
// Make newform 1805.2.a.v in Magma, downloaded from the LMFDB on 23 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_v();".
// 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_v();".
// 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 := [-1, 0, 15, -8, -27, 12, 16, -6, -3, 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], [0, -7, 4, 10, -4, -3, 1, 0, 0], [0, 0, -7, 4, 10, -4, -3, 1, 0], [-1, 1, -1, -14, 7, 13, -5, -3, 1], [2, -2, -14, 7, 13, -5, -3, 1, 0], [0, -3, 2, 14, -7, -13, 5, 3, -1], [0, 8, -4, -17, 8, 13, -5, -3, 1], [2, 7, -1, -34, 5, 30, -7, -7, 2]];
    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 := [[1, -1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 0, 0, -1, -1, 0], [0, -1, -1, 0, 1, 0, 1, 1, 1], [0, 2, 0, 0, 1, 2, 1, -1, 1], [-1, 0, -1, -1, -1, 0, -1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 2, -1, -2, -2, 0, 0, -1], [0, -1, 0, -2, 0, -2, 0, 1, -2], [1, 2, 1, 0, 1, -2, -1, 0, -1], [4, -3, 1, -1, -1, -3, -2, 1, 1], [2, -2, -2, 0, -2, 1, 0, 1, 0], [-1, 1, -3, 0, 1, 0, -1, 0, 2], [1, 1, -3, 1, -1, 3, 3, 0, 0], [0, 2, -1, -1, 0, 1, 3, 1, 1], [1, 3, -2, 0, 0, 2, 0, 2, -1], [-3, 4, 1, 2, 2, 0, 2, -2, 1], [7, 2, 0, 0, 0, 1, -1, -1, 2], [2, -4, 3, 1, -3, -4, -3, 1, 1], [4, 3, 0, 6, 0, 1, 2, 0, 2], [0, 0, -4, -3, 2, 1, -1, 2, -1], [-2, 0, 0, 2, 5, 1, 1, 3, -2], [-3, 0, -4, 1, 0, 3, -1, 1, 2], [1, 2, -3, 3, 3, 3, 0, 1, 0], [3, -4, 2, 0, -1, -2, -2, 0, 0], [2, 4, 0, -2, 3, 3, 2, -2, 3], [7, 0, -5, 3, -3, 0, 0, -1, 0], [-1, -2, 3, -4, -2, -2, -2, -3, -2], [0, 0, 3, -1, 3, -6, 0, 5, -1], [7, 0, 6, 2, -2, -1, -2, -3, -4], [-1, 3, -1, 2, -3, 4, 0, 0, 1], [0, -2, 0, 1, 3, -3, -3, 3, -4], [-4, 5, -1, 6, 2, 2, 3, 2, 1], [2, -3, 4, -4, -5, -6, -4, -5, -1], [3, 6, 0, 3, -1, 5, 4, 0, 0], [-5, 6, 6, 0, 1, -6, 0, -1, -1], [-5, 2, -4, 0, 0, 3, 1, 2, -3], [11, 3, 0, 0, -2, -1, -1, -3, 2], [2, 1, -7, 2, -2, 2, 4, 4, -1], [6, -5, 6, 1, -4, -6, -3, -1, 2], [0, -3, 7, -6, 4, -6, -4, 3, -5], [-1, 3, -3, -3, 4, 3, -1, 1, -2], [-5, -2, -7, -2, 5, 2, 3, 3, 4], [0, 2, -3, 1, 0, 0, 0, 2, 2], [-8, 10, 6, 1, 0, 6, 