
// Make newform 1805.2.a.u in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1805_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_1805_2_a_u();".
// 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_1805_2_a_u();".
// 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 := [19, 27, -51, -72, 27, 48, -4, -12, 0, 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], [-3, -1, 1, 0, 0, 0, 0, 0, 0], [-7, -2, 17, 4, -8, -1, 1, 0, 0], [0, -7, -2, 17, 4, -8, -1, 1, 0], [14, 4, -41, -10, 33, 6, -10, -1, 1], [22, 21, -72, -42, 60, 21, -19, -3, 2], [-38, -32, 123, 72, -96, -36, 29, 5, -3], [-53, -33, 165, 81, -129, -42, 39, 6, -4]];
    Rf_basisdens := [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_1805_a();"
function MakeCharacter_1805_a()
    N := 1805;
    order := 1;
    char_gens := [362, 1446];
    v := [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_1805_a_Hecke(Kf)
    return MakeCharacter_1805_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, 1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 1, 1, -1, 0], [-1, -1, -1, -1, 0, 1, 1, 0, 1], [0, 0, 1, -2, 0, 1, 1, 0, -1], [0, 1, 1, 2, 1, 0, -1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -1, -2, -1, -1, 2, -1, 0, 1], [0, 0, 0, -2, 0, 3, 0, 0, 2], [4, 1, 1, 1, 0, 0, 1, -2, 1], [4, 0, -1, 2, 0, -1, 1, -1, -1], [3, 1, 0, -1, 2, -1, 1, 0, 0], [-1, 0, -1, 1, 0, 0, 0, 2, -2], [3, 0, 3, -5, 1, 2, 3, 0, 0], [-2, -2, -2, -4, 0, 1, 1, -1, 1], [4, 1, 0, 4, 0, -2, -3, 2, -3], [3, 2, 0, 1, 0, -2, -3, 0, -3], [5, -2, 0, -2, 3, -1, 4, 1, 0], [0, -2, -1, 2, 1, -3, 1, 2, -1], [-1, 3, 0, -1, 2, 4, 1, -2, 0], [3, -1, -4, 1, -2, -2, -1, 3, -1], [2, 0, 1, -3, 2, -1, 3, -2, 0], [-3, -6, -2, -1, -2, -1, 2, -1, 4], [6, 3, 1, -6, 2, 1, 2, -3, 2], [-3, 2, -1, 1, -3, -2, -1, -1, 2], [2, -2, -1, 2, -1, -2, -3, -1, -3], [0, -3, -1, 3, 0, 1, -1, 1, 2], [8, 3, 0, -1, 0, 3, 2, -2, 2], [-3, -3, -5, -1, 0, 3, 4, 1, -1], [2, 3, 2, 3, -2, 1, -7, 2, 0], [-4, 1, -3, 2, -5, -2, 0, 1, 3], [4, 0, 1, 0, 1, -1, -5, 2, -2], [0, 0, 2, -2, 1, 0, 4, -1, 3], [0, -6, -3, -3, -1, -3, 4, -3, 1], [5, 4, 1, -3, 2, 4, 2, -1, 2], [-8, -3, -5, 3, -5, 1, 2, 3, 3], [-4, 5, 2, -2, -3, 0, -1, -1, -1], [-8, 3, -2, -2, -1, -1, 4, 1, 2], [-1, 5, 6, -5, 2, 4, 1, -6, 1], [5, 3, 4, 2, 2, -3, -7, -1, 0], [6, 2, 2, -3, 2, 1, 3, -8, 1], [-4, 1, 0, -4, 4, 1, 0, -5, 4], [1, -2, 1, 3, -2, 1, 0, 2, 0], [4, 0, 4, -2, 7, 2, 2, 0, -2], [-1, 3, 