
// Make newform 6018.2.a.v in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6018_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_6018_2_a_v();".
// 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_6018_2_a_v();".
// 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 := [-1, 26, 27, -141, -127, 121, 42, -21, -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], [0, 1, 0, 0, 0, 0, 0, 0, 0], [-289639, 499745, 637856, -1475547, 298486, 398353, -96263, -25576, 5907], [75129, 560961, -788640, -1597499, 961014, 461489, -182951, -31080, 9203], [20861, -161915, 294624, 963353, -617266, -306155, 124669, 21400, -6401], [60241, 1014297, -286688, -2690819, 1066438, 752521, -236943, -49640, 12987], [-125641, -19745, 720192, 561227, -602150, -196913, 103367, 14312, -4883], [120975, -1078105, -85312, 3591075, -1332326, -1016233, 312239, 67112, -17371], [-159013, 2036531, -1443360, -7760065, 3836354, 2291651, -810245, -154744, 42985]];
    Rf_basisdens := [1, 1, 61504, 61504, 30752, 61504, 15376, 30752, 61504];
    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_6018_a();"
function MakeCharacter_6018_a()
    N := 6018;
    order := 1;
    char_gens := [4013, 1771, 1123];
    v := [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_6018_a_Hecke(Kf)
    return MakeCharacter_6018_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], [1, -1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, -1, 0, 0, 0, 0, 0, 1], [0, 1, 1, 1, 0, 0, -1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, -1, 0, 0, 0, -1, 0], [0, 0, 1, 0, 0, 0, 0, -1, -2], [0, 2, 1, 0, -1, 1, -1, 1, 0], [0, 1, -1, 0, 0, 1, 1, 1, 0], [0, 1, -1, 0, -1, 1, 1, 0, 0], [2, 1, 0, 1, -1, 1, 1, 1, 1], [-2, -1, 0, 0, -2, -1, 0, 0, 1], [0, 0, 1, -1, 1, -1, 1, 1, -1], [2, -1, 0, -2, 2, -2, 1, -1, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, -1, 1, 0, 2, 0, 0, 2], [0, -5, -1, -2, 2, -2, 1, 0, 1], [-1, 4, 2, 2, 0, 3, -1, -1, -1], [-1, 0, 2, 1, 1, 2, 0, -1, -2], [-4, 2, 2, 0, -2, 3, 0, 1, 1], [-3, 3, 0, 0, 0, -1, 0, 0, 0], [1, -5, -2, -4, 3, -5, 0, -1, 2], [0, 3, -1, 0, -1, -2, -2, -1, 