
// Make newform 3381.2.a.bl in Magma, downloaded from the LMFDB on 25 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_bl();".
// 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_bl();".
// 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 := [14, -36, -33, 108, 26, -100, 1, 36, -6, -4, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -1, 1, 0, 0, 0, 0, 0, 0, 0], [12, -9, -37, 12, 22, -7, -3, 1, 0, 0], [-28, 26, 91, -40, -82, 27, 28, -9, -3, 1], [37, -26, -123, 45, 103, -33, -31, 10, 3, -1], [46, -25, -176, 30, 173, -30, -63, 15, 7, -2], [-70, 30, 271, -26, -266, 34, 98, -21, -11, 3], [108, -67, -385, 83, 362, -69, -128, 31, 14, -4], [-149, 106, 514, -136, -466, 103, 159, -41, -17, 5]];
    Rf_basisdens := [1, 1, 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_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, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, -1, -1, 0, 0, 1, 0], [-1, 1, 1, 0, 0, -1, 0, 0, 1, 0], [1, 1, 0, -1, 1, 0, -1, 0, 0, -1], [2, 1, 1, 0, 0, 1, 1, 1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -2, -1, 1, 0, 1, 1, -1, 0, 1], [2, -1, -2, 0, -1, -1, -2, -2, -1, 1], [0, -1, -2, 0, 0, -1, -1, -2, -1, 1], [1, 0, 2, 2, -2, -1, 2, 3, 1, 1], [-2, 1, 0, -1, 2, -1, 1, -1, 2, 1], [0, 2, -1, 1, -1, -1, 2, 0, -1, 2], [1, 1, -1, -3, 0, 1, -2, -1, -1, 1], [4, -1, 0, 2, -1, 1, 0, 2, 2, 0], [1, 1, 1, 0, 0, 0, 1, 0, 0, -2], [1, -3, -1, 1, -2, 1, -2, 0, 1, 0], [2, -2, 2, -3, 3, 1, 1, 0, 1, -1], [2, -1, -3, 0, 0, 1, -2, 0, 0, 1], [-1, 0, 2, 2, 1, 0, 3, 3, 0, -1], [4, 1, 0, -2, -1, 0, -1, 1, -2, -2], [4, -2, -2, 1, 0, 1, -2, 0, 1, 1], [0, -3, -2, 3, 0, 0, 3, -1, 3, 4], [-1, 3, 1, 1, -1, -3, 1, 1, 3, 1], [2, 1, -3, -4, 0, 2, -3, -2, -2, -1], [2, -3, 0, -3, 3, 2, -3, -2, -1, 2], [-1, -1, 1, 1, -2, 1, 2, 4, 1, 2], [1, 0, 3, -1, 1, -1, 2, 1, 2, -2], [-4, 2, -1, 0, -2, -3, 0, -3, -1, -2], [5, -4, 0, 0, 2, 1, -2, -1, 1, 1], [4, 2, 4, -1, -1, -1, 2, 4, -1, -2], [6, -1, 1, 0, 4, -1, 2, 1, 1, -3], [1, -1, -3, 1, -1, -2, -3, -1, -2, 0], [-5, -1, -1, -3, -1, 0, -3, -5, -2, 0], [5, 1, -1, -1, 0, 2, -2, -1, -4, 0], [-6, 2, 0, 2, 1, -2, -4, 0, 2, -3], [6, -2, -3, -7, 1, 1, -7, -2, 0, 0], [-1, -2, -3, -1, 1, -2, -2, -1, -4, -1], [7, -3, -3, 2, 0, 1, 1, 4, 0, -2], [7, -7, -2, 0, 2, 0, 2, -2, -3, -3], [-4, 