2, -1, 3, 3, 4, -4, 2], [9, 2, -1, 1, 0, 1, 3, -3, 1], [2, 0, 2, -4, -4, -1, -5, 2, -2], [-1, -1, 6, 3, 2, -3, -3, 1, 0], [2, 6, 1, 8, -6, 0, -6, 5, -4], [-7, -1, -3, -4, -5, -1, -6, 4, 1], [-6, -4, -3, 12, 1, -3, -7, 1, 1], [1, -4, -6, 6, 0, -3, 1, 3, 1], [1, 2, 0, 6, 5, -4, -3, -3, -3], [-13, 1, 1, -5, -5, 5, 7, 3, 4], [10, -3, 2, -4, -2, -3, -5, -1, 0], [6, -2, 2, 1, 2, -5, 7, -2, 1], [4, 5, 3, -8, 2, -1, 5, -3, -4], [1, -5, 0, -1, -2, 0, 0, -3, 7], [7, 5, 2, 2, 2, -1, -1, 1, -7], [4, -8, -6, -4, -1, -6, 1, 5, -3], [-2, 3, -1, 4, 1, 1, -6, 2, 1], [7, 11, 8, 4, 3, 1, -9, 0, -1], [3, 3, 5, -2, 3, -5, -5, 1, -1], [8, 5, 5, -6, 0, 1, 0, 1, 1], [-2, -3, 1, -1, -2, -5, -7, 3, -7], [8, -7, -4, -8, -2, 2, 0, -2, 4], [-4, 6, 1, 0, -3, 2, -4, 1, -3], [2, -6, 2, 0, 1, -6, 6, 5, -4], [3, 11, 13, -1, 4, 6, 2, -10, 0], [-5, 4, 4, 6, -1, 4, -7, 5, -3], [4, -4, -3, -4, 1, -2, 12, -2, 2], [-5, -6, -8, -4, -3, 3, 2, 3, 2], [-4, -4, -7, 0, -2, 4, 0, -3, 7], [2, -4, -7, 4, -8, -1, -1, 4, -1], [-9, 0, 4, -10, -1, 4, 7, -6, 1], [5, -12, 1, 2, -2, -7, 3, 4, -3], [3, 3, 5, 2, 3, 3, -16, -1, 0], [8, 4, 7, -5, 5, 5, 1, -2, 0], [5, -5, -4, 9, 0, -5, -3, 11, -6], [5, -6, -2, 8, 2, -2, -6, 7, -1], [-13, -7, -8, 1, -4, 0, -3, 4, -1], [-2, -4, -1, -1, -1, 1, -2, -5, 5], [-3, -1, 5, -2, -4, 5, -10, 2, -3], [2, -1, 7, -11, -3, 4, 6, 5, 1], [14, 3, 4, 7, 3, -4, 0, -6, 0], [-2, 1, -6, 0, -10, -3, -6, 7, -1], [-11, -9, -10, 7, -3, 1, 8, 2, 5], [1, -1, 4, 5, 6, 1, -2, -1, 1], [-5, -9, 2, 8, -4, -4, -6, 8, -6], [7, -9, -2, -1, -1, -9, 5, 4, 0], [6, 2, 7, 1, -3, 0, 0, 4, -3], [0, 5, 4, 1, 3, 1, 4, -6, 2], [3, -8, 0, -2, 0, -4, 2, -4, 2], [-7, -8, -10, 1, -1, -1, -1, 0, -2], [-4, -3, 0, 1, 0, -8, 8, 2, 2], [7, 9, 5, -1, 6, 8, -6, -11, 1], [-6, 2, 2, 9, 0, -3, -5, 1, 1], [3, -1, -3, -2, -7, 5, -1, 0, 0], [-7, -5, -12, 12, -3, -4, -7, 3, -1], [7, 2, 4, 14, 2, -4, -3, 5, -2], [5, 4, 5, -1, 0, -9, 0, 1, -10], [-1, 5, 9, -18, 0, 0, 6, 2, -5], [-9, -4, -5, -8, -8, 3, 6, 1, 10], [2, -7, 1, 17, 4, -9, -13, -1, 4], [-11, 4, 2, -7, -5, 3, 7, 1, 8], [-8, -5, -4, 1, -3, -2, 2, 1, 0], [0, -1, 1, -6, 3, 0, 7, 2, -3], [16, 1, 14, -8, 0, -6, 1, -4, -1], [-5, -10, -8, 3, -6, -10, 6, 11, -6], [6, -3, -3, 6, 1, -8, -8, 4, -4], [16, 11, 13, 2, 8, -5, -5, -9, 3], [7, -12, -8, -2, -6, 2, 5, -8, 8], [-20, 0, -4, -11, -5, 10, 4, 9, -1], [13, -13, -4, 7, 4, -3, 1, -4, 1], [11, -1, -1, 12, 2, -12, 0, 1, 0], [-9, -12, -1, -5, -1, -5, 