
// Make newform 2014.2.a.g in Magma, downloaded from the LMFDB on 29 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2014_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_2014_2_a_g();".
// 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_2014_2_a_g();".
// 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 := [-10, -52, -61, 21, 50, -1, -13, 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], [6, 20, 1, -10, 0, 1, 0, 0], [-9, -20, 0, 10, 0, -1, 0, 0], [-4, 4, 21, 1, -10, 0, 1, 0], [-7, -29, -16, 30, 11, -11, -1, 1], [9, 43, 18, -39, -11, 12, 1, -1], [18, 46, 10, -39, -10, 12, 1, -1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 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_2014_a();"
function MakeCharacter_2014_a()
    N := 2014;
    order := 1;
    char_gens := [743, 267];
    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_2014_a_Hecke(Kf)
    return MakeCharacter_2014_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], [0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, -1], [-1, 0, 0, 0, 0, 1, 0, 1], [-1, 1, 0, -1, 0, -1, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 0], [0, 2, -1, -1, 2, 1, 1, 0], [-2, 1, -1, 0, 1, 1, -1, 0], [-1, 1, -2, -2, 1, 2, 2, 1], [0, -2, 1, 0, 1, 1, -1, 0], [0, 1, -1, 2, 0, 1, 1, 1], [1, 0, 1, 1, -1, 1, 0, 0], [-2, 1, 1, 0, 1, -1, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [-2, 1, 1, 0, 1, 1, 2, 0], [0, 0, 1, 2, -2, 0, 0, 0], [1, 3, 1, 2, 1, -1, 0, 0], [-5, -4, 3, 1, -2, -1, -1, 0], [3, 2, -3, -1, 0, 3, -1, -1], [0, 0, 3, 0, 0, -3, 0, 1], [0, 4, -1, -1, 2, -1, 2, 2], [-8, -3, 3, -1, -1, -3, -1, 2], [-2, -2, -1, 0, 2, -2, -3, 0], [-1, -2, 1, -1, 0, 1, 2, 0], [3, -3, 0, 4, -2, -1, 0, -2], [-3, -2, -3, -1, -1, -3, -1, 1], [-1, 1, 0, 1, -3, 0, 1, -2], [-3, -1, -1, -2, 2, 1, -3, 0], [0, 0, 3, 6, -3, -1, 0, 0], [-4, 3, -2, -3, 2, 3, 0, 3], [-2, 1, -3, 1, 2, 4, 5, 1], [0, 0, -3, -3, 3, 3, 0, 2], [-7, -3, 0, 0, -2, -2, -1, 0], [1, -1, 0, -2, 2, 1, -2, -1], [2, -3, 0, -1, 2, 4, 2, 0], [4, 2, -1, 2, 1, -2, 1, -2], [-4, -5, 3, 3, -4, -4, 0, 1], [-2, 1, 4, 0, 0, 1, -2, -2], [-2, -1, 2, 2, -5, -1, 3, -2], [-4, 0, -3, -2, 2, 0, 