
// Make newform 1911.2.a.x in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1911_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_1911_2_a_x();".
// 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_1911_2_a_x();".
// 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 := [46, -80, -159, 300, 64, -212, 16, 52, -10, -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], [-4, 0, 1, 0, 0, 0, 0, 0, 0, 0], [1178, -870, -3578, 1314, 1800, -766, -301, 149, 16, -9], [-1426, 298, 5042, -891, -3978, 794, 1064, -247, -86, 22], [2950, 1284, -10962, -2971, 7486, 1001, -1788, -22, 135, -10], [2182, -113, -8409, 1994, 5778, -1560, -1365, 396, 102, -31], [2672, -2107, -8368, 6292, 5960, -4229, -1564, 981, 128, -72], [-4442, 2898, 15114, -6535, -10747, 4177, 2679, -1010, -209, 78], [-6538, 514, 26262, -5315, -19013, 4788, 4717, -1355, -365, 113]];
    Rf_basisdens := [1, 1, 1, 211, 211, 211, 211, 211, 211, 211];
    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_1911_a();"
function MakeCharacter_1911_a()
    N := 1911;
    order := 1;
    char_gens := [638, 1522, 1471];
    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_1911_a_Hecke(Kf)
    return MakeCharacter_1911_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], [1, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 1, 1, 0, 0, 1, -1, -1, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 1, -1], [-1, -1, 0, 0, -1, -1, -2, 0, -1, -1], [-1, 1, -2, 0, 1, -1, 2, 0, -1, -1], [1, -1, 0, -2, 1, -1, -2, 0, 1, -1], [-3, -1, 0, -2, 1, -1, 0, 0, -1, -1], [4, 0, 0, 0, -2, 0, 0, 0, 2, 0], [2, -1, 0, 0, 1, 0, 0, 1, 0, 0], [-3, 3, -2, 0, 1, 1, 2, 1, -2, 0], [3, -2, 2, -2, -2, -1, -2, 0, 0, 0], [1, -3, 0, -2, -1, -1, -2, 0, -1, 1], [2, 0, -1, 1, 0, 0, 1, -1, 1, -1], [2, 2, 0, 2, -2, 2, 2, 0, 0, 2], [-2, 2, -2, 2, 2, 2, 4, -1, -1, 1], [2, 1, 0, 0, 1, 0, 2, 1, 0, 2], [1, 3, 0, 0, 1, 1, 2, 0, 1, -1], [5, -3, 2, 0, -1, -1, -1, 0, -1, 1], [3, -1, -1, 1, 1, -1, -3, 0, 1, -1], [-3, 3, -3, -1, 3, 1, 3, 0, -1, 1], [-3, 3, -2, 0, 3, -1, 2, 0, 1, -1], [2, -2, 2, 0, -2, 0, 0, -2, -2, 2], [-4, -2, 0, -2, 0, -2, 3, -1, 1, 1], [0, 0, -2, -2, -2, 0, -2, 0, 0, -2], [-2, 4, -2, 0, 2, 2, -2, 1, -1, 1], [3, -1, -2, 0, 1, 1, -2, 2, -1, -3], [2, 0, -2, 0, 0, 2, 3, -1, 1, -1], [4, -4, 