1, 2, 2, 3, -4, 4, 0, 1, 1], [-3, -2, -5, 0, -1, 1, 4, -1, 1, 5], [3, -4, 2, 2, -3, -3, 1, -1, -3, -2], [3, 2, 1, 3, 0, -2, 2, 5, 3, -2], [-3, -2, -4, -2, -4, 2, 0, -4, -2, 2], [-4, 1, 2, -2, 3, -1, 9, 1, -2, -1], [3, -2, 5, 1, 4, -2, 8, 4, 3, 3], [3, -5, -5, 0, -2, 1, -3, -5, -2, 5], [8, 2, -2, -4, 1, 4, -8, 1, -6, -6], [1, 0, -1, 1, -3, 2, 0, -2, -1, 3], [11, -3, -5, 1, -2, -1, -4, -4, -4, -3], [6, 1, 2, -3, -1, 1, 1, 2, -5, -5], [-2, -4, -5, -3, 3, 3, -2, -2, -1, 2], [7, 0, -1, 4, 2, 1, 5, -1, 0, 1], [8, -7, 1, -3, 1, 2, -1, -2, -4, -2], [5, 2, 2, -4, 4, 0, -3, 3, -2, -1], [0, -2, -7, 0, -2, 1, 0, -5, -3, -2], [3, 2, 1, 1, -4, 0, 1, 3, 1, 1], [6, 2, 0, -5, 0, 1, -15, -4, -4, -3], [-7, -7, 2, 6, -5, 5, 5, 5, 3, 6], [6, -1, 3, -2, -2, 1, 5, 0, -2, -1], [1, -5, -1, -5, 2, 7, -2, -5, -2, -1], [4, 1, 9, -2, 2, 0, 1, 2, -4, -3], [8, -4, 4, 2, 1, -2, 4, 5, 2, -2], [-4, -4, -3, 3, -1, 1, -1, 2, -1, 4], [-2, 7, 9, -1, -3, -5, 0, 5, 5, -1], [-1, -1, -5, -7, 3, 2, -4, -10, -7, 1], [1, -6, -8, -2, 0, 6, -3, -4, 3, 5], [3, -8, -5, -5, 3, 0, -4, -1, -2, -7], [5, -8, -5, -3, 0, 7, -4, -9, -4, 2], [-1, -8, 1, -1, 4, 1, -2, -2, -3, 1], [-10, 9, 2, 4, 0, -1, 3, 6, 5, 3], [-3, 11, 4, 2, -5, -5, 5, 5, -1, -4], [4, 1, 7, 2, 5, -2, 3, -1, -1, 2], [-14, 7, 4, 3, 4, -5, 12, 2, 6, 3], [0, 0, 6, 8, -6, -2, 4, 10, 4, 0], [13, 6, -3, -2, 2, -1, -1, -3, -4, -1], [3, 2, -1, -2, -5, 2, 1, 3, 1, -3], [7, 3, -1, 6, -5, -3, 1, 2, 0, 0], [-6, 7, 5, -3, -1, 1, -5, 7, -3, -5], [8, 0, -4, -3, -3, -4, -1, -4, -4, 3], [-4, 1, 3, 0, 4, -2, 3, 2, 2, 1], [-4, 3, 5, 8, 0, -3, 10, 2, 6, 7], [-4, 12, 2, -6, 2, -2, -2, 2, 0, -6], [1, -4, -2, 7, -5, 0, 2, 6, 4, 5], [-6, 2, 3, 1, -3, 0, 4, 1, 4, 4], [2, 3, 11, -3, 3, -4, -4, 0, -3, -6], [3, -1, -4, 1, 0, 3, -7, 2, 3, 1], [5, 5, 2, -3, -1, -2, 3, -4, -2, 3], [-4, 12, 8, -1, -2, -5, -2, -3, 1, -3], [-12, -1, 11, 1, 7, -2, 3, -3, 4, 2], [4, -6, -3, 1, -1, 1, -9, 2, -4, 1], [-6, 10, 5, 1, -4, 1, -3, 4, -5, -3], [11, -7, -9, -8, 3, 1, -9, -8, -6, -6], [-1, 0, 2, -8, 7, -1, -1, -3, 4, -3], [1, 3, -4, 8, -7, -1, 7, 3, 3, 6], [7, -3, 1, 1, -4, 2, 0, 3, 2, 0], [0, -4, -1, -5, 5, -1, -11, 0, -3, -6], [5, -9, 1, -1, 2, 2, -9, 5, -4, -3], [7, -4, -5, 5, -3, -2, 3, 11, 1, 5], [4, 1, -4, -8, 7, 2, 1, 0, -2, -2], [12, 