5, -4, -4], [1, 2, -13, 5, -4, -1, 7, 3], [-11, 10, -4, 4, -3, 2, 7, 3], [-15, 0, -19, 0, -4, 2, 3, -6], [-4, 4, 0, 5, -3, 1, 1, -3], [-7, -9, -10, -6, 4, 6, 0, 0], [-8, 13, 4, -5, -4, -4, -2, -4], [6, -2, -6, -2, -4, -1, -2, 0], [2, 0, -14, 1, 9, 6, -1, 1], [-11, 6, 2, 0, 7, -1, -1, 1], [-1, 6, -8, 6, 1, 7, 2, -1], [24, -6, -4, 3, -7, -1, 3, -3], [-6, -5, -5, 7, -4, 1, -2, 0], [2, -7, 1, -4, 3, 1, -10, -5], [8, -6, 9, 8, -4, -2, -1, 0], [-7, -3, 0, 3, -7, 3, -10, 1], [-5, -8, -5, -5, 1, -2, 0, -3], [2, 0, 2, 8, -4, 3, -2, -4], [-3, 7, -6, 4, 2, 4, -1, -2], [-28, -2, 0, 1, -1, -1, -2, -1], [-3, -2, 15, -2, 3, 2, 0, -4], [-14, 3, 6, 0, -7, -2, -5, -5], [8, 5, -1, 7, 4, 3, 2, 0], [-19, -6, -2, -5, -2, 2, -1, 4], [14, -3, 4, 0, 3, -6, 5, -3], [10, 1, 16, 3, 2, -4, -3, -4], [-8, -1, 14, -2, 3, -1, 1, -3], [24, 3, 18, 1, 4, 1, 4, -4], [-10, 0, -6, -1, -5, -1, 3, 7], [8, 10, -4, -4, 6, -1, -3, -2], [-2, -11, -12, -1, 4, 1, -3, 6], [-2, 5, -8, 4, -1, -1, 7, 1], [-8, 3, 2, -8, 9, 0, -8, -5], [21, 1, 4, 3, 1, 4, 7, 1], [12, -1, 14, -3, 6, 5, 1, 6], [-10, 3, 0, 0, 1, -13, 5, -1], [-3, -5, -19, -5, 4, 0, 0, -6], [26, 0, 10, 3, -1, 0, -2, -5], [-19, 10, -12, -4, -1, 0, -2, 1], [0, -1, -13, 6, -5, 4, -5, 3], [0, 8, 4, 7, 3, 5, -4, 3], [14, 6, -2, -10, 12, -2, -5, 2], [15, 5, -3, 1, -4, -3, -2, -2], [15, 3, 26, 1, 1, 2, 4, 9], [3, -1, 0, 0, 12, 4, 0, 8], [0, 1, -2, 2, 1, 0, 1, 11], [2, -2, -2, -6, 8, -8, 1, -4], [-22, 5, -22, -6, -3, 8, 2, -1], [8, -11, 4, -6, 1, 2, -1, 3], [16, 3, 9, -2, -7, 3, 2, -1], [-15, 10, -12, -10, 9, -5, -5, -1], [16, -7, 4, -2, -7, -2, 4, 3], [-2, 1, -6, 8, -7, 9, 3, 7], [-14, -6, -12, 7, -3, 1, 4, 5], [3, 1, -3, 8, 0, 3, 4, -3], [2, -3, -14, 7, -10, 8, -2, 8], [-12, 2, -12, -1, 1, -4, -2, -1], [12, 9, -4, -2, 1, -2, 4, 3], [-8, 1, 10, 3, -6, 1, -11, -6], [-3, -2, -20, -6, 5, -3, -2, -1], [20, -1, -8, -6, -1, -4, 7, 5], [-8, 6, -7, -4, -2, 2, 2, 2], [-22, -7, -6, -1, 2, 0, -2, 0], [3, 0, 1, -6, 6, -13, 1, -4], [-8, -5, -10, 5, 2, 0, -9, -4], [22, 3, 12, 8, -1, -5, 3, 7], [20, 0, 20, -4, 0, 3, -5, -8], [-13, 3, -10, -1, -1, -3, -8, -5], [-6, 