2, 2, -2, -2, 0, -1, -1, -1], [2, -3, -2, -2, 1, 0, -4, 3, 0, -2], [-2, -4, 0, 2, -2, 0, -5, -1, -1, -1], [0, -1, -2, -2, 3, -2, 0, 1, 0, -2], [0, -6, 2, 0, 0, -2, -4, 0, 2, -2], [-6, 2, -4, 0, 2, 2, 5, 0, -2, 2], [0, 0, -2, -2, 2, 2, -2, 3, 3, -1], [1, 1, -3, -1, -1, 1, -5, 2, 1, -1], [0, 4, 2, 2, -2, 2, 2, 0, 0, 2], [9, -3, 2, 2, -1, 1, -4, -2, -1, 1], [2, 0, 2, -2, 2, 0, 1, -1, -1, -1], [4, 0, 0, 0, 2, 0, 2, 2, 2, 0], [-2, 0, -2, 0, 4, 0, 8, 1, 1, 1], [2, -2, 5, 1, -4, 0, -3, -1, -1, 1], [-2, -4, 2, -2, -2, -4, -6, 1, 1, -3], [11, -5, 6, 4, -3, 1, -1, -2, -1, 1], [1, -5, 2, 0, -1, 1, -4, 0, 1, 3], [4, 7, -2, 2, 1, 2, 2, 1, -2, 2], [6, 0, -2, 4, 2, 2, 4, -2, -2, -2], [-1, -1, -2, 4, -3, 1, 6, 0, 1, -1], [2, -6, -3, -3, 0, -2, 5, 0, -2, 0], [1, 1, 2, 0, -1, 1, 2, 0, -3, 5], [2, -2, 2, 2, 2, -2, 0, -4, -2, 2], [-2, 8, 0, 2, 0, 4, 2, 0, 2, 2], [-11, 5, 0, 2, 1, 5, 0, 0, -1, 3], [4, -4, -2, 0, -2, 0, -6, -1, 1, 1], [-2, -2, -2, -4, 0, -4, 0, 2, 4, -4], [-1, 1, 4, 0, -1, -3, 0, -1, -4, 4], [14, -5, 4, 4, -7, -2, -2, -2, 1, 3], [-8, 0, 2, -2, 0, -2, -8, 2, 2, -2], [13, -5, 1, 3, -1, 1, 3, 0, -1, 1], [-1, 1, 4, 2, -3, -1, -4, -2, 1, -1], [10, -4, 4, 0, -6, -4, -8, -1, 3, -1], [-2, -6, 4, 2, -6, -2, -2, -2, -2, 0], [-6, -4, 1, -3, 4, -6, 5, 0, -2, -4], [6, 0, 6, 2, -4, 2, -4, 0, 0, 4], [9, 5, 0, 4, 3, 3, 3, 0, -1, 5], [-4, 2, 2, 0, -2, 4, 6, -2, -2, 2], [-1, 5, 0, 4, 1, 1, 4, -4, -3, 5], [4, 0, -1, 1, -2, -2, -3, 1, 5, -1], [-10, 2, -3, -3, 4, -2, -3, 2, 0, -2], [-4, -8, 2, -4, 0, -4, -3, -2, 2, 2], [0, 6, 4, 4, -2, 0, -3, 1, 1, 3], [2, 0, -2, -2, -2, 4, 2, 2, 0, 2], [4, 1, -2, 2, -1, -2, 10, -1, -2, 2], [-2, 6, -2, 2, 2, 4, 6, -2, 4, 4], [1, -7, 0, -4, 3, -3, -6, 0, -1, -3], [22, -8, 7, -1, -4, -2, -1, -2, 0, -2], [7, -7, 2, -2, -3, -3, -12, 0, 1, 1], [-2, -6, -2, -8, 0, -6, -10, 2, 2, -4], [4, -6, 4, -2, -2, -6, -10, -1, 1, -7], [8, -7, 4, 0, -3, -2, -6, -1, 2, -4], [2, 0, 2, 4, -2, 4, -4, -3, -1, 1], [2, -4, 0, -2, 2, 0, 5, 1, -3, -3], [-7, -1, -6, -4, -1, -3, -2, 2, -1, -1], [14, 0, -3, 1, 0, 2, 9, -2, -4, 2], [-4, 0, -4, 0, 2, -4, 4, 1, 5, -3], [8, 4, -1, 1, 2, 2, 5, -1, 3, -3], [-10, 0, -4, -6, 0, -2, 4, 1, 1, 1], [10, 6, 2, 2, 0, 4, -2, -1, -3, 5], [13, 4, -2, 2, 0, 1, 10, 0, -2, 6], [-10, 4, -6, 2, 6, 4, 6, 2, 0, -2], [2, 