2], [3, 4, 0, 2, -3, 3, 0, 1, 1], [-4, 1, 0, 1, 1, -1, 0, 2, 3], [0, 1, 0, 0, 2, -2, -1, -1, -1], [0, -2, 2, -2, 1, -2, 0, -1, 0], [1, 1, -2, -1, -3, 1, -1, 1, 2], [-2, 1, 0, 1, -1, 0, 0, -2, -1], [3, 1, 0, 2, -1, 1, 1, -1, 1], [0, 2, 3, 3, -3, 1, -1, -1, 1], [-6, 0, -5, -2, 0, -4, 0, 0, 1], [2, 2, 1, -2, 0, 2, -1, 1, -3], [-5, 1, 0, -2, 1, -1, -1, 0, -2], [-3, 3, 4, 3, 0, 2, 0, -1, -2], [-7, 0, 1, -1, 1, 3, 2, 1, -1], [-7, 4, 1, 1, -1, 2, -2, 0, -2], [5, -3, -3, 0, 2, 0, 2, 0, -1], [-1, 0, 0, 2, 1, -2, 2, -3, -1], [4, 1, 3, 0, 0, 1, 1, 1, -4], [0, -4, -7, -2, 3, -2, 1, 2, 1], [-3, -2, 0, 1, 2, 0, -1, 2, -1], [0, 2, 1, 3, -3, 0, -1, -2, -2], [-2, -1, -1, 1, 0, 1, -1, -2, -4], [-3, 2, 2, 4, 2, 3, 1, 1, -1], [-5, -2, 0, 1, 0, 0, -1, 2, 1], [1, -10, -2, -2, 3, -5, 1, 0, 1], [6, -3, -5, -6, 1, -4, 2, 1, 2], [-1, 1, -2, 0, 3, 2, -1, 4, 2], [-2, 7, 3, 3, -1, 2, -4, 0, 0], [0, -1, 2, 1, -3, -2, 0, -2, 1], [10, -2, -1, -3, 1, -6, 1, -4, -2], [0, -7, -3, 0, 2, -5, 1, 1, 2], [0, -2, 0, 1, 1, 2, 3, -1, 4], [1, -1, 2, -3, 0, 0, 2, -1, -8], [4, -3, 0, -1, -1, 0, 2, -2, 1], [-2, -6, -4, -4, 3, -1, -1, 0, -2], [3, -1, -2, 0, 3, -6, -3, 0, -2], [-14, 0, -2, -4, -1, -3, 1, 0, -4], [0, 1, 1, 1, -4, 2, 3, -2, 2], [-10, 5, 3, 2, -3, 2, -4, 1, 4], [-7, 5, 3, 1, -4, 3, -2, 3, -1], [-12, -1, -3, -2, -1, -3, -2, -1, 4], [-1, -3, 1, 4, 1, 2, 0, 1, 1], [-4, 0, -1, 3, 2, 4, 0, 3, 3], [-6, 6, 2, 2, 0, 1, 2, 2, 0], [-7, -1, 2, 2, -2, 2, -1, 3, 4], [2, 3, 4, 2, 2, 6, -3, 1, -3], [8, -2, 1, 1, -2, 3, 2, -2, -4], [-10, 12, 3, 5, -6, 7, -4, 2, 0], [-12, -5, -3, -5, 0, -4, -3, -2, 0], [-10, 1, -4, 0, 0, -3, -3, -1, -3], [-8, 4, 3, 1, -4, -1, 0, 1, -1], [-7, 6, -2, 4, -2, 2, 0, 0, -1], [3, 4, 3, 2, 3, 3, 2, -1, -2], [-4, 2, 2, 7, -1, 3, -3, 1, 0], [0, -5, -1, -4, 8, -3, 3, 1, 2], [-2, 5, 2, 4, -1, -1, 2, -1, 1], [4, -6, 3, -2, 4, -4, -2, -4, -1], [9, -1, 1, 0, -1, 1, -3, 0, 1], [-5, -3, 0, 0, 3, 1, -2, 3, -2], [2, -2, 2, -4, 2, 3, 1, 1, 0], [-14, 5, 5, -2, -3, -1, -1, 0, -2], [-5, -3, 4, -1, 4, -2, 0, 3, 2], [2, -4, 1, -4, 4, -1, 0, 0, 1], [-14, 6, 1, 6, -1, 0, 1, 0, 1], [-7, 