2, -2], [-4, -2, -5, -1, 3, -1, 1, 2], [-2, -3, 6, 1, -4, -5, -2, -1], [1, 0, 4, 1, -2, 1, 1, 0], [1, 1, -2, -1, 3, -3, -1, -3], [-3, 3, 1, 7, -2, -1, 6, 2], [-1, -5, 5, -4, 1, 1, -4, 1], [2, 2, -3, 4, 2, 5, 5, 1], [-2, 0, -3, 1, -6, -4, 5, -4], [0, 1, -3, -4, -1, 3, -3, 0], [-7, 4, -3, 0, 1, 2, -2, 1], [-11, -4, 1, 1, -4, -3, 0, 0], [-2, 4, -2, 0, -2, -2, 2, 0], [-3, 0, 1, -3, 2, 0, -4, -1], [0, 4, 0, -1, -3, -2, -2, 1], [2, 0, 3, 2, -1, -4, -6, -2], [5, -4, -2, 1, 3, 1, -3, -4], [-2, 3, 2, 1, 2, -7, 1, -1], [-3, -2, 2, 1, 1, 0, 3, 4], [9, 4, -2, 4, -4, 3, 4, -1], [-3, 2, -2, 1, 2, 5, 7, 4], [4, -5, 3, 2, -5, -7, -4, -4], [4, 3, -7, -5, 4, 6, -1, 5], [4, 0, -2, 0, 3, 3, 3, 1], [-13, 1, -2, -1, 2, 2, 4, 4], [6, -3, -2, -5, 2, 3, 0, 0], [-2, 2, -5, -2, -2, 4, 8, -1], [3, 1, -4, -4, -2, 2, -2, 0], [4, 5, -4, -5, 1, 2, -6, 1], [-4, 1, 6, 0, -1, -9, 1, 0], [5, 2, -2, -4, -5, 0, -4, -3], [4, -4, -1, 5, -1, 2, 3, 2], [-4, -3, -5, -4, 3, 2, -5, -1], [1, -3, 3, -4, -4, 1, 0, 1], [8, 1, -1, -3, 1, -2, -8, -6], [-6, -10, 2, -2, -2, -2, -5, 1], [13, 3, 2, 2, 1, -4, 0, 2], [4, 2, 1, 2, 6, 9, 1, -1], [4, 1, 4, 9, -1, 4, 2, -1], [4, -8, 4, 4, -4, 4, -4, -3], [10, -2, -1, 1, -1, -6, -4, -4], [-6, 4, 0, -4, 2, 2, 2, -3], [4, 8, -13, 0, 5, 8, 6, -1], [4, 6, 0, -2, 1, 2, 0, 2], [3, -5, -1, -6, 5, -3, -6, 3], [-8, -4, 7, 0, -2, 1, -1, -2], [3, -11, -1, -1, 1, -1, -7, 1], [10, -6, 3, -3, 4, 0, -5, -3], [14, 7, -1, 0, 0, -2, 3, 0], [-12, -2, 2, -6, -2, -3, 1, 4], [-2, -1, 5, 3, -1, 5, 4, -2], [2, -4, 0, 11, 1, 3, -2, -1], [-4, -4, 9, -5, -3, -2, -3, 0], [4, -7, -2, -3, 3, -6, -1, 3], [14, 2, -1, 2, -6, -1, 3, -2], [4, 13, -9, -1, 4, 5, 2, 2], [4, -5, -13, -5, 3, 6, 6, 0], [6, -2, -1, 3, -5, 7, -3, 1], [3, -1, -1, -1, -7, -1, 5, -2], [8, -1, 0, 1, 0, 9, 2, 5], [-7, -3, -3, -6, -4, 0, 4, 0], [-7, -7, 3, -5, 0, -6, 0, 3], [-6, 7, -6, -4, 4, 1, -1, 5], [-4, -1, -9, -13, 3, -1, 4, 4], [5, -5, 10, 3, -4, -2, -5, -5], [21, 1, 2, 4, 1, -4, -4, -4], [-3, 6, -6, 5, 0, 9, 3, -2], [-10, 0, 0, 3, 2, -2, 3, 1], [-11, -9, 11, 4, -3, -3, -4, 5], [6, -5, 1, 6, -1, 6, -6, -2], [-2, -1, -1, -7, -6, -10, 4, 1], [0, 3, -8, -2, 1, 4, -4, -5], [6, 8, -5, 0, 2, 