4, 0, 2, -4, 2, -2, -2, -4, 0], [8, 4, 0, -2, 2, 6, 2, 1, 7, 1], [6, -8, 4, 2, -2, -2, -6, -2, 0, 2], [17, -6, 4, 4, -2, 1, 0, 0, 2, 2], [-4, 8, 2, 4, 4, 6, 4, 0, -2, 6], [-14, 4, 0, 2, -2, 4, 9, 1, 1, 1], [-12, 2, -4, -2, 4, 0, 6, 4, -2, -2], [-11, 7, -4, 0, 9, 3, 4, 3, -2, -2], [2, 3, 2, -2, -1, -4, -4, 1, 4, -2], [-8, 4, -4, 0, 10, -2, 6, -1, -5, -1], [11, 5, 0, 0, -3, -3, 0, 0, 5, -7], [7, -1, 6, 8, -7, 1, 3, -2, 3, 1], [2, -4, 8, 4, 2, 2, -2, -2, 4, 2], [5, 6, -2, -2, 2, 5, 2, 4, -4, 0], [-3, 7, -3, -1, 3, -3, 11, 4, 3, -3], [1, 11, -2, -2, -3, 1, 0, 0, 3, 1], [-12, 12, -6, -2, 2, 0, 6, 5, 5, 1], [6, -4, -4, 2, 2, 4, -1, -1, 3, 3], [4, -4, 2, 0, -2, 4, -8, 1, 3, -3], [10, 6, 1, -3, 0, 4, 5, -1, 3, 1], [-4, 2, 0, 4, -4, 6, 0, -4, -4, 2], [-2, -2, 2, 0, -2, 2, 6, 2, 2, 0], [17, -1, 6, 6, -1, -1, -8, -2, -3, 5], [8, -6, 6, 4, -2, 0, 0, -4, 0, 6], [14, 10, 0, 4, 0, 0, 12, -3, 3, 3], [-14, 0, -2, -2, 0, 4, 0, 2, 2, 6], [11, -5, 6, -2, -3, -1, -6, -6, 1, 3], [5, -7, -2, -4, -3, -5, -8, 0, -1, -3], [-1, 3, -4, -4, 1, -1, 0, 7, 6, -2], [-6, 6, 4, 2, 4, -2, 0, -3, 1, 1], [-10, 3, -8, 4, -1, 0, 18, 1, 2, 0], [-15, 7, -10, 2, 5, -1, 12, -2, -9, 1], [-5, -3, 2, 0, 7, 5, -2, -2, -3, -1], [6, -8, 2, -4, 0, -8, -8, 2, 0, -2], [-2, -2, 6, -2, 0, -4, 0, -3, -1, 3], [-16, 2, -12, -4, 0, 0, 2, 1, -5, -1], [1, -7, 6, 2, -1, 1, -2, 2, -3, 3], [-6, 2, 2, 0, 0, -8, -6, -2, 0, -4], [-2, 2, -1, -1, 0, 2, 7, -2, 0, -2], [3, 1, -6, 4, 3, 3, 14, -3, 0, 2], [7, 3, 8, 4, 1, -3, 3, -6, -5, 5], [7, -6, 8, 4, -2, -1, -4, -4, -6, 2], [5, -1, 4, 2, -1, -1, 2, 0, -3, 7], [12, -4, -1, -3, 2, -2, -11, 1, 9, -5], [-7, 3, -2, 4, 1, -5, 12, -2, 1, 1], [-2, -4, 2, 2, -4, 2, 16, 1, 1, -1], [15, -3, 4, 2, -5, -3, -6, -2, 5, -1], [-4, -2, 2, 4, -4, -2, -6, -4, 2, 2], [2, 3, 8, 4, 1, 6, 6, 0, -5, 5], [-6, 10, -4, 0, 6, -2, -4, 0, 2, 2], [-24, 3, -6, -2, 1, -4, 0, 4, -3, -5], [-4, 12, 6, 6, 2, 4, -2, 1, 1, 1], [-6, 8, 1, -1, -4, 0, -5, -3, 3, 5], [5, 3, -2, 8, 1, 5, 14, 2, 1, 1], [-10, -2, -10, -2, 4, -4, 6, 2, -4, -10], [-8, 2, -8, -4, 4, 4, 2, 5, 7, -1], [16, -6, 3, 3, -2, 2, 3, -1, 1, -1], [-16, 6, 0, 0, -4, 0, -4, -1, 1, 5], [6, 6, 2, 0, 0, -2, 2, -1, 7, -1], [-8, 2, -6, 0, 6, 0, 3, 1, 1, -1], [8, -2, 10, 8, -6, 2, 0, -3, -3, 11], [-5, 7, 2, 0, 5, 1, -3, -2, -1, 