
// Make newform 3381.2.a.be in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3381_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_3381_2_a_be();".
// 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_3381_2_a_be();".
// 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 := [80, 37, -141, -35, 75, 11, -15, -1, 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], [-4, 0, 1, 0, 0, 0, 0, 0], [-14, -13, 25, 8, -10, -1, 1, 0], [-34, -11, 41, 8, -12, -1, 1, 0], [-30, -21, 39, 10, -12, -1, 1, 0], [-10, -36, -9, 41, 8, -12, -1, 1], [-34, -32, 41, 18, -12, -2, 1, 0]];
    Rf_basisdens := [1, 1, 1, 2, 2, 2, 2, 2];
    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_3381_a();"
function MakeCharacter_3381_a()
    N := 3381;
    order := 1;
    char_gens := [2255, 346, 442];
    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_3381_a_Hecke(Kf)
    return MakeCharacter_3381_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, 1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0], [2, 1, 0, 0, -1, 0, 1, 0], [1, 0, 0, 1, -1, 0, 0, 0], [-2, 1, 0, 0, -1, 0, 0, 1], [-2, 0, 0, 0, 1, -1, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, -1, 0, -1, 0, 0], [1, 1, -1, -1, 0, -1, 0, 0], [1, 2, -2, 0, -1, -1, 0, 0], [-2, -1, 1, -1, 1, 0, 0, 0], [2, -1, 0, -1, -1, 0, -1, 0], [0, 2, 0, 1, -2, -1, 0, 0], [1, 0, 0, 1, -1, -1, 1, 1], [-2, 2, -1, 0, -1, -1, 2, 2], [2, 1, 0, -2, 1, 0, 1, 0], [3, 1, 0, 0, 2, 1, -2, -1], [0, 0, -2, 0, 1, -3, 2, 0], [-1, 3, -1, -1, 1, -2, 0, 0], [2, 0, -2, 2, 1, -1, 1, 0], [-4, 2, -2, 0, 0, 0, 2, -2], [-1, 2, 4, 0, 0, -2, 1, 2], [-2, 0, -4, 1, -1, -1, 1, 0], [4, 3, 0, -1, 0, -1, 2, 0], [2, 0, 0, 1, -4, 2, 1, -1], [4, 3, 0, 0, 1, 0, -1, 0], [1, 1, 0, 1, 1, -2, 2, 2], [-2, 0, 0, 0, -2, 0, -1, -1], [3, 0, 0, 0, -3, -3, 2, 0], [-4, 0, 2, 0, 1, 3, -2, 0], [2, -5, 6, -2, 1, 2, -2, -1], [3, 0, 0, 0, -3, 3, 0, -2], [-2, 1, -2, 0, -1, 0, 0, -3], [2, 1, 0, -3, 0, -1, 0, 2], [10, 0, 4, -1, 1, 1, 1, 2], [4, -1, -1, -1, 1, 0, 0, 2], [-2, -1, -1, 1, 0, 1, 0, -4], [0, 0, -1, 2, -3, 1, 0, 0], [2, -4, 0, -1, 2, 3, 0, -2], [3, -3, -2, -1, 3, 0, -2, 