2, -14, 3, -7, 5, 6, 9], [9, 2, 17, -4, 3, -1, 1, 2], [0, -14, -12, 0, -12, 7, -5, 6], [-14, 10, -9, 6, 6, 5, -6, -10], [9, -5, 17, 2, -6, -4, -3, 1], [20, 10, 16, -2, -4, -1, 1, 0], [-4, 6, 18, 4, -10, -2, -4, -2], [-4, 10, -6, -1, -3, 0, -4, 5], [2, 10, 10, -1, 1, -4, 8, -3], [-24, -2, -30, -5, -3, -4, -6, -7], [-10, -10, -4, -3, 3, -5, 5, -1], [1, 6, 4, -14, 11, -4, -2, 5], [0, 4, -5, -5, 9, -9, -3, 3], [-16, -10, -10, 0, -6, -3, -7, 2], [6, -6, 14, -2, 0, 3, -9, 0], [7, 4, -9, 0, -4, 3, -8, -6], [-20, -2, -16, 2, 2, 1, -1, 0], [-8, -5, -6, 5, -10, 2, 3, -4], [-12, -4, -22, 5, -11, -3, 2, 1], [-8, 2, -10, 10, -8, 4, 2, -2], [2, 8, -6, -6, 10, 2, 4, -2], [-2, 8, 11, 3, 9, 4, -6, 9], [-21, -6, -7, -2, 11, 1, -1, 0], [4, -2, 18, -5, 1, -1, 8, -1], [-7, -13, 5, 0, 8, 5, 0, 7], [0, -10, -2, 1, 11, -1, 0, 3], [-8, -2, -4, 2, 4, 8, 2, 0], [12, 7, 18, 5, 10, -2, 6, -4], [5, 9, 19, 4, 4, 7, -3, 1], [9, -2, -3, 4, -6, -1, 2, -4], [8, 3, 15, -2, -7, -3, -2, 1], [0, 0, 8, 10, -6, -5, -7, 2], [14, 1, 9, 9, -12, 2, 6, 6], [-2, 9, 8, 7, -2, 4, -3, 4], [18, 2, -12, 3, -13, 2, 4, -7], [33, -7, 7, 0, -5, -7, 4, -9], [28, 5, 12, -11, 8, -2, -3, -2], [-1, -7, 9, -1, -11, 2, -4, 0], [-6, 6, -8, 8, -6, 3, -1, -8], [-37, 7, -9, -12, -2, -2, -5, -5], [-7, 13, -1, -3, 6, 4, 2, -4], [31, -2, -1, 3, 2, -1, -1, -11], [-6, 1, 0, 1, 8, 1, 0, -6], [3, -6, 5, -9, 3, -4, 0, 13], [-6, 1, -2, -3, 4, -2, -14, -14], [18, 5, 24, 7, -8, 2, 1, 8], [-6, 2, 0, -14, 16, 0, -5, -4], [4, 3, -10, 2, -13, 6, 2, 9], [9, -2, 11, 0, 7, 0, -7, 2], [0, -10, 2, 7, 1, -1, -4, -5], [7, -4, 19, -10, 3, 2, 3, 2], [-2, 9, 6, 7, 2, 0, 4, 8], [4, -7, 16, -3, 10, -8, -10, -10], [-3, -2, -9, -2, 4, -3, 5, -10], [22, -6, 2, 1, 11, 8, 2, -5], [-3, -4, 3, -4, 9, 7, 0, 4], [-17, 7, 7, -5, -1, -3, 1, -2], [-5, 10, 5, -1, 10, 8, -5, 1], [-10, -9, -24, -6, 3, 1, -6, 3], [8, 6, 4, -4, 16, 2, -7, 0], [5, -7, -3, -5, -8, -3, -2, 2], [-16, 9, -8, 10, -3, 3, -1, 7], [8, 3, 8, 1, 12, -6, -12, -8], [-14, 3, 2, 0, -3, -4, -9, 7], [14, -6, 15, 7, -1, 7, 10, -1], [-24, -10, -4, 2, 2, 5, 5, 4], [-22, 4, -18, 3, 3, -7, -1, -1], [-20, -6, -2, 