
// Make newform 3381.2.a.bh 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_bh();".
// 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_bh();".
// 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 := [-2, -28, -17, 100, 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], [-220, 1, 441, 193, -202, -161, 44, 35, -4, -2], [48, 163, 114, -783, -73, 453, -19, -76, 6, 3], [-66, 5, -4, 166, 353, -194, -203, 81, 27, -10], [-35, -535, -324, 850, 581, -486, -228, 122, 25, -11], [-20, -346, -165, 996, 426, -647, -184, 157, 21, -13], [-42, 627, 194, -1471, -318, 855, 140, -192, -17, 15], [-54, -295, 1129, 358, -1745, 166, 744, -173, -89, 26]];
    Rf_basisdens := [1, 1, 1, 47, 47, 47, 47, 47, 47, 47];
    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, 1, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, -1, 0, 0, -1, 0, 1, 0], [-3, 1, 0, 0, 1, 0, 1, 0, 1, 1], [0, -1, 0, -1, -1, 0, 0, 1, 0, -2], [-4, 1, 0, 0, 1, 1, 1, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -2, -2, 1, 0, 0, -1, 0, 0, -1], [-5, 3, 1, 0, 1, -1, 1, 0, -1, 2], [1, -1, 1, -2, -1, 0, 1, 1, -1, -2], [-2, -2, 1, -1, -1, 1, 1, -1, -2, 0], [-1, 1, 1, 1, 0, 0, -1, 0, 2, 1], [2, -2, 0, -1, -2, -2, -3, 2, 1, -2], [-2, -1, 0, -2, -2, -1, -1, -2, 2, -1], [-6, 3, 1, 1, 1, 0, -1, 1, 1, 1], [-1, 1, -1, -2, -2, 3, 2, -4, 0, -2], [-5, 3, 4, 1, 2, -2, 1, -1, -1, 4], [-1, 2, 1, -1, 0, 0, -1, 1, 3, 1], [-1, -1, 2, -1, -1, -5, -1, 1, -1, -4], [2, 0, 0, -1, 0, -4, -2, 2, 3, 3], [-2, -1, 0, 1, -2, 1, 0, -1, 3, 2], [1, 0, -3, 1, -2, 3, 1, 0, 1, 3], [2, 1, -2, 1, 1, 1, 0, -1, 1, -1], [-6, -1, 0, 0, 0, 0, -1, -1, -2, -1], [-1, -1, -3, 2, 0, 2, 0, -1, 0, 1], [-4, 3, 0, 4, 5, 1, 0, 0, -4, 3], [3, 1, 2, 1, 2, 0, 1, 1, -1, -2], [-9, -2, 0, 1, 2, 2, 1, 0, -2, 2], [4, -4, -4, -1, -1, 4, -1, 0, -2, -3], [-10, 2, 1, -1, -1, 1, 3, 1, 0, 0], [-2, -2, -1, 3, 2, 0, -1, 4, 1, 4], [-5, -3, 0, -1, -3, 5, 1, -1, 0, -2], [-3, -1, -1, 2, 1, -3, 2, 0, 1, -1], [1, -3, 1, -4, -3, -1, 0, 0, 1, -5], [-5, 3, 3, 0, 1, 2, 0, 1, 1, 3], [5, 0, -2, 2, 0, 1, 2, 0, 2, 5], [-6, 4, 0, 2, 4, 1, -1, 0, 1, 6], [-6, 0, 1, 2, 1, -3, -2, 5, -1, 2], [-1, 3, -2, -3, 1, 1, 3, -2, -1, -5], [6, -3, 2, 3, 0, -1, 0, -3, 0, -1], [-3, 1, 4, 1, 2, -5, 0, 4, 0, 1], [2, -2, 