3, 6, 1], [5, -9, -5, -4, 0, 5, -8, 2], [-6, 1, -5, -8, 1, -2, -6, -5], [13, 0, -2, 3, -4, 4, -3, -5], [-2, -4, 1, 2, -10, -2, 4, 2], [9, 2, 1, 0, -8, -3, 6, -2], [-19, -1, 3, -2, -1, -1, -2, -4], [4, -1, 6, 3, 2, -11, -8, -9], [0, -7, 2, 7, -6, -10, -4, -2], [6, -12, 11, 4, -8, -9, -13, -6], [0, 13, -6, -2, 6, 3, 10, 3], [-2, 0, 5, 2, 5, 2, -8, 2], [2, -6, 8, 6, -5, -4, -4, -10], [1, -4, -8, 1, 2, 6, 7, -3], [-7, 4, -7, 3, 2, 6, 2, 4], [19, 8, -10, -3, 0, 7, 1, -1], [10, 7, -14, -7, 7, 9, 6, -1], [-2, -2, 4, 1, 2, -1, 0, 4], [0, -16, 4, 1, -10, 1, -2, -2], [-13, -5, 10, 1, -1, -7, -7, 2], [-2, 13, -7, -1, 9, -7, 4, -1], [-11, -8, 0, 8, 2, 8, 3, 0], [-16, 1, 2, -10, 2, -11, 5, 4], [-8, 8, -5, 3, 0, 8, 0, 0], [-7, -7, 8, -5, 3, -14, 2, 8], [-5, 4, 2, 0, -1, -4, -2, -7], [-15, 5, 4, 11, -6, -3, -5, -3], [-6, 5, 8, 4, -1, -8, -9, -9], [12, 12, -5, -4, 6, -3, 2, 5], [-18, 5, 2, -9, 6, -3, 2, 0], [-6, -2, 20, 5, -2, -6, -7, -1], [-4, -3, 4, 2, -3, 7, 4, -1], [10, -4, 12, 3, -1, -15, -4, 1], [19, 4, -2, -3, 10, 2, -4, 5], [-25, -6, 4, 1, 4, -3, -6, 0], [-8, -6, -3, 4, -5, -11, -4, 0], [1, -14, 5, -3, 2, -9, -9, 1], [-3, -3, 11, 3, -5, -13, -3, 2], [12, 9, -8, 1, -6, 2, 5, -5], [-4, 0, -1, 3, 2, 11, 6, 1], [-11, 11, 7, 14, -4, -4, 8, 2], [-4, -2, -6, -1, 3, -3, -5, 1], [-16, -6, 1, 6, -4, -12, -2, 0], [13, 0, -3, 2, 3, 11, -1, 0], [-4, 0, 12, 9, -7, -6, -4, -7], [-10, 3, 2, -2, -1, 4, -1, -4], [-10, 1, -4, 0, -4, 0, -2, 2], [3, 6, -2, 5, -3, 0, 12, 2], [-18, 0, 4, 4, -10, -3, 1, -1], [-5, -5, -6, -3, -2, 1, 7, 4], [-11, 1, -8, -3, -1, 5, -4, 6], [-12, -3, -8, 1, 0, 7, -8, -3], [1, -13, -1, -5, 2, 2, 0, 1], [-4, 0, -5, 3, -4, -8, -1, 0], [-19, 1, 7, 7, -14, -9, -1, -3], [-14, 4, -3, -8, 1, -2, 4, 0], [0, 0, 3, 1, 11, 5, 8, 0], [-13, 5, -3, 1, 8, 12, 16, 3], [-8, 1, 11, -3, -10, -11, -8, -4], [17, -4, 5, 3, 1, 3, -5, 2], [14, 6, -2, -7, -6, -9, 2, -3], [-14, -4, 15, 2, -8, -11, -8, -9], [-10, -9, 2, -7, 4, 5, 4, 3], [-1, 0, -10, -17, 4, 2, 3, -5], [9, 5, -14, -2, 12, 13, 8, 6], [21, 12, -3, -1, 1, -6, 3, 3], [-14, -5, 15, 6, 0, -7, -5, 5], [6, -10, 1, 7, -2, -6, 4, -6], [14, 8, -2, -4, 5, 2, -6, -10], [1, -16, 14, 4, -15, -6, -2, 