2, -2, 1, -1, -3, -2, 0, -3], [-10, -7, -4, -5, -2, -2, 4, 1, 5], [5, -5, -1, 1, 4, -3, -4, 3, 7], [-9, 7, 2, 0, -3, 2, -5, 4, 4], [-1, -4, 1, -6, 5, -4, 1, -2, -6], [-10, 8, 3, 2, -4, 4, -4, 2, -1], [-16, 1, -3, 3, -3, 0, 1, 5, 6], [-5, 1, 2, 4, -6, 3, 0, 4, 6], [-3, -7, -7, -7, 3, -9, 3, -1, 1], [1, 1, 4, -3, -2, 3, 2, -2, 3], [1, 1, 1, 3, 4, 5, 0, 5, 3], [-10, -4, -7, -6, -2, -4, 4, 2, 1], [-3, 0, 2, -4, 5, -3, 4, -3, -11], [6, 9, 4, 4, -7, 9, -2, -1, -1], [-9, -4, -3, -3, 0, -3, 3, -2, -4], [0, -3, 2, 1, -2, 6, 1, 1, 0], [-1, -10, -1, -1, 4, -3, 3, 2, 6], [3, -6, -1, -2, 2, 1, 4, -2, -2], [-12, 11, 0, 3, -4, 1, -3, 2, 3], [-5, 0, -5, 3, 1, 0, 0, 0, 2], [2, -7, -8, -4, 4, -1, 5, -6, 2], [-4, 0, 3, -1, -1, 4, -2, -8, -7], [-16, -1, -3, -3, 1, -4, 0, 1, 3], [0, 3, 3, -3, -3, -5, -2, -2, -2], [-6, -5, -3, -6, 3, -5, 0, 0, 7], [-9, 5, 1, 3, 3, 3, 3, -1, -5], [-10, -1, 1, 0, 2, -5, -1, 1, -2], [19, -8, -7, -7, 3, -5, 4, 0, 2], [-10, 4, -1, 2, 3, 3, 1, 3, -2], [-3, 6, -1, 5, -1, -3, -5, 1, 2], [-7, -3, -7, -4, -1, -6, -1, 0, -3], [2, -6, 0, -1, 3, 0, 1, -3, -8], [-7, 4, -4, -1, -2, -1, 1, 3, 2], [-10, 6, 4, 0, -4, 5, -2, -4, 2], [1, 6, -2, -3, -1, 4, 1, -1, -11], [8, 0, -2, -2, -2, 2, 2, -2, 2], [-8, -4, -4, -2, 1, -5, -3, 0, 4], [1, 4, 4, 5, -8, 5, -5, 1, -2], [-1, 5, -4, 1, 1, -3, -3, -3, 4], [1, 9, 3, 8, -3, 6, 1, 2, 1], [2, -1, 4, 6, 5, 3, 2, 5, 1], [-5, 3, 1, -1, 3, 5, 5, 5, 3], [0, -2, -9, -4, -3, -3, 1, 1, 8], [1, -5, 2, 3, 1, -3, -5, -3, -2], [-8, -1, 6, 9, -1, 4, -4, 0, -1], [1, 1, -1, -4, 2, -5, 4, -5, 0], [18, -2, -5, -4, -1, -4, 3, -2, 1], [-1, -3, -2, -4, 5, -4, -3, 2, 0], [4, 7, 2, 3, -4, 4, -3, 2, -7], [-8, -17, -7, -11, 7, -10, -1, -1, 4], [4, -7, -4, -3, 1, -9, 0, -1, 6], [8, -9, -3, -2, 4, -11, -3, -1, 2], [-4, -7, -9, -4, 5, -12, 4, 3, 2], [14, 10, 5, 5, -2, 4, -2, -1, -3], [-22, 0, 3, 1, 3, -3, 1, -3, -5], [0, -6, 2, 1, -1, -6, 1, 1, 14], [22, -6, -4, -8, -1, -6, 0, 1, 6], [2, -9, -3, -6, 4, -9, 5, 1, -6], [1, -2, -1, 5, 1, -6, 0, 4, 14], [-10, -8, -3, 1, 5, -7, 1, 3, 3], [2, 14, 5, 5, -7, 10, 1, 0, -4], [11, -9, -3, -7, 2, -7, 4, -5, 1], [6, 