5], [5, -5, 0, -8, -3, -3, 0, 6, 1, -5], [10, -4, 2, 0, 0, -2, 1, -5, 5, 3], [-25, 17, -15, -5, 9, 3, 7, 6, 3, 1], [0, -6, 8, 0, -2, -6, 4, 0, 2, -2], [-1, -3, 5, 11, -7, 3, -1, -6, -5, 9], [14, -9, 2, 2, -3, 6, -4, -4, -3, 3], [-11, 3, -4, -2, 5, -1, 16, 4, 1, -1], [-10, -8, 4, -2, 4, 2, -10, 0, -2, -4], [30, -8, 10, 8, -6, 2, 0, -4, -2, 8], [-2, 18, -7, 1, 4, 4, 5, 1, -7, -1], [-1, 12, 2, -2, 0, 3, 2, 4, 6, 2], [2, 10, -2, 0, 2, 8, -6, 4, -4, 6], [0, -2, -6, -6, 8, 2, 1, 1, 7, -5], [-6, 18, -6, 4, -4, 6, 4, 4, -2, 2], [-4, -8, 2, 2, 2, 2, -8, -2, -6, 4], [20, 8, 3, 11, -2, 8, 3, -2, 2, 8], [-6, 22, -2, 6, 0, 14, 18, -2, -6, 8], [-11, -3, -4, -4, 3, -7, -4, 4, 5, -1], [-4, -2, 0, -2, 4, -4, 14, 4, 2, 2], [14, -10, 2, 4, -6, -2, -5, -4, 6, -2], [-12, 0, -4, 6, 2, 0, -6, 3, -3, -3], [0, -4, -2, 6, -2, -6, -4, -6, -4, 6], [10, -11, -4, 0, -1, -2, -10, -2, 7, -3], [12, 0, -2, -4, -8, -4, -15, 4, 0, -4], [-8, -8, -2, -10, 2, -12, -10, -1, 3, -9], [12, 0, 10, 2, -4, -4, -2, -5, 5, 3], [-14, 8, -2, -2, 0, 2, 2, 6, 10, -2], [8, -14, 4, 4, -6, 0, -1, 1, 5, -1], [-19, -1, 7, -3, 7, 1, -11, 2, 1, -1], [0, -4, -1, -1, 2, -4, -5, -2, 2, -12], [-22, 10, -4, -6, 2, 0, 0, 8, 8, -4], [11, -7, 4, 4, -9, -3, -8, -9, -8, 4], [-19, -7, 2, 0, 3, -1, 6, 0, 1, 5], [14, 4, 0, 2, -4, -8, -4, 1, 5, -3], [-9, 8, -6, -6, 0, -1, -2, 10, -4, -4], [4, 2, -2, 6, 4, 6, -3, -1, -7, 1], [-22, 13, -10, 2, 7, 6, 4, 0, -5, 1], [-7, -1, -8, 0, 9, -5, 8, 5, 4, -12], [-1, 9, -7, -1, 5, 7, 7, 4, 1, 3], [-11, 3, 4, -8, -3, -3, -10, 4, 3, 5], [4, 0, 6, 0, 0, -2, 10, 0, -2, 2], [34, -14, 12, 8, -14, -4, -8, -7, -3, 3], [-15, 9, -4, -4, 3, -1, -1, 2, 5, -5], [11, -12, 4, -4, -8, -9, -12, -4, 8, -8], [-14, 12, -6, 0, 8, 8, 22, 1, -5, 1], [10, -4, 1, -3, 4, 6, -11, 0, -6, -4], [12, 2, 0, 2, 0, -2, 1, 1, -5, 3], [7, 17, -4, 4, -9, 7, 0, 0, -1, 5], [10, -6, 0, -4, -6, -6, 1, 4, 2, -6], [19, 0, -6, 6, 0, 3, 2, 2, 4, -4], [-4, -8, 4, 6, 0, -2, -12, -6, -4, 0], [-19, 3, 6, -2, -1, 1, 0, -2, 1, 5], [-9, 9, -2, -2, 3, -1, 6, 6, -1, 5], [-8, 11, 0, 0, -3, 0, 2, 4, 7, -7], [4, 14, 0, 8, 0, 12, 14, 3, 5, 1], [1, 5, 4, 8, 7, 7, 17, -4, -1, 1], [4, -4, 12, 6, 0, 2, 14, -2, -4, 12], [18, -12, 6, -4, -4, 2, -5, -9, 1, -1], [10, -15, -2, -10, -5, 0, -6, -1, 6, -2], [5, -3, -9, -3, 