0], [6, -1, -2, -1, -4, -1, -2, 0], [-3, 6, -3, -1, -2, -3, 0, 4], [-6, -2, -3, 0, 3, 1, 0, 0], [1, 0, 0, 0, 2, -2, 3, -2], [-2, -3, -3, 1, 1, -2, -2, -2], [2, -1, -3, 3, -3, -2, 2, 0], [-2, 0, 2, -1, -3, -3, 3, 0], [-7, -2, 0, 0, 5, 3, -4, -2], [-2, 3, -3, -1, -1, 0, 0, 0], [0, -6, 4, 4, 2, 4, -6, -4], [8, 4, -4, 1, 1, 1, -1, 0], [6, 5, 0, 1, -2, -3, 2, 2], [-4, 0, -2, 0, 4, -2, 2, 0], [0, -2, 2, 1, -1, -1, -1, -2], [4, -5, 0, 1, 2, 1, -2, -2], [4, -2, 5, 0, 1, 1, -2, -2], [1, -2, 2, 2, 0, 4, 0, -2], [3, 0, 0, 1, -1, -1, 3, 3], [0, -2, 2, -1, 0, 2, -1, 1], [-4, 0, -4, 0, -2, -2, 5, 5], [-1, -1, 5, -1, 6, 3, 0, 2], [-2, 0, -1, -1, -1, 0, -2, -2], [9, -2, 6, 1, -4, 2, 1, 0], [-8, 0, 2, 2, 2, 0, 0, -2], [13, 2, 2, 0, -3, -3, 4, 6], [7, 2, 0, 1, 0, 2, 1, -2], [2, -8, 1, 3, 3, 2, -2, -2], [6, 3, -1, -2, -1, -7, 2, 2], [-10, -5, 2, 1, 4, 3, 0, 0], [14, -2, 6, -2, 0, 0, 4, 2], [-1, 7, 0, 2, -6, -3, 2, 3], [15, -8, 4, -3, -2, 0, -1, 2], [0, 2, 0, 2, 1, -3, 0, 5], [4, -4, 2, -3, 1, -1, 3, 2], [3, -4, 2, 0, 2, -2, 1, -2], [1, 4, -2, 0, 0, 0, 4, -4], [-2, -1, -2, 4, 1, 6, -4, -5], [7, 0, -3, 1, -2, 1, 0, 4], [8, 0, 2, 0, 0, 2, 4, 2], [1, -6, 6, -5, 4, 0, -1, 2], [4, -3, 4, 2, -3, 0, 1, 0], [-3, -1, 0, 6, -4, -1, -1, 0], [10, -3, -2, 3, 0, 1, 2, -4], [8, -5, -3, 3, 4, 1, -2, 4], [-4, 7, 0, -3, -2, -5, 6, 6], [9, -3, -4, -2, 2, 3, 3, 2], [-18, 0, 0, 0, -2, -4, 0, 2], [-3, 8, -3, 0, -4, 0, 6, -4], [16, -1, 2, 3, -2, 5, -2, 2], [0, -7, -4, -1, 2, -1, 0, -2], [-13, 4, -3, 2, 2, 2, -2, -8], [18, -6, 3, 3, 3, 4, -2, -2], [7, -7, 1, 3, 0, 1, -4, -2], [0, -5, 8, -1, -4, 1, 2, -2], [-2, -3, -6, 1, 0, 1, -6, 2], [-2, 6, 0, 2, 4, 0, -1, -1], [13, 5, -6, -2, -2, -11, 4, 7], [1, 0, 6, -1, -3, 4, -2, 0], [-8, 6, -1, -2, -5, -5, 2, 4], [-3, 12, -2, 0, 0, -4, 1, 2], [2, -2, 0, 3, -1, -1, 3, 2], [2, -2, 2, 0, 2, 0, 1, -5], [24, 6, 0, 2, -7, -3, 2, 1], [11, -3, 1, -1, -2, -1, 0, 2], [-8, 4, -6, -4, -1, -3, 2, 2], [-6, 3, 1, -1, -1, 2, -4, 2], [14, -1, 7, -1, 0, -1, -2, -6], [5, 5, 1, -3, -2, 3, 2, 4], [7, -3, 4, 3, 5, 0, 4, 0], [-6, 5, 2, 3, -6, -3, 4, 2], [8, 0, 2, -4, 2, -2, 5, 1], [3, 0, -2, 0, -1, 1, 1, 5], [-5, 