0, -1, -1, -1, -1, 0, -4, -4], [-1, 4, 3, 1, 1, 3, 3, -1, 0, -1], [-9, -2, 5, -1, 1, -2, 0, 3, -3, 3], [-5, 6, 2, 2, 0, 2, 6, -4, -2, 2], [-5, 7, 1, 0, -1, 4, 1, -2, 0, -2], [2, -2, -3, 0, -1, -7, -6, 2, 0, -4], [0, -3, 2, 2, -3, 0, 1, -3, -2, -2], [-10, 2, 2, -3, 0, -8, -4, 5, 3, -2], [4, -8, -1, 2, -1, 1, -2, 1, 0, 4], [6, -7, -4, 2, 2, -5, -1, 2, -3, -5], [-7, -7, -3, -1, -2, 8, -3, -3, 1, -1], [-4, 0, 0, 1, 2, -2, 5, 2, -1, 2], [-2, 6, 4, -4, -1, 4, 3, -1, 0, 2], [0, -1, 1, 1, -1, 5, -2, 0, -4, -1], [-6, -4, -2, 3, 4, 4, 0, -3, 1, 3], [4, -4, -2, 7, 3, 0, -1, 4, -4, 5], [-4, -2, 3, -3, 3, -4, 0, 4, -1, -4], [1, -2, 1, 0, 0, -4, -1, -4, -1, -5], [3, 1, -3, -5, -1, -3, -3, -1, 6, -3], [1, -1, -4, -1, -3, 6, -1, -1, -1, 4], [-2, -3, 0, -5, -4, -3, -3, 2, 6, 1], [3, -1, 3, 0, 0, 4, 6, -7, -2, -1], [-2, 0, -6, -3, 0, 0, 2, -1, 1, -2], [8, -4, 0, 5, 2, 11, 3, -2, -3, 4], [-1, 5, -6, 0, -2, 5, 1, -1, 0, 5], [0, 1, -5, 0, -1, -5, -2, 1, 8, 2], [0, -6, 2, 1, 4, 4, 2, 3, -2, 1], [4, 2, -3, 2, -1, 5, 0, -3, -7, -2], [5, 4, 2, -7, -4, 0, -1, -1, 2, -6], [-2, 2, 0, -5, -2, -5, 1, 2, 3, 1], [1, 1, 5, -4, -1, -10, 1, -1, -5, 0], [5, -1, 1, -3, 1, -13, -5, 5, 4, -3], [-2, 3, 3, -2, 4, 1, -2, 1, 3, 4], [-3, 3, -3, 2, -2, 5, 3, -4, -1, -1], [4, -8, -8, -4, -2, -4, -6, 2, 4, -6], [-2, 4, 0, 2, -1, -8, 1, -1, 4, 4], [2, -8, 1, -4, -4, -6, -2, 0, 3, 1], [-5, -1, 0, -5, -1, -13, 1, 1, 3, 3], [7, -3, 0, 2, 0, 4, 3, -5, -10, -3], [5, 0, -2, -3, -1, 2, 0, -5, 2, -2], [11, -1, 3, 6, 0, 2, 0, 1, -4, 7], [-13, 3, 6, 3, 3, -3, -1, -1, -3, -2], [2, 0, 0, 2, 4, -4, -6, 6, -8, 2], [-2, 0, 8, 5, -5, -3, 4, -3, -6, -2], [8, -8, 1, 4, 7, -6, -4, 6, -3, -1], [-10, -5, -3, 0, 5, 4, 0, 0, 0, 5], [-2, 5, -3, 5, 2, -2, 5, -2, 3, 5], [0, -5, -4, 5, 3, 8, 2, -3, -6, 2], [5, 2, -3, 4, -2, 3, 3, -8, 0, -1], [-4, 1, 3, 2, 3, 7, 2, 2, -7, 3], [-1, 2, 0, 2, 4, 4, -1, -7, 1, 3], [1, 4, 0, 5, 0, 6, 3, 2, -1, -1], [9, 1, -4, 5, 1, 11, 5, -11, -3, 3], [4, -8, -3, 0, 1, -12, -7, 6, 0, -6], [7, -9, 1, 1, -1, -9, -5, 3, -2, -7], [-15, 11, 5, 4, 3, 0, 4, -1, -3, -3], [2, 4, -8, -1, 0, 3, -3, -2, 1, -4], [2, -5, 1, 0, 1, -2, 2, 2, 1, 0], [-10, 6, 1, 5, 5, 2, 0, 3, 3, 0], [-10, -1, 6, 2, 10, -5, 2, 5, 0, 2], [5, -5, -4, 5, -4, 11, 2, -5, -4, 7], [4, 2, -9, 6, 2, 