1, -4, 2], [10, 12, 9, 3, 10, -2, -5, -7, -6], [-5, 0, -7, 9, 4, -2, 4, 1, 9], [21, 3, 7, 3, 9, 2, -2, -11, 0], [8, 7, -1, 5, 6, -2, -8, -1, 3], [-1, -11, -5, -2, -4, 4, 3, -1, 5], [9, 11, 10, 6, 0, 4, -5, 1, 2], [-1, -6, 3, -8, 4, -3, 1, -4, 5], [3, -11, -2, -1, -4, -4, 1, 8, -1], [-10, -5, 10, -2, 3, 2, 0, 0, 1], [-9, -2, -13, 0, -5, -2, -5, 6, -4], [7, 11, 1, -10, 0, 12, 3, -3, 3], [-5, -6, -5, -11, -6, -4, 15, -7, 2], [-5, -7, -7, -9, -2, -4, 8, -12, 2], [5, 1, 3, -4, 4, 6, -3, -13, 11], [-4, 0, 5, 6, -4, -6, 3, 2, -6], [-3, -11, -12, -11, -13, 5, 3, 8, 0], [-16, 2, -4, 3, 1, 0, -4, 5, 6], [10, 8, -5, -1, -3, 2, -1, 1, 3], [3, -1, -2, 20, 10, -6, -14, -4, -2], [-12, 6, 0, 7, -8, -4, 5, 7, -3], [11, -5, -1, 8, -2, -14, -3, 10, -3], [-2, -14, -11, -7, -5, -6, 18, 9, -12], [-8, 2, 1, -12, -3, 7, 6, -5, 8], [5, -2, -4, -7, -10, 4, -3, 5, 6], [-5, 5, 6, -10, -4, 6, -13, 10, -2], [2, -1, 4, -3, 9, -3, 6, 1, -5], [8, 8, -3, -4, 6, 4, -2, -9, -2], [-9, 5, -1, -1, 7, 13, 11, -11, 3], [-4, 2, 3, -16, -6, 3, 2, 1, 1], [17, 9, 9, 11, 2, -1, -10, 3, -6], [-1, -11, -4, -5, 0, -4, -10, -2, 2], [-6, 7, -6, 5, -6, 9, -5, 6, 3], [-3, 7, 8, -7, 0, 3, -5, 6, 1], [2, -5, -2, 4, -1, -9, -13, -2, 0], [-1, 1, -3, -23, -8, 11, 21, 8, -6], [4, 1, 6, -6, -3, 0, -7, 0, 2], [-20, -4, -7, -11, -8, 2, -5, 8, -2], [11, -4, -2, 2, 7, 7, 7, -2, 0], [15, 2, 8, 3, 1, -3, -1, -13, -1], [-1, 8, -12, 3, 2, 1, 7, -10, 2], [-15, 7, 1, 3, 0, 0, -11, 0, 2], [-8, 9, -2, -17, -6, 11, 6, -2, 10], [-5, -13, -14, -2, 5, -5, 5, -4, 7], [-6, 4, -5, 10, 1, 3, 5, 6, -6], [-18, -12, -8, 9, -6, 3, -13, 13, -3], [14, 2, 6, 5, 11, 1, 0, -6, -1], [17, 15, 9, 2, 2, 4, -17, -4, -1], [-5, -2, -1, 18, -3, 2, -11, 4, 4], [-32, 6, 1, 1, 5, 2, 3, -8, 2], [15, -2, -9, 14, 5, -6, -16, -5, -6], [18, 7, 7, 6, -2, 0, 1, 0, -9], [1, -22, -7, 8, -5, -7, 0, 5, -1], [-1, -1, -4, -2, 3, -4, -4, 7, 0], [11, 9, 7, -9, -7, 6, 10, -5, 2], [-3, -5, -3, -16, -12, 4, 27, 5, 1], [-2, -2, 1, -6, -11, 3, 20, 7, -6], [-3, 0, -2, 22, -4, -10, -14, 17, -12], [5, 3, 5, 8, 4, 1, -18, -8, 7], [14, 0, 3, -1, -3, -1, -4, -3, 9], [-2, -9, 4, -5, 7, 0, -4, -7, 7], [-1, 9, 6, -4, 2, 6, 14, -8, 5], [-16, -3, -14, 9, 2, 2, -10, 9, -1], [2, -4, 2, -5, -1, 1, 6, -5, 11], [21, 15, 15, 16, 5, 0, 4, 0, -7], [18, -3, 7, 9, 12, 2, 5, 0, -5], [7, -5, 18, -2, 18, -2, 3, -15, 5], [5, -1, -3, 3, -1, -13, 2, 8, -10], [-21, 16, 7, -6, -7, 18, -9, -9, 7], [2, -18, -9, 