1, -2, -3, 4, 1, -4, 1], [-1, 0, 2, 3, -1, -1, -5, 3], [-8, -1, 2, -5, -1, 2, 1, 4], [-6, -8, 1, -2, 9, -1, 2, 2], [-4, 5, -1, 3, -9, 0, 0, -2], [0, 9, -6, 1, -6, -5, 0, 2], [11, 8, 0, -2, -5, -1, 4, 10], [-9, -5, -5, 1, 4, -1, 0, 0], [-13, -4, -4, 3, -11, -3, 5, 3], [4, -4, -2, 0, -3, 5, -2, 2], [3, 4, -7, -4, 0, 0, 2, 4], [11, -4, 4, 0, 0, 0, 2, 3], [10, 1, 2, 6, 5, 0, 1, 0], [8, 0, 1, -1, -1, 4, -4, -2], [2, 0, 0, 0, 1, 1, -3, 6], [7, -7, 2, -1, 1, 4, 0, 0], [7, 2, -6, 0, -3, -1, -2, -2], [-8, -4, -10, 3, 5, -1, -5, -2], [10, 3, 0, -3, 5, 6, 1, -4], [-16, 3, 8, -2, 1, 4, -5, 0], [-8, 1, -1, -3, -1, -6, 4, 6], [9, 6, -2, 2, -1, -5, 4, -4], [3, -12, -4, 3, -5, -1, -1, 1], [12, 1, 2, 3, -6, -3, 3, 3], [14, -1, -4, 0, 1, -2, -2, 1], [-20, 7, -2, 1, 6, 1, 2, -2], [0, -13, 1, 1, -1, -2, -2, 2], [-14, 1, -6, -1, -2, -3, -2, -4], [-14, -3, 5, -1, -1, 0, 4, -6], [2, -2, -2, 2, 0, -6, 4, -6], [3, 0, 1, 1, -2, -3, -2, -2], [-2, -12, 12, 0, 6, 8, -6, -4], [-21, 7, -2, -5, 1, -4, 0, 0], [-12, -6, 1, -6, 3, 1, 2, -2], [16, 11, -2, 1, -8, -1, 6, 6], [-10, 10, -12, 1, 0, -7, 2, 4], [-12, 2, -7, -2, -4, -8, 0, 0], [-1, -4, -4, 1, -7, 7, -3, -1], [1, 4, -1, 2, -4, -6, 10, 4], [2, 0, -7, 4, -9, -7, 0, -2], [-14, 10, 0, -3, 6, -8, 3, 5], [0, 6, 0, -1, -3, -7, 3, 8], [4, -4, -2, -4, 7, -1, 0, -5], [-16, -9, -7, 1, 1, 2, -8, -6], [17, 4, 4, 1, 0, 2, 5, 0], [15, 2, -6, 2, -2, -4, -3, 2], [4, 12, -9, -5, -5, -6, 6, 6], [1, -2, -10, 2, -4, 0, -1, 2], [13, 10, 1, 4, 0, -6, 2, 2], [18, -4, 4, 1, -3, 5, 3, 2], [-13, 16, -2, -2, -2, -4, 2, 3], [4, -14, 4, 4, 6, 4, -4, -2], [5, 2, 9, 0, -6, -2, 0, 4], [11, -1, 9, 7, -10, -1, -2, 8], [-1, 3, 4, -1, 5, -6, 4, 8], [-6, 15, -1, 1, -3, 0, 2, -4], [24, 3, -4, -6, 7, 0, -3, 0], [4, -6, 5, -6, -4, -2, 0, 6], [14, 0, 10, -7, 3, 7, 3, -2], [7, 0, 2, 1, -2, 6, -3, 4], [3, 3, -6, 3, -3, -4, 8, -6], [-11, 11, -12, 5, 0, 3, 1, -6], [9, -15, 9, -1, 0, 7, -4, -4], [-14, -7, 5, 1, 7, 2, -2, -4], [3, -1, -2, -1, 10, 3, -10, -3], [-3, 4, -6, 5, -2, 2, -1, -8], [9, 17, -1, 3, -6, -1, 2, 2], [-8, -3, 4, 3, 4, 13, -2, -8], [-19, -7, -1, -1, 11, 0, -2, -2], [-10, 10, -4, 