1, 11, -7, -5, 3], [38, -10, -6, 9, -1, 4, 3, 15], [-24, 10, 5, -7, 11, 3, 0, 5], [9, -16, -6, -2, -1, 1, -3, 1], [-4, -9, 10, -1, -8, -1, 5, 4], [6, -7, 8, 1, 12, 6, -4, 14], [4, 8, -4, 8, -8, -5, -4, -8], [-16, -1, 2, -12, 9, -1, -1, 11], [-4, 5, 0, 7, -2, 6, -1, -2], [-1, -1, 13, -7, 12, 1, -9, -14], [-16, 0, 7, 1, -3, -4, 4, 1], [25, 6, 22, -7, 6, -1, -9, -8], [-5, 12, 9, 0, -6, 0, -2, -10], [12, -11, -8, -4, 7, -1, -4, 1], [10, -9, -8, 3, 6, 3, -7, 16], [-8, 7, -7, 1, -6, 4, 1, 4], [24, 6, -6, -2, -4, -5, -6, 0], [2, -6, 17, -1, 1, 1, -7, -11], [18, -16, 10, 3, -5, -7, -1, -7], [22, 7, 36, -4, 9, -9, 2, -11], [-10, -9, 0, -1, 4, 5, -11, 2], [5, -11, -1, 0, 5, -5, -7, -3], [-22, -5, -32, 6, -5, 3, 0, 15], [-20, -1, -22, 8, -3, -5, 1, -1], [-7, -3, -27, -3, 1, -3, 0, -2], [-19, -4, -8, 14, 1, 12, -4, -1], [-12, -5, -6, -8, -3, -3, -6, -13], [-3, -8, 3, 6, 3, -6, -8, 2], [0, 12, 0, -8, 6, -11, -3, -4], [12, 7, 2, -4, -11, -4, 7, 3], [-4, 3, -8, 5, 6, -11, -6, -4], [6, 2, 18, 12, 0, -2, 8, -2], [-3, 2, -1, 6, 2, 16, 3, 16], [-18, 2, -10, -1, 13, -5, -5, 13], [-29, 12, -13, -6, 3, 2, -8, -6], [-31, -5, -11, -6, 1, 11, -4, -7], [11, -6, 33, -1, 4, -1, -2, 1], [18, 2, 12, 7, -19, 5, -4, -3], [8, -1, 6, 7, 0, 10, 5, 10], [-16, 12, 0, -12, 22, -3, -3, 4], [18, 1, 0, 13, -8, 7, 9, -6], [-9, 6, 9, 13, -8, 4, 2, 7], [18, 19, 14, 7, -12, 0, 0, -10], [5, -10, -3, 6, 2, -2, 2, 8], [10, -8, 7, 4, 2, 2, -4, 2], [-3, -11, -12, 6, -18, -3, 6, -2], [1, -6, 10, -7, 6, -5, -3, -12], [5, 16, 18, 10, -3, 8, 3, 7], [10, -10, -2, 5, -1, 4, 7, 11], [-26, 1, -14, -2, -11, 0, 0, 1], [2, 3, -24, 2, -1, -7, 10, 1], [-22, -3, 0, -16, 5, 7, -5, 1], [23, 5, 7, 1, 2, -8, -1, -8], [-7, 7, 5, -3, -7, -2, 8, -8], [10, 0, -2, -5, -9, -1, -11, -5], [-8, 7, 10, -3, -2, -6, 4, -8], [30, -6, 20, 7, -3, -2, -1, -1], [10, -5, 12, -11, -2, 6, 8, 2], [-2, -23, -14, -9, -2, 0, -3, 8], [-8, 9, 2, 4, -11, 2, 8, -3], [-15, 8, 10, -12, -3, -1, 5, -1], [22, -1, -20, 2, -5, -9, 12, -3], [-25, -3, -1, 0, 3, 7, 0, -7], [-12, 11, -14, -7, 6, -5, -2, -4], [-12, 15, 4, 13, -2, 10, 0, 8], [16, -6, -5, 