1, -1, -1, 5, -6, 12], [22, 0, 18, 2, 2, -1, -5, 6, -6], [17, 20, 4, -9, 10, 7, 11, 0, -11], [-7, -3, -3, -7, 16, 8, 10, -10, 7], [10, -3, -5, 1, 2, 6, 11, 1, -9], [-7, -6, -14, -1, 6, 1, 3, 5, 8], [1, -4, -9, 14, -3, -7, 8, 8, -8], [3, 3, -5, 7, -9, 7, 3, -2, 7], [-7, 13, 11, -11, -11, 5, -3, 0, 6], [-10, 9, 2, 22, -9, 5, -23, 8, -9], [-27, -5, 1, -2, -3, -7, 9, 2, -2], [0, -3, -11, 19, -4, -11, -14, 16, -5], [16, 15, 18, 12, 9, 5, -1, -2, 3], [12, -7, 3, -17, -7, -6, 10, 3, 7], [1, 15, 6, 9, 3, 3, -11, -12, 13], [4, -6, 7, -1, 15, -3, 1, -7, 2], [20, 13, 9, -5, 12, 4, 15, -7, -7], [-1, 16, -2, 14, -5, 4, -11, 8, -8], [-12, 12, 5, -4, 1, -3, 10, -8, 2], [-5, 7, 11, -5, 8, -2, 6, -2, 5], [-5, 5, 1, -11, 2, -6, 7, -11, -1], [3, 20, 7, -2, 3, 11, 8, -13, 7], [-7, -7, -6, 9, -1, -14, -15, 9, -3], [-5, 2, 6, 26, 7, -16, -13, 10, 0], [17, -3, -1, -7, 5, 4, -2, -5, 1], [4, 14, 10, -9, 7, -2, -3, -5, -9], [10, 13, 13, -5, 10, 7, -13, -10, 4], [7, -8, 0, 15, -6, -5, -21, 4, -3], [-3, 3, 7, 4, 3, 0, -3, -15, -1], [1, -11, -10, 4, -5, -19, 0, -6, 6], [6, -2, 1, 5, -6, -4, 13, 11, -8], [0, 3, -1, -17, -1, 2, 20, 2, -7], [1, -9, -8, 23, 0, -12, -14, -2, 11], [8, 8, 3, 5, -1, -1, -11, -5, 0], [0, 11, 1, -24, -4, 13, 2, 6, -2], [-4, 8, 8, -16, -1, 13, 6, -6, 7], [18, 8, 13, 19, 13, -4, -17, 8, -12], [1, 3, -7, 22, -5, 4, -21, 7, -11], [-13, 12, -5, -21, -21, 13, 7, 6, 10], [-6, -3, -5, 16, 3, 5, -6, 2, 0], [9, -6, 7, -2, 12, -8, 10, -5, 6], [1, 1, 14, -7, 15, -11, 11, -2, -15], [-10, 19, 16, 7, -6, 10, -11, 12, -7], [-3, 7, -8, 31, -4, -8, -21, 6, -2], [12, 16, -1, -1, -5, 12, -13, 2, 5], [18, -6, -3, 9, -7, 3, -2, 0, -7], [-6, -18, -11, 10, -12, -10, -5, 6, 11], [13, 21, 16, -5, 11, 1, 4, -15, 0], [7, 21, 10, 19, 0, -3, -11, 1, -12], [4, 0, 0, -1, 7, 1, 21, 5, 2], [6, 3, -10, 5, -10, 0, 0, 12, 4], [-30, 2, 0, -8, -1, -5, -5, -7, -4], [1, 5, 3, 15, 3, -5, -28, 6, -4], [17, 15, 14, 0, 20, 4, -5, -1, -14], [9, 11, 4, 7, 7, 2, -24, 6, -3], [13, 7, 22, -5, 5, -15, -3, -4, 1], [6, 13, -4, -9, -1, 9, 0, 0, 15], [-18, -18, -9, 0, -2, -2, 14, 0, 2], [-33, 13, -3, -5, -2, 9, 5, -3, 3], [-6, 9, -3, 0, 4, -8, -6, -9, -4], [-32, -9, -21, 4, -11, -4, 13, 8, 3], [-37, -10, -7, 22, -9, -10, -7, 10, -6], [2, -16, -15, -11, -9, -7, 3, 4, -4], [27, 0, 11, -6, -2, -12, -1, 4, -3], [-2, -4, -19, 3, -11, 4, 5, 4, -2], [24, 0, 12, 6, -7, -8, -8, 0, -14], [11, 27, 10, 10, 0, 11, -13, -3, -9], [0, 