2, 8, -4, 2, -8], [-4, -8, -2, 3, 1, 1, -1, 4], [-8, -11, 6, -2, 5, 0, -6, 9], [-22, 1, 2, 5, 1, -4, 4, 4], [-2, 13, -2, 5, 1, 6, 1, 0], [5, 6, 2, -2, 0, 2, 8, -4], [-20, 2, -2, -3, 10, -3, -2, 2], [17, 3, 6, -3, 1, 0, 10, 4], [10, 2, 0, -6, -6, -4, 4, 8], [14, -9, 6, 1, 4, 7, 7, -1], [-28, -1, -5, 1, 5, 4, -4, -10], [18, 7, -4, -4, -1, -12, 9, 8], [0, -6, 4, 4, -2, 4, -1, -3], [37, -3, 6, 1, -1, 0, 6, 4], [-14, -7, -6, -1, 6, -5, -2, -2], [-5, 2, -6, 3, 5, -1, 3, -1], [16, -14, 8, -2, 1, 7, -6, -10], [12, 4, -6, 8, 6, 2, -3, -3], [11, 11, 4, -3, 9, -4, 0, -2], [8, -3, 6, 1, 3, 6, 0, -6], [-6, 14, -8, 1, 7, -3, -3, 2], [6, 5, 6, -2, -7, -8, 3, 0], [4, 0, -9, -2, -5, 1, 2, 8], [6, -12, -8, 3, -7, 3, -4, -5], [14, 16, -4, 3, -6, -5, -2, 6], [0, -2, -5, -4, -1, -5, 2, 8], [22, -3, 8, -1, -8, 7, 2, 6], [-9, -1, 13, -3, 3, 4, -6, 4], [-14, -10, 3, 0, 7, 5, 0, 4], [5, 5, 0, 5, -2, -5, 9, 2], [0, 10, -8, -1, -9, -7, 5, 4], [-2, -12, -2, 0, -6, 0, 4, -2], [-13, -2, 8, -4, 3, -5, 2, 6], [2, -18, 8, -2, 4, 4, 4, -4], [-18, -2, -6, 10, -8, -2, -2, -2], [-8, -2, -10, 3, 1, -1, -1, 0], [-10, -8, 6, 6, -4, 4, -4, 0], [17, -4, -1, 7, -6, -1, 12, 4], [-17, -7, -4, 2, 2, 5, -6, -3], [0, 5, 5, -5, -1, 6, 4, -2], [-6, -11, -2, 3, 8, -3, -7, -3], [4, 6, -12, 4, 0, 4, 6, -2], [-4, 3, 4, 0, 2, 0, -8, -8], [-1, -12, 2, 4, 2, 4, -1, 2], [16, 4, -1, -7, 3, 0, -2, 6], [6, 12, 4, -4, 1, 1, 6, 5], [6, -1, 9, 3, -4, -5, -6, 8], [-11, 8, -8, 1, 7, -4, -4, 2], [3, -3, 4, -8, 1, 2, 5, 7], [14, -13, 0, 3, -3, 16, -5, -16], [12, 0, -3, -2, 13, 3, -6, 0], [1, 6, -2, 10, -7, -7, -3, 1], [20, -4, 2, 3, -9, 3, 1, -4], [-29, -1, 0, -4, 0, -1, -1, -2], [-3, -5, -6, -3, 2, -9, 3, -4], [-14, 27, -9, -5, 6, -7, 2, 0], [-6, 2, -5, -2, 15, 1, -8, 2], [2, 3, -2, 1, 2, -3, -2, 10], [17, 2, 8, 6, -15, -3, 0, 0], [-1, -6, 2, 6, 9, -1, 0, 4], [8, 9, -7, 2, -5, -3, 4, 10], [-18, 3, 6, -9, -4, -3, 0, 4], [18, 4, -6, -2, -4, -2, 6, 2], [15, 13, -1, -3, -10, 3, 2, 2], [6, 3, -4, 1, 10, 3, -4, 2], [-11, 13, -2, -3, 0, -1, -5, 6], [10, -3, 6, 5, 2, 9, -2, 10], [5, 7, -2, 1, 7, 6, -4, -6], [-13, -8, -8, 5, -5, -3, -3, -3], [10, -15, 