5, 5, 7, -3, 1, -2, -4, -4], [8, -2, -4, -1, -2, 5, 8, 3, 0], [18, -4, -3, 0, 4, -1, 6, 0, -7], [-6, 5, 3, 4, 2, 2, -2, 8, 6], [4, -14, -10, -3, 2, -6, 0, 2, 5], [-4, 5, 3, 5, -2, 12, -5, 0, 0], [2, -1, 3, -4, 0, 8, -1, 2, -3], [1, 0, -5, -3, 3, -4, 4, 2, 2], [9, 7, -2, 3, 3, 11, -1, 5, 6], [-7, 8, 9, 6, -2, 16, -1, 5, 0], [13, 0, 7, 2, -8, 12, 1, -1, 4], [-13, 0, -2, 1, 4, 0, 1, 4, 3], [-3, -7, 3, -5, 3, -8, 3, 3, 3], [6, -1, 4, 3, 2, 6, -4, -3, -7], [11, 0, 4, 2, -3, 3, 1, 4, -9], [18, -3, 0, 5, -2, -2, -1, -9, -4], [8, 4, 13, 4, -7, 8, 1, -4, -5], [-7, 6, 3, 2, -10, 14, 1, -1, 0], [27, -4, -2, 5, 4, 0, 1, 0, -1], [3, -12, -12, -7, 6, -10, 1, -8, 1], [0, 3, 0, 0, -2, -7, 3, -3, -7], [11, 2, 2, 1, -3, 5, -4, -4, -5], [6, 12, 3, 10, -1, 6, 1, 2, -3], [12, 4, 6, 4, -8, 2, -8, -2, -2], [-2, -6, -3, -4, -5, 0, -3, 0, 5], [9, 11, 11, 7, -4, 8, -9, -4, -9], [0, 2, -4, 3, -2, 5, -2, 1, 6], [-4, -5, -3, 0, 1, -2, 4, -3, 0], [1, -8, -2, -4, 6, -3, -3, 1, 5], [3, 12, 6, 8, 4, 4, -5, 0, 0], [1, -10, -9, -4, 9, -9, 6, 1, -4], [11, -4, -2, 2, 5, -8, -4, -5, -3], [13, -9, -4, -7, 10, -1, 10, 1, -2], [-8, -2, -2, -7, 0, -5, -2, 3, 8], [5, -12, 1, -3, 8, -10, -1, 3, 5], [-11, 8, 7, 8, -5, 10, 2, -4, -7], [1, -6, -1, 7, -5, 2, -2, 4, 10], [7, 0, 9, 1, 1, -5, -2, -7, -9], [-14, 12, 1, 9, -9, 0, 3, 5, 3], [16, -12, -8, -3, 8, -6, 4, -7, 2], [3, 10, 7, 3, -6, 12, -4, -1, 4], [-9, -2, 2, -4, 2, 3, 1, 3, 3], [5, 6, 1, 6, 2, 9, -2, 0, -8], [6, -5, -2, -2, -8, 2, -3, 1, 5], [-1, 15, 7, 10, 0, 2, -2, 2, -1], [-1, -7, 3, 4, 2, 6, 6, 2, -5], [-17, 10, -5, 6, -5, 5, -4, 7, 12], [9, 8, -7, -5, -1, -4, -4, -2, 2], [0, -1, 1, 4, 0, 3, -3, -1, 0], [-5, -11, -1, -10, 0, -4, -2, -2, 5], [-8, 6, -4, 0, 9, -6, -5, 6, 0], [-3, 2, -2, -4, -5, 7, 2, 7, 7], [18, 7, 3, 4, 0, 6, -1, -4, -5], [-1, -2, -3, -1, 1, 12, 2, 6, -6], [5, -5, 7, -6, 1, 4, -1, -4, -1], [6, -3, -5, -2, 1, -9, 3, 0, 4], [20, 14, 0, 8, 0, 0, -2, 0, -10], [-28, 9, 6, -2, -6, 8, -3, 5, -1], [13, -13, -1, -3, 8, -7, 2, -9, -9], [5, 13, 8, 5, 0, 3, 2, 3, -2], [1, -29, -8, -12, 5, -13, 7, -3, -1], [5, -14, -6, -11, 3, -5, 4, 