1, -4, 0, -4, 0, -10, 2, 3, 3], [-3, 4, -1, -2, -6, 6, 3, 3, -1, 1], [-8, 8, 9, 4, -5, 0, 2, 0, 1, 2], [-6, 9, 4, 0, 2, -1, -5, -2, -1, -8], [3, 0, -7, -3, 4, 0, -11, -11, -1, 1], [-7, 4, 3, 2, -3, -1, 9, 1, 2, -4], [0, 3, 4, -4, 3, 1, -2, 9, 2, -7], [-8, 8, 11, 2, 1, 0, 1, 3, 4, 0], [3, 8, 9, -8, 6, -2, 9, 3, -7, -7], [0, 7, -4, -3, 7, 0, -5, 2, 1, 4], [3, -7, -4, 13, -8, 1, -1, -2, 1, 6], [6, 3, -2, -12, -6, 3, -19, -2, -7, -1], [3, -4, -1, -2, 0, 4, 0, -3, 5, 2], [4, -4, -3, -6, 4, -3, -3, 1, 1, -4], [-5, 9, 0, -4, 2, 0, -13, -1, -8, -7], [-3, 8, 2, 1, 3, -4, -8, 2, 2, 3], [-2, -5, -2, -3, -5, 7, -10, 0, 2, 1], [-2, -2, 3, -5, 0, -2, 2, -5, -5, -3], [-11, 1, 0, 5, 6, -7, 7, -6, -3, 3], [-1, 0, -9, -2, -6, 7, 0, 5, -8, 4], [1, 6, -1, -4, -2, 1, 3, 4, 4, -2], [3, -1, 2, 4, 3, -5, 3, -4, 8, 8], [9, -3, -15, -7, 0, 3, -10, -3, -5, -3], [-7, -4, 7, 5, 0, -6, -2, 0, 4, 3], [4, -4, 0, 6, 0, 0, 1, 0, 11, 4], [-5, 12, 5, -2, -2, -4, 3, 0, 5, -2], [6, 2, -6, -8, 6, 6, -6, -2, -8, -2], [-14, 5, 9, 7, 1, 2, 13, 9, 8, 6], [-2, 5, 0, 8, -7, -1, -11, 2, 2, 8], [-5, -3, -2, 7, 3, -3, 9, -6, -2, -2], [-9, -4, 6, -6, 5, 6, 1, 3, 0, -1], [-9, 4, 2, 6, -2, -1, 3, 5, 2, 5], [0, 3, -4, 1, 0, -1, -7, 4, 1, -5], [-5, 4, -5, 4, 2, 1, 0, -4, -2, -2], [-7, 2, 1, 11, -8, -2, 6, 5, 11, 2], [-11, 12, -4, -5, -2, 4, -10, 5, 0, 1], [2, 1, -3, -2, 4, -4, 0, -12, 3, 0], [10, -5, -8, -4, -5, -5, -3, -14, -7, -1], [0, 3, 3, 5, -5, 1, -13, -4, -2, 1], [-2, 9, 8, -1, -1, 5, 11, 10, -1, 6], [12, 1, -4, 0, 8, -3, -5, 2, -4, 0], [5, 8, -2, -14, 2, 6, -4, -3, -8, -7], [-13, 1, -11, 11, -7, 0, 3, 3, 3, 13], [-3, 3, -3, -4, 2, 9, -10, -1, -4, -4], [-5, -4, 4, 2, 4, 1, 9, 3, -6, -5], [22, -14, -12, 0, 4, 0, -7, -4, 0, 0], [-1, 0, -3, 5, -4, 6, 0, 3, -1, 9], [-14, 9, -1, -2, -5, 0, 0, 0, 2, 3], [5, -2, 12, 9, 0, -2, 16, 13, 8, 7], [-10, 5, 2, -2, 9, 3, 4, 3, 3, -3], [1, 1, 2, 8, -8, -2, 10, 1, 3, 5], [12, -14, -13, -5, 9, 0, -14, -11, -2, -4], [-10, 16, 8, 8, -11, -8, -4, 6, 0, -6], [-1, 4, 7, -4, 0, -4, 12, 5, -1, -5], [-7, 15, -4, -6, -4, 0, -1, 3, -4, 3], [-22, 5, 9, 6, -2, -3, -3, 5, 10, -2], [-13, 9, 9, -4, 7, -1, 7, 7, 6, 4], [-10, 1, 10, 5, -1, 5, 15, 4, -3, -3], [17, 16, 7, -2, -5, 2, -2, 7, 1, -10], [-3, -17, -10, 16, 0, 3, 4, 1, 3, 6], [17, 6, 4, 