2, -2, -1, 7, 1], [6, 4, 1, 3, 4, -4, 4, 5, -4, -2], [-8, 3, -5, 0, -3, 1, 1, -6, 3, 5], [4, -8, -1, 5, 1, 0, -4, 0, -5, -4], [7, 4, 2, 4, -5, -3, -7, 3, 0, -5], [5, -7, 5, 0, -1, -7, -5, 6, 2, 4], [-4, -2, -1, 7, 2, 5, 4, 1, -3, 10], [-6, -2, 1, 2, 4, 12, 6, -7, -5, 3], [-8, 5, 6, 4, 3, -3, 2, 2, -10, 3], [-7, -3, -7, -3, -2, 3, -1, 1, -1, -8], [1, -5, -3, -4, -5, -8, -3, 9, 3, 0], [-3, 6, 1, 2, -2, 8, 2, -2, 1, 2], [10, -6, -4, -3, -5, 7, -3, 6, -2, -7], [10, 1, -12, -3, 0, 8, 4, -3, 1, 1], [-8, -4, 0, -3, 1, 1, -4, 1, -2, -6], [-11, 1, 1, 4, 4, -11, -3, 9, -5, 5], [-9, 16, -1, -1, 9, -1, 4, -4, 3, 2], [-6, -1, 7, 5, 0, -6, -1, -2, 1, 5], [7, -2, -6, 4, 1, 2, -1, -4, -2, 3], [-7, 10, -4, 3, -1, 7, 3, -2, 3, 5], [1, -5, -3, -3, -7, 1, -3, -1, 5, -3], [-14, 3, 0, -2, 0, -9, 5, 2, 6, 9], [-2, -10, 1, -1, 1, -7, 0, 4, -4, 2], [-16, 10, 8, -5, -2, 15, 4, 0, 0, 2], [9, -6, -1, -1, 0, 3, -2, -4, -6, -10], [4, 10, 0, -4, 2, -12, -2, 4, 0, -4], [-10, -1, 9, -8, 9, 1, 2, 6, -3, 1], [12, 5, -1, -5, -7, -5, -1, 5, 9, -5], [9, 5, -5, -4, -2, 11, -1, -2, 1, -6], [10, -4, 6, 1, -2, -6, 0, 10, -3, 1], [4, 0, 9, -4, -5, 4, -3, 1, -8, -12], [5, 13, 3, 11, 8, 0, 13, -4, -5, 3], [5, 2, 4, -1, 1, -4, -3, 4, -5, 5], [13, -10, 5, 3, -3, -2, 0, -1, -7, -1], [10, -4, -4, 4, -1, 1, -4, 0, 3, 2], [-2, 3, 5, -1, 2, 8, -4, 0, -8, -6], [-11, 7, -1, -6, 1, 10, 5, 2, 3, -4], [-1, -7, -10, 10, 0, 8, -1, 5, -1, 3], [16, -7, -5, -7, -6, 1, -9, 0, 9, -2], [-8, 3, 3, -7, -5, 1, -1, 2, 9, -1], [8, -2, 0, -5, -8, 15, -6, 7, -3, -7], [-12, 3, -1, -1, -5, 2, -4, -5, 7, 8], [-21, 5, 6, 0, 3, 16, 11, -8, -4, 7], [6, 4, -7, 0, -3, 0, 3, -3, -4, -8], [-18, 10, 0, 8, 4, -5, 0, 0, -4, 12], [8, 4, -3, 9, -1, 21, 6, -16, -5, 2], [3, -1, 3, -4, -2, -1, 0, -4, 8, -3], [32, -18, -6, 0, -1, 9, 2, -2, -11, -8], [3, 5, 1, 5, 7, -17, 5, 4, -2, 4], [-8, 7, -8, 6, 2, 5, 6, -11, 5, 3], [10, -8, 5, 0, -3, 4, 4, -8, -7, -7], [-14, 10, 2, 0, 4, 2, 10, 5, -4, 4], [20, 2, -7, -4, -4, -3, 0, 1, 1, -5], [-6, 11, 4, 3, -2, 10, 4, -5, 1, -3], [10, -7, -4, 0, -13, 3, -7, -6, 0, -7], [7, -3, -12, 4, -5, 9, 3, -1, 6, -1], [-3, -5, -3, 3, 4, 4, 3, -9, 3, 7], [1, 8, 5, 4, -4, -2, -2, 7, -5, -10], [13, -7, 1, -3, -3, -4, -3, -6, -8, -5], [-7, 0, -1, -11, 1, -2, -3, 4, 