2], [-13, -5, 9, -2, 3, -2, -4, -7], [-10, 2, -1, 2, -2, 9, -1, -6], [3, -9, 9, 16, -8, -10, -17, -8], [7, 1, -6, -4, 0, 7, 6, 0], [-12, 8, -2, -12, 3, 9, -3, 5], [-3, 8, -3, 7, -8, 6, 9, 4], [3, -6, -8, -5, 0, 20, -2, 3], [-20, 12, -2, -4, 4, 5, 5, 9], [13, 1, -5, 6, 10, 1, 4, 2], [1, -10, 2, 0, -5, 0, -1, 4], [6, 6, -7, -6, 2, 14, 8, 6], [-22, 12, -5, -2, -2, 10, 6, 8], [-5, -11, 7, 1, -6, 1, -11, 1], [-18, 9, -11, -3, 10, 5, 0, 4], [-8, 16, 2, 2, 6, 3, 7, 6], [3, -1, 5, 1, 1, -2, -2, 1], [16, -6, -2, -7, 10, 4, 3, 5], [-28, 10, -5, -3, 3, 12, 11, 11], [1, 7, 11, 6, -2, -14, 8, -2], [-3, -1, 2, 13, -9, -6, 5, -6], [-12, 9, 0, 4, -1, -11, 3, 2], [1, 0, 13, 3, -3, -18, -17, -1], [7, 7, -9, -12, 11, 9, 6, -5], [-19, 2, 7, -4, 8, 3, -2, 7], [2, -7, -6, -10, 2, 1, 6, 2], [-19, -18, 18, 0, -3, -14, -8, 7], [-6, 2, -6, -3, -8, 0, -5, -6], [-7, 3, -5, -5, -7, -7, -2, -7], [3, 0, 15, -1, -4, -7, -5, -2], [16, -15, 3, -5, 6, 4, -9, -7], [-6, 0, -4, -4, -13, -7, 13, -1], [-6, -4, 0, 10, -8, -9, -13, -8], [-17, -15, 6, 8, 0, -8, -10, -1], [-13, -2, 2, 3, 9, 6, -6, 2], [-2, -9, 10, 15, -15, -2, 0, -6], [5, -11, 3, 7, -1, 5, 3, -4], [30, 0, -7, -10, 10, 2, -3, -4], [-11, 14, -11, -2, 5, 9, 13, 14], [-11, -4, 10, -1, 7, -8, -11, 2], [-19, 16, -8, 2, 3, 12, 7, 9], [25, -4, -9, -1, 5, 9, 7, 9], [-13, -3, 5, -1, 5, 5, 10, 0], [28, 9, -8, 9, 0, -6, 2, -3], [2, 6, 5, -5, -3, -5, 6, 4], [-24, 0, 6, 2, 0, -10, -1, -4], [-2, -6, -10, -7, 3, -2, -1, -3], [18, -4, 4, -1, 1, -9, -6, 7], [14, -11, 11, 15, -11, -6, -17, 1], [7, 11, 5, 7, 8, 4, -11, -6], [7, 26, -9, 1, 3, 4, 20, 1], [-7, 7, -1, -2, 1, -8, 13, 6], [-9, 7, -4, 4, -9, -7, 12, -2], [24, 10, 3, 4, 3, 2, 8, -1], [5, -7, -1, 1, -12, -1, -7, -6], [-1, 11, -6, -11, 1, 12, 8, 4], [8, 13, -10, -1, 11, 0, -2, -3], [13, 3, -6, -10, 10, 4, -9, -9], [0, -14, 3, -6, -1, -5, -11, -4], [4, -5, -7, 5, -3, -1, -1, 2], [-11, 1, 1, -8, 3, -10, 5, 2], [10, 12, -12, 1, 3, 8, 6, -3], [-3, 0, -8, -3, -8, -3, 6, -1], [-6, -9, -7, -12, 2, -5, -4, 2], [-12, -6, 7, 8, -6, -1, -11, -5], [-7, 6, -3, 5, 2, -9, 1, 2], [-13, 3, -2, -6, 2, -6, 3, -9], [10, 2, -10, 6, -10, -6, 2, -11], [-8, 8, -4, 0, 7, 6, 