4, -8, -3, -12, -3, 1, -4], [1, 4, 15, 6, 2, -4, 8, 7, 3, -4], [5, -14, -11, -2, 2, 5, -10, -8, -6, -2], [-10, -4, -11, 4, -2, -3, 4, -3, 5, 2], [-2, -13, -5, -4, 4, -3, 3, -14, -4, -5], [-3, -5, -5, 5, 4, 3, -6, -2, -5, 0], [10, 9, 8, 4, -7, 3, 5, 18, 3, -6], [-9, 10, -1, -12, 4, 10, -12, 1, -15, -6], [-19, 11, 9, 0, 2, -3, 4, 3, -2, -2], [-1, 2, 9, 10, -1, 0, 12, 3, 3, -8], [-1, -12, -3, -2, 2, -8, 7, -7, 3, -7], [1, 11, -12, -2, -5, 5, -4, 1, 1, -2], [15, -4, -2, 16, -14, -1, -6, 6, -1, 0], [5, -4, -9, -5, 6, 3, -1, -8, -13, -1], [3, 2, 1, -3, 7, 0, -2, 0, 5, -3], [16, -6, -1, 4, 4, 2, 11, 0, 1, 7], [5, -8, -8, 8, -6, 9, -4, -1, -4, 2], [-8, -7, 2, -5, -1, -3, 5, -8, 5, -2], [17, -5, -1, 7, 3, 4, 3, -5, 5, 7], [-5, -8, -2, -13, 13, 0, -1, -3, -2, -10], [-17, -3, 11, -1, 1, 5, 2, -3, -2, -1], [-3, 14, 1, -2, -4, -3, 1, 1, 0, -3], [12, 8, -4, 3, 0, -1, -2, -4, -1, 7], [-10, -12, 3, 7, 1, 7, 5, 4, 1, 6], [15, -2, 12, 5, 8, -5, 7, 4, -2, -4], [-10, 11, 4, 13, -11, -5, 1, 2, 9, 16], [1, -8, -3, 7, 0, -14, 0, -13, 0, -3], [1, -8, -10, -12, 8, 8, -3, -2, -5, 7], [15, 0, -1, 5, 1, 10, -4, -6, -5, 1], [-8, -6, -15, -6, -5, 2, -22, -2, -11, 2], [2, 0, -15, -7, -1, -5, -3, -4, -6, 4], [8, 0, -10, -9, 13, 6, -13, -10, -14, -3], [17, -22, -9, -12, 8, 13, -6, -13, -12, 4], [13, 2, -8, -9, 1, 2, -15, -4, -11, -13], [17, -19, -6, 2, -7, 9, -15, 5, -7, -10], [-25, 7, -2, 4, -5, 1, -11, -7, -3, 8], [-4, 9, 6, 8, 0, -7, 8, 8, 12, -4], [-7, 1, 5, 3, 8, 4, 5, -3, 4, 4], [2, -6, 6, -2, -6, 6, 7, 4, -4, 6], [-3, 4, -5, 6, -1, -6, -18, 4, -4, -5], [18, -5, -5, 9, 0, -2, 10, 9, -2, 9], [4, 4, 10, -5, 6, -1, 6, -5, -7, -1], [-7, 4, 8, 0, 3, -5, 7, 5, 2, -3], [26, -10, -4, 10, -10, -1, -8, 9, 4, 7], [9, -2, -2, 9, -7, 2, 14, 12, 2, 11], [9, 1, 11, 8, 2, -5, 9, -8, -2, 6], [12, -11, 2, 12, -13, 3, 5, 20, 7, 9], [19, -12, -9, -2, -9, 7, -20, -3, -11, -9], [-2, 9, 2, -4, 10, -5, 6, 0, 2, 0], [-8, 3, 11, 0, -5, -3, -2, 4, 4, -5], [3, 18, 1, -5, 4, -4, 6, -13, 3, 5], [12, -13, 5, -9, 8, 4, 5, -2, -6, -2], [-2, -5, -4, 2, -2, -4, 0, -6, -1, -8], [-8, 13, 2, -14, 10, -7, -10, -7, -7, -4], [10, 1, -3, 7, -10, 6, -17, 0, -4, 0], [7, 1, 3, 10, -8, -9, 19, 3, 9, 2], [-13, -1, -5, 3, 11, -4, 18, -6, 5, 3], [0, 5, -9, 7, -6, 2, -9, 1, 14, 7], [14, 10, 5, 1, 