2, -3, 2], [9, -3, -2, 3, -5, -3, 2, -5, 1], [2, -3, -5, 6, 0, 1, 6, 3, 2], [-2, -3, 1, -3, 2, -6, -1, 3, -6], [0, 2, -13, -1, 1, 3, -2, 0, 2], [10, -5, 0, 1, 1, -3, -1, 3, 3], [-6, 3, -6, 2, 3, 1, 8, 3, -1], [3, 0, 5, 5, -2, -3, 1, -5, 5], [7, -1, -4, 4, -4, 5, 3, -2, 3], [4, 2, -2, 8, 3, 12, 6, 1, 4], [-1, -9, -2, -1, 0, -8, 1, 7, -4], [-12, 5, 1, -4, 0, 0, -2, -7, -1], [-1, 4, 2, -3, 5, 0, 0, 3, -2], [8, 4, 5, 3, -6, 0, 1, 2, 3], [2, -1, 3, 1, 0, 5, 3, -3, 5], [2, -9, 2, 4, 0, -7, -1, 4, -5], [5, -7, 6, -1, -7, 1, -1, 5, -5], [10, 0, 13, -1, -2, -6, -4, -1, 1], [12, -8, 2, 4, -7, -7, -4, -7, 1], [-9, 0, 0, -1, 5, -2, 0, 3, 1], [-3, -6, 4, 3, -1, -11, 4, 1, -5], [9, 3, -4, -4, 2, -2, 2, 2, 0], [14, 0, -7, 2, -5, 2, -4, -8, 3], [0, -7, 4, 6, -6, -3, 1, 6, -8], [8, 2, -5, -2, 1, 4, 3, 2, 0], [1, -3, -6, 4, -2, 5, 3, -4, 1], [-8, 7, 3, -5, 11, 0, -1, 4, -2], [-5, 0, -6, -4, 0, 9, -1, -3, 8], [-4, 4, 2, 3, -1, 4, 0, -2, 1], [6, -4, 0, 0, 3, 8, 1, 1, -3], [7, -8, 3, -3, -6, -6, -2, 4, -5], [-3, 6, 6, 2, -3, 10, -1, -3, 1], [-12, 8, 4, -2, 9, -1, 4, -1, 0], [-8, -2, 7, -1, 3, -13, -1, -3, 0], [-2, 1, 11, 3, -4, -8, 1, 1, -4], [3, 1, -2, 3, -6, 8, -5, 2, 5], [2, -1, 5, 2, 4, 2, -6, -12, -3], [-2, 9, 2, 10, 3, 6, 12, 1, 1], [2, 5, -1, 10, -1, 1, -1, 3, 5], [9, -9, 4, 0, -3, -12, 2, 2, 5], [5, -15, 8, -7, -6, -16, -6, 4, 0], [-1, -5, -2, -1, -4, 8, 1, -5, 3], [2, -2, 3, 3, -8, -3, 2, -9, 5], [10, 5, 9, 9, 2, -2, 1, -3, 11], [-2, -11, -2, -8, -2, -4, -4, 10, -4], [10, -2, -1, -1, -2, -9, -10, -1, -6], [-8, -5, -9, 3, -5, 5, 5, 6, 5], [9, -4, -4, -1, -10, -2, 2, -1, 2], [-5, 4, 2, 8, -8, -4, 4, -4, 2], [1, -5, -5, 1, -6, 11, 2, 9, 2], [9, -1, 5, 4, -2, 6, 2, -4, 6], [2, -9, -1, 7, -5, -4, -4, -4, -3], [0, -12, -2, -5, 6, 1, -1, 3, 1], [14, 3, 4, 1, -3, -5, -1, -11, 0], [-6, -6, 2, 2, 2, -7, -3, 6, -11], [17, -7, -4, 3, 6, -2, 1, 8, 0], [-5, -5, -2, -3, -1, 1, -2, 3, 2], [8, 4, -3, -2, 3, -1, -5, 4, -1], [-7, 5, 9, 1, 13, -9, 4, 9, -4], [5, -4, -11, -3, 7, 8, 4, 7, 2], [9, -4, -8, 10, -2, 0, 5, 3, 2], [-5, -8, -3, -8, 2, 3, -1, 10, 0], [5, 8, 21, -5, -1, -8, 