12, 9], [7, 6, -4, 8, 5, 9, 13, 9], [4, 2, 3, 5, -5, -6, 11, -9], [-14, 3, -4, -6, 2, 17, 15, 10], [23, -2, 0, -1, -9, 7, 1, 4], [0, 7, -8, 13, -4, 1, 9, -1], [-8, -3, 3, 9, -2, 7, 6, 1], [0, 1, -4, -5, 9, 2, 7, -1], [-10, 1, 8, -3, 2, -17, -2, -6], [-6, 2, 0, 4, -2, 13, 5, -1], [3, -1, -4, 2, 4, -3, -14, -2], [-6, 3, 0, 0, 11, 9, -8, 3], [11, 5, -6, -6, 1, 10, -12, -1], [-7, -13, 4, 5, -8, -4, -11, 6], [8, -4, 2, 8, 1, -5, -5, -5], [26, 2, 2, 2, 6, -9, -13, -6], [13, 1, 1, 10, 4, -1, 4, 8], [3, 2, 2, 7, -7, 10, 0, 0], [-1, 2, 0, 3, 1, -12, -4, 2], [-7, 14, 4, 6, 6, -4, 7, 3], [-17, 11, -6, -9, 8, -2, -5, 1], [18, -12, 19, 2, -13, -12, -18, -10], [24, -8, -3, 3, -13, 1, -9, 0], [-16, -12, -1, 3, -6, -5, 0, -7], [-29, 12, 1, 1, 8, 7, 7, 3], [-9, -18, 14, -2, -7, -7, 5, 9], [12, -23, -6, 1, -4, 10, -9, 4], [-13, 22, -1, -1, 11, -1, 5, 3], [-1, 16, -11, -7, 4, 17, -6, 2], [-17, -1, 8, 10, -14, -15, -3, -5], [-4, -3, 8, -1, -3, 9, -1, 9], [-7, 7, -10, -11, 2, 7, 17, 3], [15, 13, -5, -3, -3, -2, 5, 5], [7, -7, 6, -8, -4, -5, 2, -1], [-13, 1, -2, 5, -15, 1, 1, -8], [-24, 4, 2, -5, -13, -9, 10, 8], [14, 4, 8, 3, 3, -5, -10, 1], [1, 16, -17, -8, 6, 20, 14, 0], [19, -3, 7, -4, -4, 6, -5, 1], [25, 7, 10, 1, -7, 0, -14, -2], [7, 12, 8, -2, -3, 5, 4, -7], [-19, -3, 4, -14, 8, -7, 3, -3], [0, 1, 9, -5, -6, -9, 6, -12], [8, 9, 15, 14, -2, -11, 9, 2], [-6, -12, 7, 5, -3, -20, -1, -3], [-1, -5, 7, 4, -9, 1, -3, -8], [-24, 8, -9, -3, 10, -9, 2, -1], [0, 9, -13, -9, 9, 10, 7, 0], [-4, -14, 10, 9, 0, 7, 0, -6], [27, 14, 3, 2, -1, 2, 3, -1], [0, -5, 10, 14, 2, -8, 2, 0], [20, 1, 4, 17, 3, 6, -13, -5], [16, 4, 10, -1, -10, -12, -7, -1], [-1, -6, 16, 0, -2, -21, -9, 1], [-3, 7, -5, -4, -13, 2, -1, -1], [0, 4, -3, -3, -19, 2, -1, 0], [-18, 0, 21, 5, -7, -26, -10, 0], [2, 14, -12, -4, -7, 17, 19, 1], [-8, -8, 2, 3, -9, 3, 0, -2], [2, -11, 4, 5, 3, -17, -11, -4], [-2, 0, -21, -9, 4, 15, 0, -4], [-2, 4, -21, -14, 10, -2, -1, 3], [5, 6, -7, 13, -6, -6, 2, 0], [-2, -2, 11, -9, -8, -8, -12, -6], [8, 4, 3, 11, 5, 7, 9, 3], [6, -9, -4, 11, -1, 10, 11, 3], [-7, 8, -2, 7, 4, 10, -8, -5], [-5, -1, 10, 20, -7, 3, -4, 3], [15, 7, -8, -17, 14, 16, -1, 