2, 1], [-11, 1, 3, 7, -3, 14, 1, 3, -1], [-14, -1, 4, -6, 0, 1, 3, 1, 7], [2, -5, -11, -3, 0, 2, -1, 4, 8], [6, 7, 3, 7, 6, 11, -1, 4, -2], [8, -7, -12, -4, 5, -3, 2, -5, -3], [4, -7, -8, -8, -1, -11, 0, -5, 3], [-22, -4, 2, 2, 1, 8, 2, -1, -2], [3, 8, 3, -4, -2, 6, 5, 1, -8], [6, 15, 7, 3, -3, 9, -6, -4, -12], [-7, 1, 7, 1, 0, 9, -4, -3, 7], [12, -6, -11, -7, -1, -6, 4, -2, 3], [9, -11, 2, -3, -2, -3, 4, -7, 0], [17, 8, 2, 2, -1, 12, 2, 7, -9], [-6, -1, -5, 2, -1, -6, 0, -1, -6], [-7, 5, -1, 7, -3, 11, 10, 2, -5], [-11, 6, -8, 0, -13, 4, -6, 5, 13], [-17, -1, 1, 7, -4, -4, -4, 1, 11], [5, -2, 10, 1, 5, -3, -6, -6, -11], [19, -5, -4, 5, 6, 0, 4, -1, -8], [12, -1, -2, -3, 6, -9, 1, 2, 1], [3, -6, 5, -6, 4, -12, -5, -1, -2], [-4, -4, 1, 2, -11, 1, 1, 2, 7], [14, -8, 2, -6, 3, -10, 2, 11, -2], [-12, -14, -11, -8, 14, -18, 6, 1, -2], [13, 15, 3, 3, -3, 9, -3, 11, -1], [-3, 5, 2, -3, 4, -4, -6, 5, -8], [-6, 9, -6, 4, -9, 6, -2, 0, 12], [6, 1, 3, -1, 3, 5, -2, -2, -16], [-8, -5, 4, 2, -6, 10, 1, -3, 3], [5, -13, -8, -3, -2, -1, 3, 2, 4], [1, -1, -4, -10, 3, -4, 3, 6, -2], [11, 10, 3, -2, -10, -1, -9, -7, -10], [10, 0, -5, 2, -2, 6, -2, 1, 6], [6, 1, 8, -4, 3, -1, 2, -1, -15], [7, 13, 8, 13, -7, 7, 1, -3, -8], [-2, 10, 4, 3, -6, 12, -2, 0, -1], [-8, 5, 1, 5, 7, -9, -4, 2, 2], [-21, -14, -2, -4, -6, 0, 1, -3, 9], [5, -10, -1, -4, 3, -8, 4, -6, -13], [10, 1, 10, -2, 1, 3, 2, -5, -15], [8, -15, -9, 2, 2, -9, 5, -1, 14], [4, -4, -6, 2, 2, -4, 4, 2, 2], [-3, 7, 3, 5, -3, 0, -7, 3, -7], [-2, -1, -1, -6, 15, -5, 5, -2, -6], [13, -5, 5, -8, 1, -9, -5, -6, -9], [-4, 13, 10, 9, -9, 10, 2, 0, -9], [-19, -7, 1, -3, -1, 8, 3, -9, 1], [-5, -14, 1, -9, 8, -10, 1, -5, 9], [2, -25, -1, -10, 10, -19, -3, -7, 0], [-16, -13, -6, -4, 0, -1, 4, -2, 9], [-2, -10, 7, -7, 6, -12, -2, -7, -7], [17, 3, -7, 6, 5, 1, 1, 2, -5], [4, 23, 12, 12, -3, 8, -6, 1, -3], [-10, 17, 5, 12, -11, 1, -6, 3, -2], [-16, 3, 5, 4, 2, -3, 3, -1, -2], [1, -4, 9, 1, -5, -5, -4, 4, 4], [-8, 12, 10, 10, -11, 4, -12, -3, 6], [10, 16, -6, 2, -11, 9, 1, 2, 6], [-7, 15, 1, 8, 2, 13, -4, 9, 2], [21, 4, -13, 2, -11, 7, 