0, 7, 0, 10, 5, -2], [6, -6, -18, -3, -9, 5, -13, -4, 1, 3], [8, 12, 1, -2, -9, -6, 12, 2, -1, -10], [22, -10, -7, 15, -15, -1, -17, 6, 2, 2], [-21, 4, 0, -7, 3, -8, 1, 3, 4, -6], [10, -9, -4, -8, 7, 4, 9, 5, 2, -4], [9, -1, -4, -6, 7, -7, -11, -3, -5, -4], [-14, -5, -2, -12, 2, 7, -20, -11, -7, -3], [-15, -13, -6, 1, 3, 0, 2, -6, 2, 13], [0, -7, -8, 2, 6, 11, -10, -7, -3, 0], [7, 12, -8, 0, 2, -6, 18, -4, -2, 9], [4, -8, -4, -7, 18, 6, -13, -5, -7, -2], [4, 3, 11, 21, -11, -3, 15, 18, 6, 9], [-4, 6, 1, -6, 8, 5, 1, 3, 11, 6], [-11, 6, 2, 10, -6, -1, 10, 5, 3, -4], [-19, 6, 7, 0, 6, -13, 3, -3, -5, -7], [-4, 7, -8, -9, 0, 5, -34, 0, -2, -3], [-3, 26, 10, -10, -2, 1, 1, 16, 1, -16], [-19, 0, 13, -7, 13, 4, 6, 1, 6, -3], [-19, 2, -11, 4, 3, -5, -9, -3, 4, 11], [10, -5, 5, -6, -2, 7, 5, -3, -9, -5], [-16, 6, 2, -4, -8, -2, 3, 2, 2, -8], [-12, 15, 14, -12, 0, -1, 2, -9, -3, -2], [-18, -8, -1, 11, -7, -3, -5, -6, 6, -4], [3, 13, 3, -5, 1, 1, 14, -1, 2, -3], [-2, 4, 5, 20, -9, -10, 13, 12, 5, -4], [6, -3, -7, 5, -9, -5, -3, -8, 4, 7], [31, -10, -12, 6, 9, 2, 2, -7, -8, 5], [15, -4, -9, -4, 7, 1, -13, -11, 2, -1], [-12, -1, -2, 2, 0, 13, 25, 6, -1, -3], [0, -5, -8, -9, -3, -3, -5, -12, -3, -5], [-8, -10, 2, 9, -6, -3, -14, 10, -5, -6], [-3, 7, -3, -3, -6, -1, -18, 2, 0, 1], [-3, -4, -3, 7, 3, -8, -6, -1, 6, 9], [0, 11, 3, 3, 1, -5, -21, 8, 2, -3], [4, -7, 3, 2, -1, -3, -2, -14, 2, -3], [5, -16, -9, -2, 2, 6, 7, -8, 1, 16], [-1, 19, 9, -1, 8, -3, 32, 3, 2, -3], [-21, 6, 0, 10, 2, -2, 3, -5, 2, 1], [20, -23, -5, 3, -5, 1, -9, -1, -11, -7], [-1, -1, 1, -5, -1, 2, 15, -3, -8, -4], [8, -5, 0, -8, 1, 5, 5, 1, -4, 1], [4, 4, -5, -1, 8, 4, 0, 5, 9, -3], [-3, 3, -1, 11, -3, -5, 14, -5, 8, 7], [24, -8, -9, -1, 8, -8, -10, -1, 3, -1], [2, -13, -2, -2, -4, 6, 0, 0, -1, -5], [-11, 1, -9, -17, 5, 0, -15, -20, -13, -1], [0, -5, 1, 16, -7, -3, 21, 8, 12, 17], [-3, 14, 8, 11, -5, -2, 12, 8, 0, 1], [-6, -21, -3, 0, 0, 1, -11, -4, -2, 7], [15, -6, 2, 4, -9, 8, -10, 17, 4, -1], [-1, 10, 10, 4, 8, -4, 12, 4, 20, 1], [16, -6, 5, 16, -5, 6, 7, 2, 3, 0], [3, -16, -12, 0, -6, 8, -6, 3, -4, 11], [-5, -3, -4, -4, -11, 1, -23, 7, -3, -2], [4, -5, -5, 0, -19, 6, -16, 1, 1, 0], [8, -7, -2, 16, -8, -3, 20, 9, 1, 12], [-7, -10, 3, 12, 2, -10, 22, 3, 3, -7], [9, -8, 