1], [-27, -2, 13, 4, -6, -10, -8, -6], [-13, -6, 11, -9, 9, 9, -9, 6], [-27, -2, 0, -3, 14, 0, 6, 3], [-11, 0, 3, -8, -7, -16, -10, -3], [18, 8, 5, 8, 6, -3, 8, 1], [13, 2, -7, -6, 9, 11, -14, -10], [-3, 8, 21, 2, -6, -3, -2, -1], [-1, 16, 3, -1, 13, -1, 5, 8], [2, -9, -10, -27, 6, 5, -12, -9], [-6, -10, 2, -12, 18, -9, 1, -1], [-3, 10, 9, 11, -11, 3, 5, -7], [16, -2, -21, -2, 5, 13, 12, 3], [-36, 4, 6, -5, 11, 0, -1, -5], [7, 8, -11, 1, -2, -4, -10, -15], [6, 5, -1, -3, -5, -1, 7, 7], [-22, -16, 11, 8, -2, -11, -18, -7], [23, -1, -12, 5, -9, 0, 8, 10], [-22, -24, 15, -9, -3, -12, -17, 3], [-6, 30, 2, -4, 1, 10, 12, -4], [-12, -19, 10, 8, -1, -5, -1, 5], [-20, 5, -11, 4, 10, 16, -7, 3], [-28, 7, 1, 3, -7, -3, -10, -11], [19, -9, -21, -12, 8, 10, 3, -7], [30, -10, 0, -5, -1, 1, 7, 0], [10, 9, -1, -5, 3, 8, 0, 13], [32, -12, 6, -1, -13, 2, -9, -3], [-5, 0, 9, 6, -4, -8, -14, -7], [-24, 1, -2, 2, -14, 0, 18, -7], [-26, -13, 16, 4, -12, 8, -1, 5], [-14, 8, -8, -6, -1, -2, 1, 2], [-15, -5, -12, 0, 5, 9, 20, 10], [10, -4, -9, -8, -6, -8, -4, -12], [-10, 0, -10, -9, 15, 1, -6, -8], [-9, -13, 2, -1, -12, -5, -7, 7], [-6, -1, 1, 3, -4, -6, 15, 6], [19, 7, -13, -12, 16, 11, -10, 6], [13, -5, -7, -7, -3, -17, -5, 0], [2, 29, -7, -5, 19, 9, 6, -1], [9, -14, 10, -1, -21, -9, -19, -15], [-11, -8, 23, 21, -10, -4, -2, 1], [3, -5, 2, -5, -19, -10, 2, -3], [-31, -12, 6, -5, -5, -9, -17, -3], [-2, 27, -12, -14, 14, 1, 17, 16], [-24, 3, 3, -1, -6, 11, -4, 2], [-9, -2, -11, 6, 7, 13, 16, 14], [-24, -21, 6, -5, -4, 10, -4, 9], [-23, 12, 9, 14, -2, 6, 7, -2], [-1, -12, 5, -9, -9, -4, -20, -8], [-12, -8, 4, -14, 2, -7, -5, -1], [-19, -5, -2, -1, 7, 2, 2, -4], [-19, -10, 3, 3, -6, 18, 15, 10], [-8, -9, 14, 17, -9, -14, 1, -4], [-29, -9, 11, 14, -1, -6, -11, -5], [8, 8, -12, -5, -3, 4, -6, -13], [-21, 19, -18, -11, 24, 10, 0, -3], [6, -15, 12, -13, 2, -20, -11, -2], [-26, -15, 15, -12, 6, -3, -9, 0], [-22, -2, 7, -2, 7, -9, -19, -10], [-15, 8, -10, -8, 7, 4, 10, 6], [-9, -8, 3, 5, -10, 16, -2, 13], [5, -7, 8, 11, -9, -8, -12, 5], [36, -10, -4, -2, -20, -6, 0, -4], [-1, -11, -24, -14, 5, 11, 12, -7], [0, 13, 5, 5, 4, 8, 13, 1], [13, -9, 6, 