7, -3, 1, 0, -5, 1], [9, -3, -4, -2, -5, -7, -16, -2, 3, -5], [-5, -9, 2, 0, -3, 5, -5, 0, 1, -1], [4, 4, 4, 8, -2, -6, 16, -6, -6, 4], [-4, 6, 7, -5, -2, 2, -5, -1, 5, 7], [-12, 11, -8, -8, -3, 0, 6, 2, 5, -9], [-5, -1, 0, -4, 9, -9, -16, -3, 0, 0], [-4, 2, 0, -6, 4, -6, 6, 0, 6, 2], [10, -3, 2, 2, -7, -4, 0, -1, 4, -2], [14, -14, 4, -2, 2, 6, 2, -5, 1, -3], [6, 12, 0, 6, 6, 2, 12, 6, 0, 6], [-1, 13, -8, 0, 11, 5, -4, 1, -2, 2], [-24, 2, -2, 2, -2, -8, -10, 3, -3, -1], [-25, 3, -4, 4, 1, -5, 8, 2, 1, -5], [4, -2, 2, 2, 4, -2, 6, 1, 3, -5], [-14, -3, -2, 2, -1, -2, 8, -8, -5, 1], [-16, 6, -12, 0, 6, 2, 0, 6, 8, -4], [-18, 18, -2, 0, 2, 2, 1, 0, -6, 6], [-8, 8, -4, 2, -2, 10, 12, 3, -3, 7], [6, 10, -6, 4, 4, 12, 2, 4, -4, -2], [-18, 4, 4, -2, 10, 8, -7, 1, 1, 5], [-11, 1, -11, -5, 11, 1, 7, 12, -1, -3], [-10, 4, -2, 8, -2, -2, 8, -4, 0, 8], [-8, -5, 4, -8, 1, -4, -2, 0, 3, 1], [-15, 3, 0, -4, 5, -3, 4, 2, 1, -5], [14, -8, 14, 8, -2, -2, -8, -4, 2, 0], [-8, 5, -2, -6, 7, 8, -12, 4, 7, -7], [-16, 10, 6, -2, -4, -6, -2, -1, 5, -3], [10, -16, 4, -4, -2, -10, -6, -1, -1, -1], [-18, -8, -6, -18, 8, -8, -4, 7, -3, -13], [-7, -4, 4, 0, -8, -3, 0, -2, 2, 10], [4, 6, 0, 4, 0, 2, 8, 3, 9, 1], [-37, 7, -16, -12, 9, -3, 14, 2, -1, -7], [19, -16, 6, -2, -8, 3, -2, 0, 10, -2], [-24, 2, 0, -2, -2, -8, 14, 2, 2, -8], [-21, -3, 2, -2, 7, 3, 8, 0, 1, 7], [-4, 8, 8, 2, -2, 12, 6, 1, -5, 17], [-8, -9, 0, -4, 1, -10, -24, 7, -2, 2], [-8, 5, -12, 4, 1, -2, 6, 3, 6, -4], [10, 16, -4, 0, 4, 4, 4, 0, -4, 0], [10, -18, 10, -8, -6, -12, -20, 2, 2, -6], [4, -5, 2, 6, -11, 0, -12, 2, -1, 1], [11, -1, 12, -4, -7, 1, -11, -4, 5, 3], [12, 10, 4, 2, 2, 8, 2, 2, 2, 4], [-12, -6, -2, 2, 2, 0, 14, -7, -5, -3], [-11, 5, -2, 0, 1, -5, 0, 0, -9, 1], [-7, -11, 6, 4, -1, 3, -13, -4, -5, -3], [-37, 13, -4, 4, -1, 11, 10, 2, -1, 1], [-7, -9, 2, -6, -13, -9, -20, 2, 1, -7], [-22, -4, -2, -2, -2, 0, 1, 5, 5, -3], [2, 7, -4, 0, -11, 4, -12, 3, 10, 2], [6, 10, -2, -6, 2, 0, -12, 2, 0, -4], [10, -2, 5, 5, 0, -4, 9, -3, 1, 7], [11, 4, 0, -4, 12, -5, 12, -4, -4, 0], [6, 4, 0, 0, -10, -4, -12, 6, 8, -10], [-2, -14, 0, 4, -2, 4, -2, -4, -2, 2], [-18, 10, -20, -2, 6, -6, 14, 0, 0, -6], [-8, -6, 2, -2, -6, 4, -14, 2, 0, 8], [8, 10, 2, 10, -4, 0, 22, -2, -2, 4], [9, 11, -4, 