1, -4, -3], [-15, 3, -6, -6, 10, 2, 4, 6, 5], [7, 5, 6, 3, -10, 5, 1, -6, -4], [5, 8, -1, 1, 7, -2, -5, -2, 2], [-5, -14, 1, -15, -1, -11, -8, 11, -5], [9, 0, 1, -6, 2, -3, 0, -2, 2], [-8, -1, 6, -10, 8, 0, -5, 0, 1], [-10, -6, -9, -12, 4, 0, -6, 5, -3], [-12, 3, 3, -7, 11, -1, 4, -4, 2], [11, 2, 7, -9, -5, 3, -1, -7, -2], [8, 0, 7, 3, -3, -4, 6, -6, 0], [-13, -2, 1, -3, -1, 16, 2, 2, 3], [-2, 1, -15, -2, 7, 9, -4, -4, 6], [-9, 7, -1, 1, 5, 14, -4, -2, -1], [-6, -5, 3, -5, -1, 1, -4, 14, -3], [-2, -4, 4, 5, 4, 0, 3, -8, 4], [3, 8, -6, 12, -1, 12, 5, -7, 9], [-6, 7, 9, -2, 2, -6, 4, -4, -1], [-5, -5, -2, -7, -2, -3, -4, -4, 1], [9, 3, 2, 17, -3, 7, 1, -8, 8], [-4, 3, 13, -1, -3, 5, 0, 4, -7], [3, -4, 3, -5, 7, 8, 2, -2, 6], [4, 4, -5, 4, 7, 6, 3, 0, 2], [4, -1, 7, 1, -5, 1, 0, 2, -3], [10, 3, 7, -2, 1, 11, -1, 2, 4], [4, -4, -5, -9, 2, -1, -4, 5, 4], [10, -9, -9, 0, -2, 11, -5, 0, 2], [-6, 0, 1, -4, -3, -11, -10, 0, -7], [-4, -5, -2, 6, 0, -4, 6, -6, -2], [-4, 11, -8, 5, -4, 3, 5, -2, 9], [8, 6, -2, -4, -1, -2, -1, 0, -7], [8, -10, 3, -6, -11, -1, 0, 4, -2], [2, 5, -2, -2, -11, 9, -4, -11, 2], [25, -11, 3, -3, -10, -2, -5, 4, 4], [-6, -13, 5, -10, 0, -11, 0, 10, -2], [15, -1, 2, 0, 6, 2, 3, 5, 5], [8, 6, -3, -3, 3, 5, -2, -6, -8], [14, -6, 9, 4, -3, -5, -2, 5, -5], [-6, 19, -14, 5, 3, 8, 2, 5, 5], [0, 0, 8, -11, -9, -15, -4, -5, -2], [-4, -5, -3, 4, 10, 4, 8, 10, 8], [15, -6, -3, 8, -14, 6, -2, -11, 3], [12, 2, 7, 0, 0, -4, -6, -15, 7], [-1, 0, -6, 9, -4, 11, 2, 7, 0], [-6, 5, -16, -2, 1, 13, 5, 3, -6], [15, -15, 8, -9, -8, -9, -12, -14, -6], [6, -9, 6, 2, -1, 7, -3, 2, -4], [13, -5, 2, 1, 0, 3, 0, 5, 2], [7, -16, -7, 4, -8, 1, 3, -3, 3], [-5, 13, -8, 6, 4, 13, 10, -7, 14], [2, 10, -10, 0, 5, 5, 5, -8, 6], [-5, -8, 11, -14, -8, -4, -8, 11, -10], [0, 18, 1, 15, 2, 18, 8, -13, 9], [2, 0, 16, -11, -3, -21, -13, -9, -4], [14, -8, 8, 9, -4, -17, -5, 11, 5], [6, -7, -1, 13, -4, -1, 4, -3, 5], [-6, 6, 16, 4, -3, 0, -3, -4, 1], [-15, 7, -2, 17, 11, 1, 9, 4, -4], [0, 9, 25, 7, -1, -13, -3, -6, 4], [3, -12, 8, -16, -15, 3, -10, 4, -4], [-1, -8, -6, 10, -3, -10, 