3, -10, 14, 9, -4, 4, -10, -8], [12, -12, -11, -13, -3, -3, -12, -18, -15, -4], [16, 3, -8, 17, 0, 5, 14, 15, 12, 13], [21, -10, -14, -4, 4, 1, -8, -10, 7, 4], [-39, 6, 10, -11, 4, 1, -7, -2, 6, -6], [4, 13, 7, 6, 4, -5, 0, 17, 5, -3], [-8, 7, -2, -2, 3, -2, 26, -3, 4, 4], [3, -8, -17, -10, 12, 6, -15, -11, -9, 1], [2, 15, 18, -8, 4, -2, 14, 6, -5, -7], [10, 0, -9, -20, 9, -2, -18, -13, -18, -14], [8, -4, -18, 0, -8, 4, 0, -14, 2, 10], [-28, 4, 8, -3, 17, -2, 11, 2, 0, -1], [9, 4, 8, 6, -14, 3, -6, 17, 1, -6], [5, -9, -6, -7, 8, 6, -8, -21, -8, 3], [35, -31, -14, -3, 7, 4, -9, -14, -10, 1], [0, -6, 2, 0, 8, -2, 16, 0, 0, 10], [9, 6, 19, 16, -14, -5, 16, 20, 0, -10], [12, 10, -8, -6, -2, 4, -8, 8, 0, -6], [-6, -7, -12, 12, 0, 3, 19, 4, 7, 0], [-1, 12, -8, -22, 10, 5, 0, -13, -9, 7], [1, 6, -4, 20, -20, 0, 11, 3, 6, 1], [-6, -5, 4, 14, -14, 5, 0, 11, 11, 7], [-5, 8, 3, 7, -6, -6, 21, 1, 5, 11], [25, -16, -6, 0, 1, 4, 5, -5, -4, 1], [-9, 0, -5, -5, 11, -2, 14, 6, 7, 5], [-8, -10, -3, 2, 7, 0, 16, 0, 5, 22], [-25, -1, 4, 1, 8, -5, 33, -6, -1, 3], [1, -6, -9, 3, -3, -2, -12, 10, -5, -1], [35, -10, -11, 4, 0, -10, -3, -4, -7, -2], [-3, -1, 3, -3, 6, 6, -1, 3, 0, -14], [-9, 3, -8, -11, -1, 1, -13, -18, -9, 6], [-20, -16, 1, 9, 1, -5, 9, -8, 12, 18], [20, 1, -4, 1, -8, -1, 2, 4, 4, -5], [5, -19, -19, 1, 1, 1, -30, -17, 1, -1], [5, 9, 15, 4, 1, -8, -4, 14, 6, -11], [29, -20, -11, 0, 4, -5, -9, -11, 0, -1], [13, 3, -15, 5, 4, 0, 2, -13, -10, 2], [19, 12, 20, 18, -13, -11, 18, 15, 5, -1], [-6, -14, -9, -5, -8, 7, -11, -9, -6, -7], [7, -8, -3, -4, 10, 4, 11, 2, -9, 2], [20, 1, 13, -1, -3, 1, -7, 6, -4, -7], [-6, -1, -1, 4, 6, 4, -5, -12, -2, 9], [-9, 26, 28, 15, -3, -8, 9, 34, 13, -5], [-20, -13, 4, -2, 8, -1, 30, -10, 2, 2], [-6, 0, 4, -1, 2, 12, -9, 9, 12, 5], [-17, 4, 3, 12, -2, -7, -1, -5, 8, 7], [2, 7, 3, -7, -12, 5, 8, 4, -2, -14], [22, -3, -15, -8, -7, -5, -17, -12, -12, 5], [0, 9, -3, 14, -14, 3, -10, 9, 3, 11], [7, 13, 1, -15, 10, 2, 21, 1, -2, -10], [13, -15, -1, -3, -4, 10, 5, 3, -2, 4], [15, -15, -18, -10, 6, -1, -6, -15, -9, -10], [7, 1, 1, -17, 3, 4, -30, -2, 1, 5], [33, -3, -4, -16, 1, 7, 1, -2, -2, -2], [-5, 9, 11, 15, -10, 3, 19, 5, 15, 5], [11, 3, -8, 12, -23, -3, -22, 7, -1, 10], [2, -4, 5, 19, 11, -9, 