2, -10, -1, -1, -12], [-16, -1, -18, -9, 6, 8, -2, -8], [-18, 10, -22, 1, -5, -13, 7, -9], [1, 12, 19, -2, -10, 1, 2, -22], [-9, -3, 13, -7, -7, -2, -15, -6], [-17, -2, -9, -2, 11, 1, -1, 4], [-11, -3, -14, -1, 1, 15, -5, -3], [16, 5, 2, 9, 8, -10, -4, -6], [-18, -1, -12, 6, 3, 0, 8, 17], [-2, 7, 4, 6, -1, -11, 2, -3], [-28, -9, -22, -2, -5, -4, 0, 9], [12, 0, 2, 13, 3, 3, 7, 1], [-9, -15, -22, -4, 6, 6, -1, 14], [15, 11, -7, 10, 0, 7, 6, 5], [13, 6, -9, 5, -19, 2, -1, -3], [2, -2, 20, 9, -5, -4, -1, -13], [6, 4, -12, 7, 3, 13, -1, 3], [14, 7, 20, 5, 14, -1, 14, 0], [0, 3, -29, 5, 8, 8, -2, -4], [-5, 0, -9, -7, -7, -4, -5, -7], [-4, -2, 2, -3, 3, -4, 5, 11], [-22, 3, 6, -10, 1, -2, 3, -15], [16, 7, 1, 9, -20, -6, 2, -10], [-5, -14, -16, -1, 6, 7, 5, 20], [-2, 0, 20, -1, -1, -7, 8, -13], [24, 21, 30, 8, 15, -1, 3, -11], [-17, -8, 3, 1, 0, 7, -11, 9], [-24, 1, -2, -10, 1, -6, -8, -9], [-18, 5, 20, 9, -2, 3, -1, 0], [-38, 5, -18, 12, -1, -8, 1, 9], [6, 6, 22, -7, 11, 2, 2, 11], [-13, -7, -7, -7, 9, 1, 8, 2], [19, 7, -15, 6, 3, -3, 2, -1], [0, -9, -17, 8, -11, 8, -7, 9], [-8, -9, -14, -4, 15, 3, -4, 9], [15, 4, 19, 4, -1, -7, 3, -8], [-2, 8, 8, -11, -7, -15, 3, -15], [6, -20, -4, 0, 2, 2, -14, -2], [-20, -5, -32, 2, 11, 9, 6, 5], [-8, 2, -30, -4, -2, -9, 3, -6], [14, -2, 2, 4, -2, 4, 7, 8], [23, 1, 34, 4, 0, -3, 8, 0], [12, -1, 32, 4, 11, -5, 6, 7], [-2, 1, -1, 1, 4, -1, -12, 14], [-20, 12, 6, 0, -16, -6, 8, 0], [14, -7, -10, 13, 6, -3, -3, -2], [-14, 10, -16, 7, -11, 9, 10, 5], [-28, -21, 2, -10, 7, -2, 5, 11], [5, -1, 29, 7, 1, 0, 7, -14], [-21, -3, -8, -5, 11, 12, 3, 7], [-11, -6, 15, 5, -5, -6, 9, 5], [8, 2, -4, -3, 21, -6, 1, -5], [10, 4, 4, -8, 20, 6, 3, -8], [27, -2, 16, -3, -6, -4, 10, 0], [0, 10, 18, -5, 13, -11, 13, -7], [36, 2, 17, 8, -6, 0, 2, 4], [-1, -4, 7, -12, 8, -4, -12, 8], [29, -14, 21, -8, 10, -8, -12, -10], [38, 18, 16, 4, -18, -10, -12, -14], [-22, 17, 14, 4, -1, 7, -9, 13], [25, -10, 15, 4, -19, -2, 4, 10], [-12, -9, 6, -11, 10, 3, 12, 0], [-1, -2, 1, 2, 9, 5, 2, -12], [-17, -2, -29, -2, 5, -1, -1, 12], [8, 4, -2, 8, -12, -2, 4, 