6, 11, -8, 11, 0, 0], [13, -3, -8, 10, 0, 9, -7, -10], [8, -2, -5, -2, -1, -3, 4, 0], [-2, 8, -2, 2, -2, 2, -6, 2], [17, 8, -8, 1, -2, -4, 5, -8], [-12, -17, 4, -9, 9, 8, -7, -10], [-16, 1, 13, 3, -1, -2, 2, 2], [-8, -4, 13, 0, 1, 7, 2, -6], [-11, 2, -10, -4, 1, -7, 11, 5], [12, 7, 0, 7, -10, 1, 6, 2], [15, -5, -9, 5, -5, -6, -2, -2], [7, -12, 6, 4, 1, 7, 0, -10], [-20, 0, 2, -4, 2, 8, -8, -6], [-26, -15, 14, -3, 7, 0, -9, 8], [11, 6, -8, 3, -3, 5, -1, -9], [9, -5, -2, 1, 2, 1, -7, 2], [-23, 7, -6, 8, 7, -2, 1, -9], [-10, -20, 1, 0, -3, 3, 0, -6], [-25, -1, -1, 5, 5, 0, -6, -12], [-1, 10, 12, -2, -8, 0, 9, 4], [5, -13, -4, -9, 19, -2, 0, -2], [-1, 7, -4, 6, -12, 5, 2, -9], [18, 2, -5, 2, -8, -2, 8, 0], [-12, 11, 0, -7, 6, -13, 4, 6], [-12, -14, 11, -3, 11, 4, 0, 0], [3, 15, -7, 1, 5, -2, 2, -4], [-1, 9, -12, 3, -1, -2, 4, 0], [-4, 3, -4, 12, -5, 4, -3, -6], [6, -14, 15, -1, -1, 0, -8, 12], [-15, -7, -14, -2, -6, -1, 0, -13], [-8, 18, -16, 7, -1, -11, 1, 2], [-19, 10, 10, 3, -2, 2, -3, -8], [-18, -9, 8, 5, 6, 7, -8, 0], [16, -6, -7, -8, -7, -1, -2, 4], [1, 13, -20, -7, -5, -8, 8, 4], [-1, 0, -12, -2, 0, -6, 10, -1], [-11, 6, -10, -6, -3, -9, 2, 0], [-12, 1, -1, 1, -3, -6, 0, 6], [-1, 7, -12, 3, -6, -3, 1, -6], [2, 7, -6, -6, 11, -4, -5, 8], [-2, 8, -2, 2, -2, -2, 6, 12], [14, 5, 8, -1, -12, -1, 0, 4], [-21, -4, 5, 1, 14, 13, -6, -10], [-2, -2, -6, 8, -6, 0, -9, -3], [3, 6, -2, 0, 9, 7, -5, -11], [4, 0, -2, -8, -10, -2, 0, 2], [-30, 9, -7, 3, 14, -7, 0, -6], [7, 2, -13, 8, -2, -6, -4, 6], [2, -15, -4, 5, -4, 1, 4, 2], [1, 20, -8, 0, -6, -4, 2, -3], [-4, 8, 2, -3, -7, -7, 8, 1], [10, -6, 7, 0, -3, 15, 2, -2], [32, -12, 12, -3, 0, 2, 3, -3], [-39, -1, -3, 3, 10, 1, -6, -8], [22, 6, 4, -2, -5, -1, 2, 12], [3, -3, 2, -1, 1, -10, 6, -8], [-22, 6, 0, 11, 3, -3, -1, -8], [-7, 1, 1, -1, -10, -7, 8, 0], [-34, 2, 5, -6, 5, -9, -4, 6], [7, 10, -8, 2, -8, 0, 6, 1], [8, -2, -4, 8, -2, 0, 7, -3], [-6, 4, -1, -2, -11, 3, -8, 6], [-24, 4, 2, -5, 11, -5, -5, 4], [7, 13, 6, 3, -21, 2, 12, 4], [22, 2, 12, -6, 8, -4, 0, 2], [-18, 6, 5, 0, 11, 9, -2, -2], [-3, 6, 10, -6, 5, 7, -2, 10], [14, 17, 3, 