0, 3, 16], [-17, 2, 11, 5, 3, -2, -2, -4, 6], [-10, -11, -5, -5, 0, 4, 3, -8, -2], [-4, -2, -8, -9, -2, -2, 1, -6, -12], [10, -5, -4, -9, 6, -4, 8, 1, -5], [0, 0, 1, 4, 0, -4, -7, 7, 13], [-8, 11, 7, 4, -5, 11, 2, 5, -2], [2, 3, -1, -6, 8, -1, -1, -5, -16], [-11, 0, -1, 7, -8, 16, 3, 0, 6], [15, -12, 0, 0, -2, -4, -1, -1, 9], [-19, 9, 4, 4, -3, 4, 1, -2, -10], [-7, 6, 8, 10, -1, -1, 0, -3, -5], [13, 5, 9, 3, 8, 5, -4, -1, -15], [10, 11, 5, 1, -10, 8, -1, -8, -2], [7, -12, 3, -5, -3, -6, 0, -2, 2], [3, 12, -2, 10, -9, 1, -4, -4, -6], [-19, 0, -10, 6, -2, 1, -5, 3, 9], [-12, 3, 3, 13, -4, 13, 3, 2, -2], [1, 2, 5, 10, -1, 17, -2, 3, 6], [10, -6, 2, -7, 2, -1, 2, 7, -6], [-11, 3, -6, -11, 10, -12, -4, 3, 0], [-17, -20, -16, -17, 10, -14, 7, 8, 3], [16, -1, 4, -4, 4, -3, 8, 4, -3], [-22, 10, -8, 1, -2, 1, 0, -1, -2], [-26, 11, 10, 4, 0, 8, -7, -1, 5], [18, -11, -8, -4, 7, -7, 6, -7, -5], [-7, -10, -3, -13, 12, -15, 3, -12, -8], [12, -22, -10, -17, 11, -15, 10, 0, 8], [-7, 0, 4, 1, -10, 12, -1, -4, -3], [7, -14, -16, -14, 7, -15, 8, -8, -2], [10, 18, -2, 8, -11, 16, 6, 5, 4], [-3, 4, 1, 4, 5, 7, -2, -5, -10], [14, -4, -2, 2, 5, -16, -6, -5, 12], [-9, -2, 8, 7, 1, 6, -10, 0, 7], [0, -9, -4, 10, 6, -3, 3, 1, -3], [-2, -8, -2, 7, -2, -4, -6, -4, 7], [40, -3, -1, -3, -2, -5, 9, -7, -7], [-10, -8, 0, 2, 4, -6, -2, -10, -4], [10, 3, 3, -7, -1, 7, -2, -4, -12], [-9, 11, 10, 9, 13, -1, 3, -1, -4], [12, -10, 2, 3, 1, 1, 10, 0, 0], [-6, -4, -2, -3, 9, -9, 3, 1, -6], [-4, 12, 9, 5, -5, 3, 3, 7, 11], [1, 1, 1, -1, 5, 3, -8, -2, 3], [-4, -1, -14, -7, -9, -6, -8, -6, 3], [21, -18, -4, -3, 7, -11, 10, -12, -13], [-2, -13, -10, -9, -7, -8, -2, 0, 3], [-11, -4, 0, -13, -4, -12, 1, -10, -1], [-11, -22, -1, -7, 13, -4, 4, 4, 0], [3, -15, 1, 2, 8, 7, 8, 1, -8], [1, 5, -3, 5, 7, 1, 2, -2, -11], [17, -12, 0, -3, 7, -11, 0, -4, 7], [-5, -10, -8, -1, -6, -4, -1, -4, 7], [2, 18, -6, 6, -9, 8, 0, 7, -4], [-15, 8, -1, -3, -9, -6, -4, -6, 6], [19, -12, -5, -3, 13, -2, 10, 0, -4], [8, 11, -4, 18, -7, 7, -2, 7, 13], [14, -22, -3, -13, 2, -13, 5, 4, 1], [-15, -4, 9, -1, 2, 7, -1, -16, -4], [29, -18, 