6], [17, -9, -29, -5, -4, -10, 10, 0], [22, 3, 10, 0, -7, 4, 7, 7], [-10, 5, 2, -14, 1, -7, 0, 3], [2, -10, 22, 0, 10, 10, -7, 6], [-6, 0, -40, 11, 3, 6, 3, -1], [32, -3, 10, 1, -2, -11, -8, -10], [20, -8, 32, 9, -1, -1, -3, 7], [22, -2, 44, -10, 8, -3, -5, -6], [6, -8, -14, 15, -5, 1, 1, 3], [-24, -12, 2, -15, 11, -10, -3, -5], [13, -5, 9, 5, -6, 7, -11, 12], [2, 14, -6, 4, -10, -10, 4, -2], [-20, 5, 15, -16, 5, -9, -4, -17], [0, 31, -6, 8, -3, -5, 3, -13], [-31, 19, 3, -13, 18, -3, -8, -6], [-14, 3, -16, -8, 1, 5, -6, -7], [-20, -9, 0, 0, -7, -10, 1, -5], [-10, -3, 18, -10, 5, -9, -1, -13], [13, 5, 29, 11, 5, -2, 17, -4], [1, 1, 19, -8, 7, -13, -4, -21], [-44, -4, -3, -1, 19, 2, 2, 9], [-38, -18, -12, -1, 5, -9, -3, 9], [5, -1, -12, 1, -17, -11, 7, -13], [10, -6, -8, -21, -3, -9, 1, -7], [-39, 12, 5, 4, -16, -9, 12, -14], [8, 8, -14, -4, -10, -10, 5, 8], [-9, -16, -15, 3, -6, -5, -5, -5], [2, 9, 17, 15, -10, 12, 2, 4], [-12, -12, -10, -21, 3, -1, -9, -5], [-2, 9, -12, 3, 10, 6, -15, -8], [-41, -5, -8, 3, 3, -9, -4, -5], [26, -1, 8, 8, -7, 15, -6, 7], [16, 8, 8, 6, 6, -14, 12, -2], [1, 7, 5, 8, -9, -8, -4, 9], [-12, -1, -32, -4, 5, 4, -4, -5], [45, -8, -16, -3, -14, -2, 5, 4], [-12, -7, -2, -4, 25, -4, -6, 17], [-24, -8, -18, -11, 9, 2, -3, 9], [-2, -3, -40, 11, -14, 7, 3, 12], [21, 14, 31, 10, -15, 0, 1, -20], [-24, 1, 22, -19, 20, -4, -5, -14], [-4, -7, -6, -4, -15, 13, -12, -5], [-18, -3, -16, -1, -22, 8, 5, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4114_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4114_2_a_bg();".
// 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_4114_2_a_bg(:prec:=8)
    chi := MakeCharacter_4114_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_4114_2_a_bg();".
// 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_4114_2_a_bg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4114_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![19, -98, 39, 115, -24, -51, -5, 5, 1]>,<5,R![80, 560, -920, -360, 289, 51, -30, -2, 1]>,<7,R![121, 431, -18, -610, -409, -50, 32, 11, 1]>],Snew);
    return Vf;
end function;