9, -13, -6, 4, 6], [-14, 10, -12, 6, -3, -7, 8, 3], [-15, -5, 4, 0, -4, -11, -1, 10], [-18, 8, -2, -6, 16, 4, -2, 4], [35, 5, 0, 1, -1, 8, -2, 8], [-7, -14, 10, -4, 18, 0, -12, 3], [-11, 9, 8, -1, 11, 4, 2, 6], [-14, -9, 8, -5, 11, 10, -7, -2], [29, -7, -3, -3, -6, -1, 2, 6], [0, 3, 6, 2, 5, 6, -9, -8], [-3, 13, 0, 1, -4, -3, 1, 0], [-1, 15, 3, 3, -5, -2, 4, 0], [-20, -18, -2, 4, 6, -4, -4, 0], [9, -3, 11, -7, 4, 7, 0, -10], [26, -9, 9, -1, 4, 13, 2, -6], [-1, -11, -4, 0, 4, 3, 4, 13], [10, 6, 2, 3, -12, 5, -4, -6], [-4, -3, 8, 1, 12, 7, 4, 6], [-13, 5, -4, -7, -2, -7, 11, 0], [5, -6, -6, -5, 3, 8, -2, -4], [-4, -4, -6, 4, 11, 5, 1, -4], [-17, -4, 15, -3, -2, -5, 8, -2], [5, -6, -6, 1, 9, 1, -7, 5], [12, 6, -17, 8, -5, -13, 10, 2], [-6, -1, 4, -1, -1, 8, -9, 4], [-5, -3, 4, 1, 0, 3, 1, -10], [22, 1, 11, 5, -2, 1, -10, 2], [-3, 10, 7, -9, -10, -9, 8, 6], [4, 18, -6, 4, -6, -8, 2, 0], [36, 11, -12, -4, -5, 0, -1, 6], [-25, -14, 2, -4, 10, -4, -15, -10], [28, -21, 4, 1, -4, 9, 0, -2], [20, 3, -8, 2, -10, 10, 8, -8], [-4, 7, 12, -5, 2, 7, 4, -8], [-46, -16, 8, -1, 7, 5, -8, -1], [19, -11, 8, -3, 5, -4, -12, 2], [4, -11, -12, -1, -2, -5, 4, 2], [-8, 4, -10, -5, 7, 1, 1, -2], [-6, -8, 6, 4, 4, 12, -2, -2], [-16, -4, -10, -4, 14, -6, 0, -2], [0, -12, -7, 1, -3, -2, 6, 0], [8, -19, 8, 2, 5, 8, 3, -4], [16, 0, 14, 0, -2, 8, 0, 4], [18, 4, 4, 0, -10, -10, 8, 6], [-22, 0, 2, -3, -3, -9, 9, 8], [-17, 2, -12, 10, -7, -7, 6, 12], [-2, -3, -2, 3, 13, 12, -6, -16], [20, -3, -5, 9, -5, -2, 4, -14], [-4, 20, 2, -2, 7, -3, -3, 0], [-16, -22, 2, -1, 7, -1, 1, 0], [10, -6, -15, -10, 9, 3, -4, -8], [3, -6, -3, -4, 4, 2, -4, 14], [-18, -1, -7, -5, 6, -1, 10, -8], [-7, 8, -2, -2, 4, 4, -3, -6], [-18, 6, -14, -2, 11, 3, -6, 3], [-17, 9, 9, -11, -5, -8, 4, 8], [-6, -6, 2, 0, 18, 4, -2, 2], [-12, -2, -6, 0, 0, -8, 0, -4], [10, 3, -4, -6, 1, 0, 3, -10], [-8, -6, 20, -2, -2, 14, -9, -1], [17, -7, 3, 9, -6, 9, 2, -4], [23, 15, -8, 7, -8, -7, 1, 6], [-5, -2, 12, -3, 1, -9, -3, 19], [-24, 10, -15, 0, 9, -5, 0, -2], [6, 16, 1, 10, -7, -5, 4, 14], [5, -5, 2, -1, 4, -3, 0, 9], [-8, -3, 1, 11, -2, -7, 4, 2], [5, -13, 