12, -1], [1, 2, 5, -6, 6, -6, 2, 6, -5, 0], [15, -4, 2, -3, -13, -2, -3, 2, 7, -1], [-6, 2, 2, 13, 11, -4, -1, -4, -2, 5], [1, 3, -8, 1, -5, 16, 3, -9, -3, -2], [15, -3, -10, -1, 4, 10, -1, -3, -5, -2], [-14, 11, -1, 4, 1, -3, 1, -7, 3, 11], [5, 4, -5, 10, 6, 20, 4, 0, -1, 6], [-7, 5, -2, 10, 11, -4, -1, 2, -4, 1], [-3, 2, 5, 4, 10, 14, 8, 1, -7, 6], [6, -2, 7, -10, 0, -6, 4, 3, 3, -9], [1, -3, -3, -9, 1, 12, 3, -3, 6, -1], [-5, -6, 3, -8, 3, -16, -9, 18, 3, 1], [-10, 6, 8, -3, 0, -6, 5, -8, 1, 1], [-4, 6, 7, 0, 1, 9, 0, -7, 4, 4], [1, 2, 3, -3, 8, 14, 0, 3, -4, -1], [1, -8, -15, -8, -1, -10, -1, -2, 10, -5], [2, -3, 5, -3, -6, 7, 7, -12, 3, -6], [4, -11, -3, -3, 3, 0, 0, 3, -9, -6], [23, -6, 0, 0, 3, -13, -16, 10, 3, -5], [8, 7, -2, 3, 2, -3, -1, -1, 2, -1], [28, -8, -10, -4, -3, 0, -5, 7, 2, -8], [7, -2, -7, -5, -2, -7, -13, 4, 11, -7], [4, 0, 0, 7, 0, -13, -11, 2, 1, 4], [-5, 4, 7, 0, 0, 11, 5, -5, 1, -2], [22, -13, -7, -5, -6, 1, -7, -2, -1, -12], [6, 10, 5, -6, 5, -19, -6, 4, 7, 8], [-2, -8, -6, -13, -4, -2, 0, -5, 10, -9], [8, 4, -9, -2, -5, 3, -2, 3, 12, 6], [8, -12, -1, -7, -12, 12, -4, 1, 2, -4], [2, 6, -10, -4, 6, 9, 1, 2, 5, 8], [-29, 0, -2, -3, -1, -8, -2, -1, -2, 0], [-3, 12, 4, -6, -3, 8, 5, 0, -2, 7], [-4, -2, 2, 4, -3, 16, 2, -15, 0, -2], [11, 1, 5, 5, 3, 5, 5, -3, -6, -7], [-17, 15, 1, 7, 5, 3, 5, -9, -2, 1], [-14, -3, -5, -13, -1, -8, -9, 6, 11, 3], [31, -13, -13, -4, -7, 13, -6, -11, 3, -11], [-20, 14, 4, 0, 4, -5, 6, -6, 6, 2], [14, -4, 5, -4, 4, -7, -2, 4, 10, -1], [-5, -1, 3, 4, 7, 13, -4, 2, 7, 14], [-25, 14, 11, -6, -2, 1, 3, -2, -6, 1], [2, -8, 3, -2, 1, -4, 1, 14, -4, 6], [1, 4, -5, -8, -1, -3, -11, -1, 10, 8], [0, 6, 6, -7, -1, -9, -6, -1, -2, 0], [-11, 7, 2, -1, -7, 17, 1, -10, 5, 7], [-19, 5, 5, -2, 5, -4, -3, 0, -7, -6], [-10, 0, 2, -7, -1, 1, 1, -12, 8, 2], [-8, 1, 1, -5, -3, 2, 9, 0, -7, 1], [13, -11, 4, 1, -9, 5, 5, 4, -5, -4], [0, 10, 7, -3, 3, -3, 0, -6, 1, 10], [-8, -3, -9, -5, 3, 3, 0, 7, 6, -5], [2, 9, 4, 1, 2, 18, 8, 4, 4, 16], [-22, 3, 3, -9, -3, -8, 7, 8, -2, 1], [-4, -17, 1, 1, 5, -5, 2, 3, 6, 3], [5, -3, 2, 9, 7, 17, -5, 0, 4, 7], [-10, 13, -7, -12, 3, -5, 0, 1, 10, -4], [-9, -3, 3, 6, -3, 4, 6, -8, -19, -2], [16, -2, -6, 13, -6, -4, -5, -1, -3, 2], [-13, -2, 