40, 4, 15, 2], [-13, 9, 1, 4, -12, 11, -4, 3, -6, 6], [-8, 3, 3, 17, 7, -3, 19, 8, 12, -1], [4, 21, 14, -2, -1, 5, -1, 11, -6, 1], [4, -6, -10, 7, 6, 5, -19, -14, 2, 11], [-6, 11, 15, -9, 9, 6, 15, 3, 0, -18], [4, -7, 2, -10, 15, -2, 0, 2, -1, 5], [-39, 19, 13, -5, 3, -1, 0, 7, 4, -9], [19, -14, 9, 12, 2, -6, 9, 5, 7, -7], [2, -5, 0, 0, -5, 12, 14, 0, -11, 5], [14, -14, -7, 3, 9, -3, 7, -2, 6, -1], [-19, 14, 5, 26, -19, -8, 23, 18, 16, 13], [-7, 14, 0, -1, 7, 4, 13, 7, 2, 6], [19, 13, 14, -12, 9, -7, 17, 8, 1, -14], [-14, 1, 7, 18, -5, 0, 3, -3, 9, 14], [-2, 4, 4, -12, 6, 6, -15, 1, -11, -14], [16, 1, -8, -12, -6, 11, -38, -5, -5, -6], [-33, 19, 12, -7, 1, -6, 4, -8, -4, -7], [16, -4, -7, 14, -4, 8, 17, 14, 11, 3], [-8, 11, -8, 6, -4, -3, 4, 10, 9, 3], [-18, -6, -1, -7, 3, 9, 2, -10, 9, 0], [11, -4, -21, -3, 2, 4, -2, -17, -3, 9], [20, -8, -2, -11, 0, 15, -12, 5, 1, -1], [24, -21, -14, -10, -1, 8, -16, -13, -26, -2], [-7, 4, -8, 12, -6, -9, -13, -2, -13, 0], [-22, -7, -5, 16, -5, -7, -4, -6, 6, 15], [-35, 12, 16, 12, 2, -9, 18, -3, 9, 7], [13, -2, -4, 6, -4, -1, 7, 8, -3, -6], [-11, 5, 5, 7, -4, 0, 1, 23, 0, -10], [-6, 0, 0, 6, 6, -14, 15, -6, 6, -6], [-1, -28, -3, -14, 7, 5, -26, -3, -9, -3], [-33, 5, 5, -1, 2, 5, 4, 13, 8, 1], [19, -7, 10, 10, -7, -3, 14, 13, 15, 0], [18, -5, -11, 21, -9, 7, 5, 15, 15, 7], [-16, -16, 23, 9, -9, 1, -1, 24, 4, -9], [-6, 9, 2, 6, -9, -3, -13, -12, -1, 3], [11, 11, 25, 8, -3, -3, 25, 7, 6, -2], [-9, -13, 1, -11, 15, -8, -7, 1, -10, -12], [-31, 2, 13, 4, 0, -9, 5, 5, -5, -7], [23, -10, -9, -1, 7, 9, 7, -4, -9, -8], [8, -5, 1, 25, -2, -2, 30, 4, 19, 14], [-3, -17, -15, -3, -9, 3, -38, -11, -1, -11], [-1, -18, 19, 0, 14, -1, 3, -3, 5, -7], [18, -3, -3, 1, -7, -1, 7, 14, -14, -3], [10, 3, -4, 4, -13, -19, 1, -4, -1, -1], [-26, -7, -3, -8, -11, 8, -17, -15, -5, -8], [-15, 1, 3, -2, 5, 0, 20, 5, 2, -5], [-6, 29, 15, 2, -5, 0, -10, 16, 8, 3], [-13, 18, 3, 4, 1, -11, 5, 3, 8, -2], [13, -7, -14, -9, 8, 4, 5, -11, -1, 9], [-14, -11, -14, 8, 8, 9, 3, 2, -9, 14], [2, 9, -5, 19, -8, 2, 23, 13, 6, 11], [-16, 22, -7, 4, -15, -2, 12, -2, 1, 14], [10, -3, -10, -20, 6, 3, -24, -14, -17, -7], [11, -17, -9, 28, -19, 7, -1, 13, 4, 16], [10, 3, 11, 14, -9, -9, 20, 12, 16, 5], [-29, 15, 1, -3, 5, 7, -5, 3, 