6, 5, 3, 6, -12, 2], [-14, -10, 2, 4, 6, 10, -6, -2], [2, -12, 8, -4, 13, 11, 0, 3], [10, -10, 6, -4, 8, 4, -4, -6], [-29, -1, 4, -10, -2, -9, 1, -2], [14, 11, -2, -1, -12, -13, 6, 12], [20, -8, -4, -5, 4, -7, 12, 4], [-15, -2, 5, -2, 0, 2, 4, -8], [-12, -10, -6, -2, 14, 0, -6, 2], [6, -9, 8, 7, 2, 3, -8, 4], [-19, -1, 6, -3, 2, 7, -6, 5], [19, 4, 16, 0, -4, 0, -2, 15], [-15, -19, 18, -6, 6, -5, -2, 5], [-4, 1, 15, 3, -11, -2, -6, 6], [-2, 12, -3, 0, -8, -12, 12, 8], [0, -3, 1, 3, 7, -8, 2, 6], [7, 13, -12, 5, -8, -11, 9, 16], [-53, -6, -2, -2, 8, 2, 2, 1], [-6, 5, 10, 3, 4, -5, 2, 12], [53, 8, -1, -2, 0, 2, 2, -4], [-20, -4, -4, 3, -7, -13, 5, -2], [2, 13, -15, 11, -13, -2, 6, -12], [29, -3, 12, 2, -12, -3, 10, 1], [-26, -20, 3, 0, 7, 3, -16, 4], [30, 23, -5, 0, -19, 3, 16, -2], [0, -6, 11, 4, -1, -1, 8, 0], [-6, -21, 18, 2, 17, 18, -9, -12], [15, 3, 8, 0, 1, 2, 9, -9], [1, 4, -10, 0, 7, 3, 2, -8], [-6, 22, -6, 0, 2, 2, 12, -4], [8, 4, -8, -3, 0, -5, -4, 0], [1, 1, 1, 5, -12, -11, -8, 8], [3, -10, 4, -8, 4, 8, 4, -9], [12, 7, -18, 5, -6, -17, 0, 10], [-7, 9, 0, -3, 10, 11, -9, -18], [-11, 27, -8, 10, -6, -3, 9, 6], [26, 6, 4, -12, 4, -6, 10, 14], [-12, -8, -18, 3, 10, -5, -6, 2], [22, -17, 18, -9, 1, 2, -4, 0], [-16, 2, -4, 2, 18, 8, -6, -14], [-34, 6, 0, 8, 0, 6, 4, -2], [-32, 6, 2, 4, -16, -10, 0, 14], [-12, 3, 0, 11, -8, -1, 6, -4], [26, 10, 1, 2, -7, 3, 12, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3381_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3381_2_a_be();".
// 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_3381_2_a_be(:prec:=8)
    chi := MakeCharacter_3381_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_3381_2_a_be();".
// 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_3381_2_a_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3381_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![80, 37, -141, -35, 75, 11, -15, -1, 1]>,<5,R![-1200, -2013, 291, 933, 81, -127, -21, 5, 1]>,<11,R![-4384, 3056, 2704, -1864, -304, 256, -4, -10, 1]>,<13,R![2127, -2370, -4602, -930, 660, 158, -38, -6, 1]>],Snew);
    return Vf;
end function;