1, -1, 0, -14, 3, 1, -9, -7], [8, -2, 15, -5, 0, -14, 0, 5, -2, 4], [-20, 16, 6, -12, 0, 1, 5, 4, 7, 4], [11, -2, 10, 2, 5, 1, 4, 6, 1, -3], [18, -3, 8, -1, -4, -3, 0, -5, 7, -2], [11, -3, 1, 5, -3, -3, -5, 11, 2, 1], [-13, 7, 9, 5, 1, -23, -3, 11, 0, -6], [18, -3, -4, 9, 1, 5, 0, -1, -2, -12], [-14, 19, 11, -11, -9, 0, 5, 0, 4, -5], [-22, 6, 2, 6, 14, -7, -2, 13, -2, 7], [-2, 4, 4, -5, 3, 7, 4, -1, 9, 1], [-13, 13, 6, 8, 4, 4, 11, -5, -3, 11], [-20, 0, 4, -3, -5, -3, -3, 4, 12, -1], [-3, -4, 3, -5, 1, 4, 11, 0, 6, -5], [-18, 10, 4, -7, -1, -12, 3, 9, 4, 2], [-19, 7, -5, 10, 8, 13, 9, -14, -1, 15], [23, -6, -9, 4, -7, 3, -1, -8, -5, -3], [-14, 6, 3, 6, -3, 11, 0, -3, -5, -4], [4, 2, -10, -10, -3, 8, -1, -4, 2, 0], [-3, -15, 4, -5, -5, -8, 1, -3, -12, -10], [-4, -10, 4, 10, -6, -11, -6, 0, 4, 6], [-4, -3, -11, -3, -3, -22, -7, 10, 14, 11], [10, -2, 10, -10, -6, -9, -5, -2, 9, -12], [-6, -1, -2, 1, -2, 19, 7, 1, -4, 1], [6, 0, -5, -3, 6, 11, 6, 3, 2, 2], [-3, -11, 8, 0, -4, 8, 11, 1, -7, -11], [0, -6, -4, 3, -4, -19, -4, 0, 1, -9], [-18, 0, 2, 2, 3, -4, -7, 4, 0, -2], [13, -1, -13, 2, 3, 18, 3, 7, -3, 2], [3, -1, -1, 1, -8, -10, 1, 1, 7, -3], [4, 8, -3, 7, 2, 4, 7, -1, -5, 6], [8, 7, -2, 8, -6, 9, -1, -10, -5, 7], [-30, -2, 7, -2, -1, -9, -4, 1, 5, 6], [-9, 21, 0, -2, 0, 16, 5, -15, 5, -1], [3, 11, 4, 3, 9, -13, -7, 8, 1, 4], [-3, 10, 3, 1, -6, 9, 0, -6, 12, 6], [-2, -3, -8, -3, -2, 1, -5, -2, 12, 17], [30, -8, 2, -5, 0, 0, -2, 7, -5, -7], [-1, -5, 10, -8, -6, -10, -17, 3, -2, -15], [15, -13, -13, 0, -1, -5, -10, 14, 3, -11], [-3, 1, 5, 6, 7, 10, 3, -4, -10, -8], [17, 10, 9, -7, -1, -11, 6, 4, -5, -8], [-14, 1, 0, -15, -8, -11, -7, -9, 6, -11], [3, 6, -7, 9, 3, 1, 6, -10, -7, -6], [-33, 1, 12, -9, -8, 25, 8, -3, -4, -5], [0, -5, 1, -2, 3, 10, 12, 1, 4, 2], [7, 13, -2, -3, 1, -2, -3, 4, 5, -2], [-14, 10, -4, 5, 5, 19, 6, 1, -2, 8], [-31, 19, 10, -3, -1, 10, 15, -9, -5, 10], [-2, 14, 4, -5, -8, -13, 6, 6, 3, -7], [14, 4, -6, 2, -4, 3, 0, 7, -4, -3], [10, -4, 3, 1, -8, 11, 0, -7, -4, -8], [-4, 26, 0, -7, -8, 11, 4, -5, 1, 7], [7, -9, 5, -9, 3, -5, -13, 5, -8, -3], [-12, 1, 1, 4, 2, 18, -6, 5, 1, 2], [6, -21, 3, 1, -9, -2, 3, 8, -10, -1], [-8, 8, 15, -4, -6, -11, 6, 1, 1, 3], [-35, 2, 4, -11, -5, 2, -7, 8, 