4, -3, -1, -4, 1, 4, -4], [-14, -4, -2, -12, -2, -4, 0, 1, -1, -11], [-4, 2, 0, 8, -8, 6, 16, 0, 8, -2], [-20, -9, 6, 2, -3, -4, -4, 2, 7, 1], [-16, 8, -4, 2, 4, 6, -6, 7, -1, -5], [-4, -2, 2, 8, 2, -2, -6, -1, -3, 5], [4, 12, -2, 0, 0, 6, 0, 7, -3, 7], [14, 2, 0, 2, -2, 8, 16, 1, -7, 7], [-10, -8, -8, 2, -10, 4, -10, -2, -6, -2], [-4, -2, -6, 0, -4, -4, -16, -2, -10, -8], [7, 13, 2, -6, 3, 3, 6, 2, 5, -1], [-8, 20, -2, -2, 2, 6, -12, 4, 10, 0], [-27, -3, -2, -2, -5, -7, -10, -4, -3, -1], [43, -17, 16, 8, -11, 5, -1, -6, -5, 9], [10, -7, 2, 10, 1, 8, 4, -3, 6, -4], [-32, 6, -6, -10, 0, -10, -8, 4, 0, -10], [-20, 2, -4, 6, 2, -2, 4, 0, -4, -4], [3, -9, -4, 4, -1, -7, -6, 0, 3, -5], [4, 4, -4, -2, 2, 0, 16, 0, 2, 0], [12, -26, 11, -9, -10, -4, -5, -4, -4, -6], [-19, 12, -2, -6, 8, 5, -2, 10, 4, -8], [17, -1, 0, 0, 5, 7, -4, 1, 8, 4], [-8, -10, 14, 0, -6, 2, 6, -2, 2, 10], [14, 4, 8, 2, -10, 8, -3, -5, -5, 7], [0, 12, 4, 2, -4, 2, -8, -2, -8, 0], [16, -2, 6, 8, 2, -2, 6, -1, -9, 1], [1, 13, 0, 8, 7, 7, 6, 2, 1, -1], [-30, 12, -10, -6, 10, -4, -9, 3, -1, -5], [-30, -7, -2, -6, -3, 4, -4, 5, 2, 0], [17, -5, 10, 4, -3, 5, -14, -5, -8, 6], [-11, 13, -14, 6, 11, 5, 6, 2, -1, 9], [1, -1, 0, 4, -11, 7, -4, -1, 2, 2], [22, -4, 6, -2, -2, 2, -32, 3, 3, -1], [-4, -12, 0, -6, -4, 4, -4, 0, 10, -6], [-20, 2, -6, 6, 4, 2, 15, -3, 11, -1], [4, 11, 8, 8, 3, 10, -2, -1, -4, 6], [-1, -21, -4, -12, 1, -3, 3, -2, 7, -7], [0, 24, -14, -2, 4, 0, 4, -2, -2, -2], [10, -6, 7, -5, 0, -8, 3, -5, -5, 13], [30, -12, 2, 4, -2, 0, -6, -4, -2, -4], [-7, -11, 2, -4, -1, -1, 3, -4, -1, -7], [-8, 7, -6, 2, -3, 0, 12, 0, 1, -1], [0, 2, -4, 4, -2, 10, 4, 4, 6, -2], [19, -9, 5, -1, -7, -1, 3, -6, -1, 9], [22, -17, 10, -2, -7, 2, 0, -4, 9, -1], [-7, -11, -4, -6, -7, -9, -18, -4, -13, -7], [-7, -1, -10, -2, -9, 3, 10, 4, -9, -1], [8, -14, 4, -10, 0, -14, -21, -3, 7, -9], [-8, 2, 2, -8, -12, -8, -22, 2, 8, -4], [14, 6, 4, 0, -6, 2, -12, -3, 7, 5], [0, -16, 0, -12, 2, -10, -2, 2, -8, -10], [-4, -4, 7, 3, 10, 2, -1, 1, -5, 5], [0, 10, 0, 0, 2, 10, 4, 10, 4, 4], [-8, 0, 6, 2, -6, -10, 0, -1, -1, 3], [21, -19, 14, 0, -7, 9, -8, -2, 5, 7], [2, 6, -12, 6, 10, 2, 2, 0, 4, -8], [-14, 12, -2, 6, 8, -4, 14, -6, -6, 2], [22, -16, 14, -2, -4, 0, -8, -4, 4, 4], [9, -3, 2, 4, 3, 7, 4, 