7, -11, 7, -8, 6, -2, -6], [5, 5, 3, 11, -8, 5, -6, 9, 3], [-4, -8, -3, -13, 1, -1, 1, -11, -13], [-17, 16, 5, 4, -2, 3, -3, 3, 10], [-25, 4, -8, 3, 3, -4, 1, 3, -1], [7, 15, 8, 1, 2, 4, 0, 3, -12], [2, -1, 6, 3, 0, 0, 5, 8, 5], [9, 12, 8, 7, 0, -2, 3, -2, -7], [16, 13, 2, 3, 1, 1, -5, -1, -15], [-1, -11, -18, -10, -1, -18, 5, 2, 6], [32, -23, -9, -9, 11, -18, 5, -9, 2], [-9, -9, 3, -3, 0, -2, 4, 3, 13], [22, 2, -5, -2, 4, 1, 0, 8, -11], [5, -4, -8, -4, -6, 6, 3, 2, 4], [-21, 3, -1, -1, 1, 5, -7, 1, 11], [10, -23, 1, -8, 14, -17, 1, 3, 0], [11, -21, -13, -12, 1, -18, 6, 7, 5], [0, -14, -5, -5, 3, -5, 1, 1, 5], [1, -8, 0, -10, 7, 1, 0, 8, -2], [-4, -5, -9, -8, 1, -2, -2, -3, 12], [6, 8, 5, 1, 2, -11, -6, -3, -9], [21, -10, 6, 6, 2, 11, 3, 3, -5], [9, 4, 8, -3, -4, 12, 7, 0, 7], [-1, 7, 6, 15, -5, 5, -9, -3, -10], [17, -5, -6, 3, 4, -11, 1, -6, -14], [20, -3, -13, -11, -3, -4, 3, 3, 8], [24, -10, 1, -1, -1, 1, 3, -9, -5], [15, 4, -2, 7, -14, 16, -1, 6, 5], [-12, 23, 1, 4, -10, 2, -7, 3, 12], [-7, -2, 7, 14, -3, 8, -2, -2, 1], [-8, 13, 11, -7, -3, 12, 1, -3, -16], [-2, 5, 6, 7, -17, 7, -6, 4, -1], [-5, 4, -5, 0, -1, 8, 7, 2, 6], [-6, 6, 1, 2, 5, 1, -3, -3, -10], [3, -2, 9, 0, 8, 6, 5, -5, -4], [3, -1, 2, -1, -15, -5, -9, 1, 10], [-14, -3, 14, 3, -2, -9, -11, -6, 7], [4, 9, -7, 3, -8, 6, 3, -4, 2], [-14, 5, 4, -12, -3, 6, 4, 5, -9], [5, 19, 14, 11, -8, 11, 1, 4, -2], [-8, 10, -9, -3, 5, -5, -9, 9, 15], [24, -15, 7, -6, 2, -4, 1, -5, -2], [-28, -4, 6, -3, -5, -2, 1, 5, 12], [19, 4, -7, 5, -13, 11, -1, 5, 4], [5, -3, -7, 1, 3, 2, -1, 3, 9], [45, 9, -2, 11, -8, 11, -1, 0, 6], [16, 16, 4, 0, -4, 7, 0, 9, -5], [-1, -3, -4, 5, 5, -5, 9, 7, 8], [-7, 17, 4, 11, -19, 15, 3, -5, -2], [-11, -12, -10, -10, 1, -17, -2, -11, 7], [24, -13, 0, -16, 7, -20, 0, -8, -2], [-13, -8, -3, -6, 9, -14, -2, -9, 10], [3, -21, -4, -19, -3, -6, 5, -5, 4], [-23, 0, -7, -2, 3, -11, 2, -9, 0], [1, -30, -18, -17, 12, -23, 13, 5, 8], [3, 10, 4, -3, -13, 12, -1, -9, -9], [-19, -9, -5, 3, -4, -1, 4, 13, 19], [6, 0, -6, -2, -11, -8, 3, -2, 0], [6, -8, 2, 2, 0, 13, 16, 5, -9], [1, 3, -3, 