3, -7], [-8, 4, 0, -9, -2, -14, -13, 12, 17, 0], [-3, 7, -1, -13, -6, -19, -11, -4, 2, -7], [25, -8, 9, 6, 5, -6, -3, 0, -3, -1], [-13, -8, 11, -16, -2, -17, -5, 13, 9, -2], [13, -13, -14, -7, -13, -17, -3, -3, 12, -16], [4, -3, 8, -1, 0, -12, 10, -1, -19, -4], [-10, 6, 3, 8, 18, 28, 2, 1, -7, 17], [21, -7, 4, 3, -10, -3, -12, 9, 6, -1], [20, 10, 1, -7, -14, -6, -6, 1, -5, -24], [-26, 24, 2, 4, 12, 14, 4, -2, 2, 10], [1, -8, 0, -15, 1, -14, -6, 11, 18, -12], [-17, 14, 13, -11, 1, -30, -9, 6, 6, -3], [-4, 1, -10, -4, -5, -29, -6, -4, 15, -8], [0, -13, -10, -3, -3, -14, -16, 3, 2, -10], [6, -2, 4, 6, 14, -10, -2, 6, -6, -8], [3, 24, -6, 3, 7, 16, 3, -8, 1, 11], [6, 2, 4, 2, 2, 2, 8, 2, -4, -4], [8, 11, -1, 6, 5, 17, 13, -20, -3, 9], [10, -6, -11, -5, 1, -19, -5, 5, 12, -8], [-36, 16, 12, 11, 0, 0, 4, -5, -15, -3], [-11, -5, -3, 5, 1, 15, -5, 11, -4, 4], [2, 0, -13, -1, 3, 6, -2, -14, 7, 14], [-14, 8, -8, 9, 18, -4, 6, -10, -5, 11], [4, -8, 3, 0, -3, 13, 2, 7, -4, 4], [0, 20, -11, -11, -2, 2, 2, 1, 6, 10], [-20, 5, 17, -3, 2, -8, 1, 0, -1, 11], [12, -10, 3, 0, -11, -7, 2, -5, -8, -2], [33, -20, -7, -1, -8, 1, 0, 0, 0, -14], [-3, -27, 3, 2, -3, -3, -16, 11, 3, 3], [-15, 19, 3, 3, 4, 5, 7, 2, -15, 8], [-26, 22, -2, -10, -2, 15, 1, -2, 21, 6], [-25, 5, 9, 4, 8, -1, 7, 2, -7, 1], [-22, 17, -8, 2, 6, 13, 5, -1, 12, 19], [-24, 11, 7, 8, 0, -7, 0, -2, -10, -7], [18, 2, 0, 12, -3, -20, 5, -5, -4, 10], [1, -11, -1, -6, -7, 4, -6, -7, -5, -15], [18, -6, -11, 1, -11, -21, -1, 0, 0, -16], [-11, 2, 4, -5, 4, 0, 1, 6, 2, -1], [33, -2, -1, 1, 2, -13, 10, -2, -12, -8], [23, -17, -4, 0, -2, 10, 3, -3, 5, -3], [-7, -11, 5, 6, 6, -2, -2, -1, -12, 1], [6, 22, 0, 8, 3, 24, 0, -7, -4, -4], [0, -21, -5, 7, -5, -4, -19, 10, 7, 1], [11, -8, -6, -6, -5, -6, -16, 10, -5, -10], [-20, -2, 8, 4, 7, 0, 3, -15, 0, -8], [8, -19, -2, 4, 4, -6, -3, 11, 3, 2], [-43, 19, 0, 7, 11, 16, 11, 0, -5, 12], [3, -9, -8, -13, -5, -9, 1, -11, 4, 2], [15, -9, 13, -6, 5, -13, -4, 7, 5, -5], [-17, 7, -13, 4, 1, 13, 4, -1, 3, 21], [-1, -1, 13, -5, -5, -10, 5, 6, 0, -11], [4, -17, 3, 4, 7, -19, 4, 3, 2, 6], [9, 21, 11, 9, 1, 9, 13, -5, -1, 7], [2, -11, -16, -8, 0, 6, -7, 10, -4, -13], [-19, -1, 7, -9, -5, -10, -11, 3, 6, -5], [-16, 4, 2, -19, -8, -12, -3, -4, 