2, 1, 3], [30, -10, 10, 6, -6, 12, 4, -2, 0, 4], [-1, -1, 1, -5, 9, -9, -1, -6, -13, -3], [-6, -14, 2, -4, -8, -8, -36, 3, 3, -3], [5, 1, 1, -1, 3, 1, 11, -8, -9, 13], [-16, 10, -8, -4, 8, 4, 14, 3, -3, -3], [42, -7, 8, 4, -9, 0, -8, -9, 0, 0], [-20, -8, 2, 2, 4, -6, 4, -2, -6, 2], [-16, 0, 6, 4, 0, -12, -11, -4, -4, -4], [0, 4, 6, 16, -8, -2, 10, -4, 2, 8], [-10, 6, 4, -2, 0, 0, 2, 8, 4, -10], [12, 13, 4, -4, -1, 6, 0, 1, -2, 14], [30, 2, 4, 8, -14, -2, -8, 0, -2, -2], [-10, 16, -1, 15, 4, -2, 11, -2, -4, 2], [-8, 20, 4, 10, 4, 12, 12, -2, 4, 8], [-36, 18, -22, -12, 20, -2, 0, 11, 11, -11], [0, 4, -12, -2, 12, 8, -8, 3, -1, -9], [17, -27, 7, -7, -5, -11, -7, -2, -3, 3], [-2, -24, -2, -2, -8, -10, -26, -1, 3, 1], [19, -7, 12, 4, -7, -9, -26, -6, -13, -1], [42, -16, 16, -4, -16, -4, -14, -3, -5, 1], [-16, 4, 1, 5, -2, -2, -11, -1, -7, 7], [28, -14, 12, -4, -16, 0, -10, -3, 7, 7], [0, -18, 8, 4, -2, -8, -9, -5, -5, 5], [14, -10, -6, 4, 4, 14, -2, -1, 11, -5], [-13, 11, -13, 1, 13, 15, 1, 6, -9, -7], [9, -2, 6, 2, -2, -3, -2, 2, 6, 2], [12, 6, 18, 8, -4, -6, -10, -6, -6, 0], [22, 1, -10, -2, 3, -2, -28, -6, -7, -5], [-6, 2, -2, 4, 4, -6, 16, 5, -3, 7], [6, 8, -6, 0, 12, 8, -4, 4, 6, 2], [5, -3, 4, 2, 1, -3, 10, -8, -13, 3], [-6, 13, -8, -12, 9, 0, 6, 3, 2, -12], [-36, 12, -14, 0, 16, -4, 17, 12, 0, -8], [19, -15, 4, 8, -13, -7, -32, -5, 0, 0], [-7, 2, -12, 0, 2, -7, 20, 6, 8, -8], [22, -4, 16, 2, -8, 6, -6, -8, 10, 8], [-1, 1, -16, -4, 15, -7, 16, 1, -6, -2], [-2, -4, 4, 6, 4, -2, 2, -2, 4, 0], [0, 8, -2, 4, 8, 12, 23, 0, 0, -4], [0, 7, 2, 2, 13, 0, 8, 0, -7, 11], [-5, 7, -12, -8, -1, -11, 18, 2, 9, -7], [-10, 28, -10, 2, 22, 16, 22, 5, -7, 5], [9, -3, -12, 0, -1, -3, 12, -1, -12, -8], [-2, 11, 2, 2, -7, 4, -22, -3, 6, 2], [8, 6, -2, -4, 6, 8, 2, 8, 12, 0], [-2, -14, -6, -2, -12, -2, 16, -2, -6, -4], [-28, 0, -4, -16, 10, -10, -2, 2, 6, -4], [14, -20, 10, -6, -10, 0, -13, 7, 15, 3], [-5, 9, -5, -3, 5, -1, -19, 0, 1, -5], [50, 9, 4, 16, -5, 6, 18, -8, -3, 3], [12, 4, 16, 0, -2, -2, -2, -3, 5, 1], [-2, 17, -14, -2, 9, 0, -4, -1, -4, 2], [-23, -9, 8, 14, 1, -9, -2, 0, -5, -1], [-7, 13, -7, -1, 19, -7, 11, 0, 3, -3], [0, -8, 4, 2, -12, 0, -20, 0, -2, -4], [-24, 10, 0, 12, 6, 2, -8, -6, -4, 4], [12, -12, 2, -6, -2, 6, 