9, -8, -5, 0, 1, 7], [7, 6, 6, 2, 11, -15, -10, 2, -12], [13, 5, 8, 16, 4, 11, -4, -4, 2], [8, 1, -3, 13, -5, 13, 0, 2, 14], [-4, 3, 0, -2, -15, 2, -10, 8, 6], [-16, -5, 2, 4, 3, 3, -6, -5, -5], [-13, -18, -9, -14, 5, -4, 0, -1, -4], [2, -27, -12, -10, 7, -7, -2, 1, 1], [-2, -6, 4, -6, -12, 0, 0, -2, 14], [-10, -20, 8, -12, 8, -10, -2, -10, 2], [-5, -13, -9, -3, -9, -14, 3, 3, 7], [-34, -1, 4, -3, 4, 9, 9, 4, -3], [9, -6, 0, 2, 8, -2, 0, 12, 13], [11, 18, 8, 12, -4, 9, -5, 5, -5], [6, -7, -2, 0, 13, -18, 2, -11, -9], [-36, 3, -12, 2, 0, 8, 3, 15, 17], [-9, -3, -9, -6, 0, -6, 0, -6, -9], [7, -3, 11, 8, 1, 5, 3, 4, -9], [20, 22, 7, 5, -9, 15, -7, 1, 1], [1, -15, 1, 1, 6, -5, -4, -3, 11], [9, -6, -1, -1, -4, -1, 9, 4, 0], [-28, -29, -7, -8, 12, -11, 5, 7, 12], [-2, -22, -5, -19, 27, -13, 11, 3, -3], [11, 17, 8, 6, -8, 7, -2, 0, -12], [-8, 3, 12, -1, 0, 2, 2, -9, -23], [-12, 3, 5, -5, -2, 3, -4, -11, -10], [5, -27, -9, -26, 6, -17, 2, 1, 10], [-2, 1, 11, 2, -5, -13, -1, -6, 8], [-1, 35, 26, 26, -11, 22, -9, 0, -18], [9, -1, 0, 0, 7, 2, 5, -2, -16], [-7, -10, -18, -8, 5, -17, -1, -12, 5], [-3, -21, -12, -18, 17, -18, 11, 8, 4], [33, 4, -9, 3, -2, -1, -5, -2, 12], [-8, 13, 13, 11, 2, 12, -5, -2, -22], [13, -27, -13, -6, 7, -6, 8, -3, 15], [-10, -1, -10, 6, -5, 15, 2, 1, 19], [-13, -18, -2, -5, 15, 1, 8, 6, 1], [-20, 2, 0, 11, -3, 3, -1, 3, -2], [-13, 14, -2, 3, -4, 9, 5, 10, 11], [2, -3, 7, -7, -2, -2, -3, -10, -14], [-32, 3, -9, -9, -10, -6, 7, 0, 16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6018_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6018_2_a_v();".
// 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_6018_2_a_v(:prec:=9)
    chi := MakeCharacter_6018_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_6018_2_a_v();".
// 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_6018_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6018_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![76, -156, -193, 582, -289, -110, 105, -9, -6, 1]>,<7,R![2, -9, -54, 309, -223, -360, -71, 28, 11, 1]>],Snew);
    return Vf;
end function;