12, -15, -7, 14, -5], [-3, 3, -2, 12, -1, 12, -1, 8], [15, 3, 4, 10, -10, -4, -15, -12], [2, -5, 3, 10, 1, 10, 11, 1], [22, -6, -10, -2, 19, 12, 10, 18], [8, -5, 10, 8, 5, -4, -2, 12], [-10, 3, 15, 16, -4, -23, -15, -11], [4, 8, -7, 2, 10, 24, 28, 6], [-11, 18, -8, 8, -5, 1, 10, 9], [-14, -9, 10, 5, -1, 17, 2, 5], [8, -12, -12, 2, 6, 10, 16, -9], [22, -14, 13, 2, 2, 0, -7, -12], [-24, 8, 6, 4, 6, -2, 15, 11], [13, 3, -14, -12, 18, 1, 0, 0], [20, 11, -17, -22, 16, 19, -11, 0], [6, -7, 7, 13, -7, -8, 5, -11], [-8, 6, -6, -2, 13, 10, 6, -1], [1, -25, 1, 2, -1, 1, 6, -4], [11, -22, 17, -6, -2, -13, 4, -1], [-14, -7, -2, -2, 0, 3, -1, -12], [-37, 3, 5, 6, 10, -5, 4, 6], [10, 0, -4, -10, -2, -24, -8, -6], [-6, -2, 2, -12, 11, 2, -8, -9], [10, -3, -15, -11, 2, 26, -9, -6], [15, -19, 14, 9, -14, -20, -2, -7], [-23, 13, -12, -16, 10, 10, -3, 9], [-16, -18, 18, 16, -23, -5, -1, -16], [9, -13, 1, -16, 9, -9, 5, 0], [-7, -8, 28, 9, 0, -18, -24, -4], [4, -10, -5, 5, 11, 3, -14, 4], [-39, 3, -4, -2, 3, 15, 3, -6], [-23, -12, 15, 16, -18, -7, 4, 1], [1, -4, 17, 16, 0, -10, -5, -1], [0, 19, -5, 11, 4, -18, 3, -5], [14, -16, 10, 2, -15, -8, -12, -12], [-10, -17, 18, -8, -6, -13, -15, -4], [26, -2, -6, 0, 11, -4, -9, -9], [-15, 18, 1, -10, 11, 20, 3, 0], [-14, 5, -1, -10, 18, 0, 11, 1], [-7, 10, 5, 9, -2, -3, -13, 1], [16, 5, -11, -23, 13, 14, -17, -3], [10, -3, -9, -19, 2, -4, -16, -8], [-12, -1, 0, 4, -5, -2, 2, 11], [17, -11, -1, 6, 17, 18, -17, -9], [7, 15, -13, -17, 16, 17, 10, 17], [46, 18, -5, 4, 2, 6, 2, -8], [-6, -7, -7, 12, 11, 19, 2, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2014_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2014_2_a_g();".
// 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_2014_2_a_g(:prec:=8)
    chi := MakeCharacter_2014_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_2014_2_a_g();".
// 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_2014_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2014_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-10, 52, -61, -21, 50, 1, -13, 0, 1]>,<7,R![52, -236, -391, 84, 179, -6, -24, 0, 1]>],Snew);
    return Vf;
end function;