12, 17, -20, 4, 3, 1, -18, 12], [0, 14, -2, -2, -3, -5, 4, 0, -3], [3, 8, 1, -5, 12, 5, -10, -18, 10], [-21, 10, 5, -8, -9, 9, 11, -3, 3], [-30, -5, -15, -5, -9, -3, 9, -3, 5], [23, 8, 8, 6, 11, 1, -3, 3, 2], [11, -18, -8, -3, -5, -9, 21, 1, 4], [-12, 8, -2, -14, -2, 3, 6, -6, 5], [-3, -3, 1, 16, -1, -9, 5, 2, 0], [-7, 8, -2, -15, -5, 6, -1, -2, 10], [-3, -18, -6, -2, -9, -10, 16, -6, -3], [-13, -3, -13, -9, -5, -7, 11, -5, 13], [-6, -6, -15, 7, 0, 1, 1, -4, 15], [8, -8, -5, -5, 10, 1, 16, -1, -1], [-9, -6, -5, -3, -5, -5, -6, -11, 6], [13, 14, 0, -21, 5, 7, 26, -15, 10], [23, -9, -13, -15, -7, -10, 3, 12, -7], [3, 3, -10, -1, -2, 6, 20, 3, 1], [3, -11, 15, -15, 8, 3, 11, -7, -1], [13, 2, -4, 9, 4, -9, -22, 0, -9], [-13, -6, -23, 2, -15, 2, -4, 16, -7], [19, 2, 4, 5, -12, 3, -26, 4, 2], [-12, -19, -18, 10, -15, -8, -7, 25, 3], [22, -11, 0, 12, 15, -2, -9, 1, -7], [12, 11, 1, -34, -8, 20, 8, -7, -2], [-29, 9, 7, 0, 5, 16, 4, -13, 6], [-8, -7, -9, 10, -2, -14, -11, 4, -16], [-4, 29, -5, 12, 1, 3, -10, 8, -9], [-15, 4, 11, -1, 10, 2, -7, -7, 8], [20, 7, 21, -20, 14, 10, 15, -16, 11], [-1, -4, 0, 4, -2, -8, 3, -1, 4], [-15, -25, -19, -4, -3, -6, 9, 10, 14], [24, -12, -6, -4, -7, 7, 9, 13, 4], [3, 19, 1, 14, -1, -1, -9, 7, -7], [17, 20, 5, -12, 7, 3, 4, -4, -12], [-6, -2, -9, -37, -2, 14, 9, -7, 10], [-6, 14, 20, -25, -6, 11, 2, 3, -14], [3, 20, 24, -14, 8, 10, 15, -13, 0], [-9, -2, -7, -6, -3, -8, 19, -8, -1], [16, 10, 6, 19, 3, -8, -20, 2, -4], [5, 10, 8, -5, 8, 15, 15, -8, 12], [20, -7, -7, 14, 2, -16, 7, 14, -9], [16, 4, 0, 21, -5, -14, -13, 17, -12], [9, -17, -13, -4, -10, -23, 16, 16, -6], [-15, 4, 1, 24, -11, -4, -21, 19, -13], [-6, 19, 21, -6, -6, 11, -8, -1, 3], [-2, -5, -1, -1, -12, -11, 22, 4, -5], [24, 8, -3, 4, -2, -13, -10, 0, -18], [-7, -12, 10, -25, 6, 1, 3, 8, -1], [2, 1, 2, -12, -2, 3, 14, -5, 3], [10, 8, -6, 13, 10, -6, -16, -1, 1], [-4, -1, -1, -18, -17, 3, -1, 9, 3], [10, 9, 14, 29, 2, -13, -4, 11, -17], [11, -1, 5, -11, 0, 13, 6, 8, 10], [-11, -12, -23, -2, -12, -5, 4, 12, 1], [-20, 17, -3, 12, 3, 19, -9, -9, 16], [-4, -5, 0, -12, 8, -1, 4, 7, 4], [10, -9, -4, -17, 6, 3, 24, -9, 0], [-20, 10, 2, 29, -1, 2, -15, -1, 0], [-24, -14, -15, 2, -12, 7, 0, 7, 13], [-21, 15, 2, -29, -9, 8, 5, 0, -2], [24, 6, -1, 17, -4, -14, -26, 6, 1], [11, 0, 5, -22, 13, 2, 22, -24, 12], [-24, -2, 10, -3, -2, -10, -21, 11, 5], [20, 9, -7, -15, -3, -7, -4, 4, -3], [18, -3, 