0, -5, -5, -5], [-12, -2, -8, -10, 6, 10, -2, 3, 11, -7], [14, 18, -14, 0, 2, 8, -22, 7, 17, -11], [21, -13, 1, -5, -13, 1, -13, 0, -1, 5], [-28, 5, -6, -2, 7, 4, -8, -2, 5, -9], [-16, -4, -2, 2, -8, -6, 0, -6, -2, 2], [4, 6, -17, -5, 18, 10, 3, 3, -7, -1], [-23, 17, -16, 4, 7, -9, 25, 0, -9, -7], [8, 18, 6, -2, 10, 12, 6, 4, 6, 10], [-22, 2, 0, 6, 0, 6, 10, 5, -11, 5], [-3, -11, -10, -12, 3, -9, -17, 8, 19, -15], [-10, 7, -2, 2, -9, 0, -8, 5, 0, 2], [-15, -5, -10, -10, 9, -3, 20, 6, -1, -15], [15, 11, 0, 0, -1, 1, 18, -4, -5, 11], [-10, 12, -4, -12, 12, -2, 8, 11, 1, -5], [13, -8, 4, 12, -16, 1, -8, -4, -4, -4], [17, -21, 9, 7, -17, -7, -21, -4, -1, -3], [-18, 8, -6, 2, 6, -8, 6, -11, -7, 5], [-4, -2, 0, 0, 10, 14, 12, -7, -3, 11], [40, 4, 9, 5, -14, 2, -3, -7, -5, 13], [-29, 7, 0, -6, 9, -7, 0, -8, -17, 3], [15, 7, 11, 5, -7, -1, -7, -4, 1, 7], [16, -14, 14, 6, -24, -14, -17, -3, -1, -5], [-37, -1, 0, -2, 5, -15, -14, 8, 1, -5], [5, -9, 8, -4, 5, 1, -22, -2, 9, 3], [10, -14, 2, -8, -6, -20, -18, -6, -6, 4], [-23, 7, 0, 0, -3, 7, 0, -3, 4, 8], [0, -20, 0, -8, 4, 4, -2, -3, 1, -13], [-20, 22, -15, -7, 10, 10, 17, -1, 9, 3], [-20, -12, 6, 2, -4, -14, -28, -2, -6, -6], [16, 4, -9, 1, 2, 10, -7, 13, 1, 3], [-4, 14, -8, 6, -4, 8, 22, 8, 8, 14], [-5, 9, -4, 4, 13, 1, -14, -2, 3, 7], [10, 8, -2, -14, 4, -4, -6, 6, -2, 2], [-16, 9, -2, -10, -1, 4, 8, 0, 11, 5], [-28, -12, -6, -10, -2, -6, -34, 3, 3, 3], [-10, 4, -14, 10, 0, 12, -12, 0, 4, 0], [15, -6, -4, 4, 14, 7, -12, 2, -4, -4], [-9, -7, 10, -2, 3, 7, 6, 2, 7, 5], [0, -15, 0, 0, 3, -12, 10, -2, -9, 9], [-13, 23, 6, 2, 9, 9, 8, 10, 3, 1], [-18, 4, -10, -8, -2, -2, 6, 8, 6, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1911_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1911_2_a_x();".
// 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_1911_2_a_x(:prec:=10)
    chi := MakeCharacter_1911_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_1911_2_a_x();".
// 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_1911_2_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1911_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![46, -80, -159, 300, 64, -212, 16, 52, -10, -4, 1]>,<5,R![4, -1288, 2948, 4704, -956, -1456, 196, 160, -23, -6, 1]>],Snew);
    return Vf;
end function;