5, -3], [6, -12, -7, 16, -4, 10, 8, 4, 11, 12], [-16, 6, 16, -14, 8, 8, -17, 14, -14, -18], [-15, -4, 13, -6, 7, 4, 28, -5, -13, 6], [-24, -4, -2, -7, 0, -16, 6, -1, 1, 6], [-2, 15, 7, -22, 14, 3, 9, -1, -10, -2], [-9, -4, -11, -14, 4, 15, -14, -6, -18, -10], [15, 10, 24, 11, -9, -10, 24, 12, 9, -3], [18, -1, -22, -22, -7, 4, -22, -9, -24, -12], [-21, 12, 17, 6, 7, -8, -18, 11, 11, -9], [-6, 10, 6, 5, -13, -15, -21, -12, -11, -9], [14, -2, -4, -8, 6, -2, 39, 2, 0, -2], [-25, 7, 8, -7, 13, 8, 15, 9, 2, 7], [17, 7, 3, -13, 7, 15, -6, 9, -12, -9], [-6, 6, 7, 4, 7, -10, 11, -15, 18, 6], [6, 10, 3, 8, -3, 4, -18, -10, -11, -2], [-12, -14, -10, -22, 16, 16, -34, -22, -2, -2], [6, 6, 10, 4, -12, -6, -2, 8, 16, 0], [15, 5, 2, -14, 2, -4, -5, -3, -4, -15], [-9, 7, -18, 0, -3, 1, -1, -12, -2, 8], [22, -2, 27, 6, 8, -6, 22, 20, 19, -12], [14, 1, 2, -12, -2, 20, -8, 0, -11, 5], [12, -11, 7, -3, 7, -11, 15, 7, -1, -11], [-36, 15, -2, 9, -4, -5, 15, 3, 10, -3], [-29, -8, -7, -3, 1, -6, -15, 1, -5, -17], [-4, -4, 4, 11, 4, -15, 17, -1, 23, 13], [-19, -22, -6, 8, 8, 6, 10, -1, -8, 11], [-15, 15, 17, -13, 13, 2, 7, 14, -3, -13], [19, -21, -9, -16, 15, 3, -27, -3, -12, 4], [-20, -11, -4, -7, -2, -5, 14, -18, -12, -5], [-25, 7, 1, 3, -18, -7, 0, 0, 0, 5], [21, -12, -2, -17, 13, 8, -20, 8, -10, -11], [13, 7, -8, -18, 6, 6, -38, -11, -17, -13], [-25, 7, 3, -18, 2, 9, -16, -14, -15, -4], [-21, -6, 2, 16, -7, -5, -3, 9, 6, -15], [20, -14, -16, 11, -1, 4, 9, 10, 0, 9], [27, -1, -8, 8, -17, -13, -1, -1, -5, 14], [21, 8, -16, -22, 12, 4, -26, -20, -9, -3], [13, 19, -9, -9, -3, 3, -12, -1, -6, -1]];
    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_bl();".
// 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_bl(:prec:=10)
    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_bl();".
// 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_bl( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3381_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![14, -36, -33, 108, 26, -100, 1, 36, -6, -4, 1]>,<5,R![392, -672, -377, 976, -24, -428, 71, 72, -16, -4, 1]>,<11,R![-12953, -28550, 8629, 19900, -6314, -2936, 1038, 140, -57, -2, 1]>,<13,R![5598, -1104, -12535, 10904, -350, -2584, 747, 96, -52, 0, 1]>],Snew);
    return Vf;
end function;