7, 0, -1, 6, 2, 5, -7], [-17, 6, 6, 9, -10, -10, -9, 12, -18], [19, 12, 18, -1, 11, -8, -4, -15, 10], [17, 0, 23, -23, 5, 2, 5, 5, -18], [-8, 6, 15, -16, 6, 9, 4, -7, 14], [1, -6, -13, -15, 7, 13, 20, 3, -3], [23, -8, 10, 7, 17, 9, -12, -1, -7], [-3, -1, 8, -42, -6, 10, 9, -11, 21], [13, -5, 12, 4, 9, 4, 15, 1, 2], [6, 12, 15, -22, 1, 6, -12, 5, 1], [21, 16, 24, 1, 16, 13, 1, -17, 20], [5, -1, -1, -6, -8, 3, -2, -13, 4], [21, -4, 13, 23, 20, -27, 0, -2, -15], [-1, -10, -5, 7, -13, -21, -12, 6, 2], [-7, 7, -8, -3, -10, 12, -3, 12, 12], [10, 20, 3, 5, -3, 6, -17, -5, -5], [-5, -3, 4, 16, 8, -13, 1, -14, 2], [-9, -23, -11, -7, -19, -1, 4, -1, 19], [5, -27, 5, 16, 4, -14, -4, -4, -4], [-38, 2, -20, -5, -16, 22, 7, 13, 5], [1, -28, -6, 25, 8, -17, -7, -3, 0], [-9, -13, 1, -28, -3, 19, 0, 1, 8], [-8, -18, -8, 9, -13, -7, -6, 5, 10], [25, -4, 26, -13, 13, -10, -1, -4, -9], [-15, -15, 2, 12, 8, -15, -1, 5, -3], [35, 7, 21, -3, 6, -15, -10, 8, -19], [-22, 2, 0, 6, -15, -3, -14, -1, 12], [-10, -13, -18, 25, 3, -15, 1, -14, 8], [-8, 10, -7, -24, -4, 24, -9, -7, 12], [9, 4, 24, -10, 2, 1, -8, -15, -3], [13, 3, 5, 7, 11, 1, -15, 4, -13], [-7, 2, 8, -3, 9, 5, 9, -11, 1], [-14, -28, -21, 7, -20, -12, -14, 23, 7], [8, 17, 15, -7, -1, 12, 2, -13, 1], [16, 14, 5, -4, -3, -3, -16, 11, -4], [6, -11, 1, 2, -21, -5, -20, 24, 0], [-9, 13, 7, 11, -4, 8, -4, -1, 5], [-21, 11, -7, 26, -14, 10, -5, -11, 13], [4, -10, -4, -20, -13, 9, 11, -14, 9], [-24, -2, -19, -7, 4, 28, 3, 3, 15], [-34, 0, 2, -16, -2, -3, 8, -14, 0], [-9, -23, -11, 9, -6, -4, -19, 16, -6], [-27, 6, -15, 14, -3, 23, 5, 7, 7], [-3, 2, 3, -5, -1, 15, -11, 9, 5], [5, -1, -13, -1, -9, -3, -29, 16, 4], [-2, -7, 7, -33, 5, 2, 27, 2, -6], [26, -19, 6, -18, -8, -4, 3, 5, -6], [-3, -9, -10, 0, 12, 9, 18, -2, 16], [-10, 3, -6, -9, -15, 5, 12, -10, 3], [19, 4, 4, -12, 18, -10, 14, -24, 15], [13, 4, 13, -3, 24, 11, 14, -8, 4], [7, 11, 6, 13, 10, -18, -12, -13, -5], [23, 15, 3, 30, 7, -14, -36, 3, 6], [-18, -7, -15, -31, -14, 10, -1, -2, 17], [-16, -2, -9, 2, -17, -14, -16, 9, 1], [6, -7, 8, -16, 6, 10, -5, -3, 3], [14, -38, -11, -7, 7, -22, 15, 4, -3], [30, -2, 10, 30, 21, -34, -18, -11, -4], [-3, -6, -14, 22, 0, 1, -7, 5, 10], [30, 8, 4, 15, 13, -4, -2, -17, 4], [48, 4, 12, 35, 7, -7, -15, -7, 1], [-4, 0, -2, 9, 9, -9, -10, -12, -9], [-19, -30, -36, 4, -20, -2, 5, 6, -2], [37, 6, 10, 22, -3, -20, -32, 6, 2], [-5, -2, 9, 28, 9, -4, -26, 