17, -4], [43, -13, 2, -2, -7, -12, -15, 8, 2, -21], [15, -17, -12, -6, -6, 6, 1, -7, -1, -9], [29, -21, -5, -4, -15, 4, -5, 8, 0, -6], [9, 6, 3, -4, -8, -26, 11, -4, -3, 1], [-2, 9, 3, 6, -3, 19, 14, -14, 3, 13], [-11, 5, -6, 7, 8, 7, -6, 0, 12, 3], [2, 17, 10, 1, 0, -1, 1, -3, -10, 9], [-22, 12, 1, 14, 4, -8, 6, 9, -7, 15], [-5, -1, 18, 7, 8, -3, 4, 2, -8, 11], [-57, 34, 16, 6, 14, -10, 7, 3, -7, 15], [16, -6, -11, 0, -6, -2, -4, 8, 6, -5], [33, -24, -10, 14, -5, 9, 0, -4, -11, -1], [5, -11, -3, 4, 9, -11, -7, 15, 13, 19], [24, -5, 1, -12, -14, -7, -4, 3, -1, -10], [-16, -6, 2, -8, -2, -21, -14, 18, 9, -8], [-12, -17, 3, 3, -7, -18, -11, 4, 2, -5], [20, -7, -8, -17, -13, 7, 6, -3, 10, -14], [1, -4, -13, 5, -4, 12, 2, -3, 0, -5], [-1, -17, 5, -4, 9, -17, 3, 13, -1, -4], [10, -2, -22, -7, -8, 28, -1, -4, 9, -6], [-4, 28, -7, 1, 9, -3, -4, 2, 13, 8], [3, -10, -1, 2, 4, -9, -13, -2, -4, 7], [-14, 5, -10, 13, 14, 24, 16, -15, -19, 8], [19, -16, -1, 16, -3, 17, -3, -10, -17, 5], [19, -21, -12, 1, -1, 7, -7, -12, 1, -4], [-28, 18, 3, 7, 21, -1, 5, 11, 4, 24], [13, -6, 15, -10, 3, -21, -3, 12, -7, -15], [21, -5, -5, 2, 9, 15, -10, 7, -3, -5], [8, 4, -14, 4, 0, 31, 2, -10, 8, 6], [12, -8, -6, -3, -19, 21, 7, -14, 10, -6], [-18, 11, -4, 1, -6, 17, -1, 0, -6, 5], [3, -3, -7, 3, -7, -4, 5, -5, 0, -13], [19, -3, -10, -2, -6, 14, -3, 7, 0, -13], [-7, 0, -2, 8, -4, -2, -13, 11, 7, 13], [-15, -11, -9, -2, 1, -8, -5, -4, -5, -8], [-29, 21, 10, 6, 15, -13, 15, -1, -12, 9], [-13, -5, -7, -10, -1, -3, 8, -4, 1, -15], [-6, 10, -6, -3, 3, 2, -1, 3, 24, 10], [7, 14, 6, -9, -10, 3, 3, -6, -9, -12], [8, 11, -3, 8, -5, -2, 2, -3, 14, -1], [16, -21, -10, 4, -4, -9, -5, 7, -8, -21], [16, -18, -4, -5, -1, -4, -5, -3, -16, -14], [11, -3, 0, 4, 0, -4, -5, 5, -1, 7], [9, -5, -15, -4, 3, 2, -9, -14, 11, -2], [34, -25, -9, 0, -12, -21, -8, 1, -5, -10], [-16, 3, -1, -15, -2, -5, 6, -2, 1, 9], [-13, -9, -9, 2, 2, -7, 8, 0, 4, 17], [-29, 18, 0, 8, 11, -5, 3, -5, 12, 13], [-18, -13, -2, -7, -3, 10, -8, 10, 3, 4], [10, -25, -9, -2, -9, -9, -5, -2, 1, -11], [-19, 1, -9, 18, 9, 0, 6, -10, -3, 6], [-10, 22, -5, 11, 10, 14, -2, -9, 16, 8], [19, -3, 9, 14, -19, 5, 1, -9, -7, -2], [15, -5, -6, 0, -1, -7, 3, 1, -4, -9], [-3, 17, 2, 1, 7, 13, 