2, 4], [-4, 10, 2, 26, -12, -30, -35, 21, -15], [-4, 0, 0, 17, -21, -8, -25, 4, 11], [27, -5, 11, -15, -7, -1, 28, -9, 11], [-20, 8, -7, -1, 6, 8, -33, -1, 5], [-8, 1, 4, 20, -4, 0, -1, 2, -1], [8, -23, -3, 4, 9, 5, -3, 1, 2], [0, 10, 10, 26, -6, -3, -25, -3, -5], [-17, 2, 8, -19, 10, 6, 9, 3, -8], [-10, -11, 0, 0, 3, -21, 16, 5, -15], [-27, 21, 3, -7, -6, 4, 12, 0, -2], [-1, 13, 12, 13, 1, 11, -19, 8, -2], [-24, 1, -12, 26, 2, 0, -18, -4, 11], [23, -12, -8, 9, 15, -17, -19, -9, -12], [23, 4, -8, 34, 16, -19, -21, -6, -22], [16, -4, -1, 12, -12, -8, -5, 12, -1], [12, -12, 2, 22, -3, -11, 14, 13, 1], [3, 1, -3, -12, -12, -2, 8, 10, -20], [-23, -13, -23, 33, -12, -2, -1, 3, 0], [27, -21, -3, 6, 4, -12, -13, -1, -12], [10, -18, 0, 5, 4, -14, 4, -6, -12], [-32, 10, 10, 21, -2, -6, -13, 12, -15], [32, 8, -3, 3, 10, -14, 13, -20, 0], [-9, 13, -1, 27, 2, 4, 0, -1, -4], [17, -5, -1, -6, -18, 2, 8, 16, -20], [15, -6, -7, 4, 1, -12, -19, 10, -6], [36, 4, 18, 5, 12, 6, 25, -11, 6], [6, 10, 9, -22, 4, 9, 18, 3, -19], [-19, 18, -3, -21, -3, 12, 8, 14, -8], [6, 7, 4, 6, -8, 7, -15, 23, -11], [23, 0, 7, 8, 6, 6, -25, -1, 5], [-3, 9, 0, -26, 0, -5, 8, -9, 7], [26, 8, -6, -7, 4, 8, 22, -5, -9], [-10, -11, -21, 0, -1, 0, 4, -4, -16], [-28, -3, -3, -32, -5, 12, 16, -5, 13], [19, -23, -22, 11, -10, 4, 4, -6, 5], [12, 22, 20, -23, 9, 3, 3, -23, 0], [13, -10, 9, -4, 6, -10, 16, -4, -4], [-25, -22, -19, 12, -3, -12, -6, 13, 7], [-3, 9, 6, -14, -9, -6, 15, -4, 9], [2, -7, 5, -13, 17, 11, 28, -23, 22], [2, -1, 7, -8, 26, 21, 9, -10, 2], [16, 19, 11, 18, -11, 2, -3, 7, -10], [-18, -21, -23, 13, -22, -23, -8, -2, 15], [33, -8, -17, 3, -11, -13, -2, 13, -8], [6, -12, -10, 13, -3, 2, 9, 5, -8], [3, -9, 3, -3, -3, 4, -8, 7, 5], [-15, 4, 4, -9, 12, 7, -6, 5, -6], [-24, -4, 3, -7, -5, 6, -30, -2, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1805_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1805_2_a_u();".
// 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_1805_2_a_u(:prec:=9)
    chi := MakeCharacter_1805_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_1805_2_a_u();".
// 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_1805_2_a_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1805_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-19, 27, 51, -72, -27, 48, 4, -12, 0, 1]>,<3,R![-19, -75, 186, -6, -147, 39, 37, -12, -3, 1]>],Snew);
    return Vf;
end function;