19, -8, -3, 14], [6, -13, -10, 6, -6, 22, 5, -7, -18, -7], [-10, 8, -13, 19, -2, 4, -2, -10, -6, 2], [-10, 18, 4, -22, -6, 3, 8, 8, 6, 0], [21, -24, 9, -12, -6, -8, 0, 15, -15, -8], [-2, 16, 4, 9, 7, -10, -6, 5, -7, 11], [34, -25, -20, 4, -11, -1, -1, 10, 4, 3], [9, -10, -8, 1, 13, -10, -9, 8, -1, -15], [-24, -10, 6, 0, -1, -19, -10, 9, -2, -4], [46, -19, -10, 11, 2, 28, 8, -15, -19, 2], [-28, 18, 17, 0, -2, -9, 16, 10, 9, 9], [19, -14, -15, 3, -4, 22, 3, 1, -9, -11], [-24, 30, -6, 0, 8, 1, -2, 4, 6, 16], [-14, 9, -16, 6, 1, 32, 4, -13, 9, 20], [-14, 25, -8, 15, 8, 1, 9, -7, 8, 17], [40, -8, -15, -9, -14, 1, 2, -5, 17, -16], [34, -16, -1, -3, 4, 0, -6, -3, 0, 4], [-10, 2, 6, -6, -2, -20, -16, 4, 2, 0], [10, 6, -8, -6, -6, 2, -2, 16, 6, 2], [-18, 9, -14, 13, 6, 12, -4, 7, 15, 15], [25, -9, 7, 5, -9, 5, -5, -3, -1, -1], [-2, -14, 5, 5, 4, -10, 10, 4, -2, 12], [-27, -1, 6, 5, -8, 23, -2, 7, -6, 5], [-29, 27, 6, -4, 8, -14, -1, 9, 6, 15], [-11, 19, 7, 3, -6, 6, 15, -12, -10, -1], [18, -16, 1, -5, -9, -18, -6, -15, 17, -8], [-17, 4, 7, -12, 10, 12, 3, 8, -4, -7], [2, -8, -4, -21, 2, 11, -2, 1, 7, -15], [24, 9, -3, 0, 1, 16, -2, 1, -4, -8], [-13, 15, 6, -4, 3, 11, 19, -5, 0, 3], [-29, 21, 13, 14, 12, -5, -1, 8, -5, 17], [22, -21, -12, 2, 6, -29, -3, 8, -1, -1], [-14, -16, 4, -15, -5, -13, 4, -1, 8, 2], [-26, -15, 2, 0, 4, -13, 10, 7, -7, 3], [9, -1, 8, 6, -3, -16, -9, 12, 5, 5], [0, 14, 9, 6, 17, -14, -7, 4, 0, 6], [3, -10, 6, 5, -11, -2, 8, 1, -16, -14], [19, -7, 3, -1, 5, -19, -7, 7, -6, -13], [-8, 0, 16, -1, 2, -29, 0, 3, 4, -1], [14, 1, 4, -9, -8, -5, -11, 7, -10, -15]];
    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_bh();".
// 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_bh(: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_bh();".
// 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_bh( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3381_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, 28, -17, -100, 26, 100, 1, -36, -6, 4, 1]>,<5,R![-648, 1512, 279, -1704, -104, 708, 95, -112, -24, 4, 1]>,<11,R![-2441, 2654, 9373, -3220, -7618, 528, 1302, -68, -65, 2, 1]>,<13,R![1358, -9704, -46991, 19552, 26186, 1616, -2541, -464, 44, 16, 1]>],Snew);
    return Vf;
end function;