1, -8, 3], [5, 3, -21, 0, 1, 16, 0, 1, -5], [14, -16, 2, -14, 0, -7, -5, 2, -6], [10, -19, 5, -8, -9, -3, -6, -4, -12], [14, -3, -9, 5, -17, 5, -8, 4, -7], [-8, -11, 4, 2, 5, -7, -4, 5, -12], [1, -3, -5, -5, 14, 9, -5, 0, 2], [14, -9, 9, -4, -15, -5, -7, -1, -3], [-10, 15, 8, -10, 21, -10, -2, -3, -5], [3, 16, 12, -2, -2, 0, 3, -10, 1], [-14, 13, -9, 12, 6, 19, 12, -2, 5], [15, 0, -7, -3, 8, -2, -5, 10, 3], [6, -20, 8, -18, -6, -24, -13, 3, -7], [-4, 2, -12, 3, 3, -5, 5, 4, -1], [3, 0, 13, -10, -5, -18, -9, -6, 1], [8, -14, 9, -13, 4, -11, -9, 2, -5], [-2, -9, -2, -3, -3, -12, 1, -3, -6], [5, 4, 12, 4, 5, -7, 8, 12, -7], [12, -2, 5, 7, -7, -3, -3, 1, -4], [-22, 1, -1, -8, 8, 13, 2, 7, -10], [7, -3, -3, 8, -8, 14, 0, 9, -1], [20, 10, -6, 8, -7, 9, 3, -8, 1], [13, -3, -6, -7, 3, -11, -2, 6, -7], [1, 8, -1, 1, 6, 4, 3, -9, -6], [19, -18, 15, -13, -13, -10, -7, 1, -8], [0, 13, -14, 19, 11, 10, 9, 7, 9], [-28, 3, -3, -11, 1, 11, -3, -11, 2], [-13, -6, 1, -11, 12, 0, -4, 3, -3], [-4, 17, -18, 7, 1, 17, 5, -3, 2], [15, 15, -2, 16, 3, 13, 7, -3, 15], [-1, 2, 1, -11, -4, 12, -6, 7, -3], [-3, 27, -4, 10, -3, 26, 12, -12, 19], [-20, 2, 8, 7, 6, -20, -2, 3, -7], [-18, 17, 2, 6, 1, -10, 0, 9, -4], [-11, 19, -8, 13, 11, 13, 8, 4, 1], [-21, 4, -6, -5, -4, 11, -2, -12, 2], [4, -2, 13, -15, 1, -12, -5, -11, 6], [-24, 12, 6, 0, 13, -3, 3, -2, 9], [2, 1, -2, -5, 1, 17, 0, -4, 1], [11, -3, 1, 2, -7, 2, 0, -7, -3], [8, -2, -9, -10, -2, -3, -9, -6, -1], [5, 2, -4, -3, -2, -5, 0, -14, 8], [-8, -5, -22, 14, 13, 6, 4, 10, 9], [4, -7, -7, -8, -6, 3, -11, -8, -3], [-7, 19, -6, -6, 5, 3, -1, 7, 0], [7, -1, -5, 16, -12, 6, 8, 7, -3], [-22, 19, 12, 0, 11, 7, 10, -6, -1], [-20, 0, -13, 13, -4, 4, 8, -3, -2], [-4, -8, 11, -5, -9, -6, -8, -4, -10], [7, -18, 1, -3, 1, -17, 1, 8, -15], [-22, -1, 0, 0, 0, -7, -4, -4, 1], [-12, 7, -4, 0, 9, 2, 2, 5, -14], [1, 9, -20, 22, -15, 4, 15, 3, 8], [-8, 7, 2, -5, -2, -11, 4, -3, -11], [-18, -7, -5, 3, -7, -5, -5, 2, -4], [2, -13, 10, 4, -9, -13, -14, 2, 5], [-33, 6, -9, -3, -1, 3, 1, -5, 6], [-5, 7, 11, -15, 1, -14, -5, -11, -2], [31, -22, 3, 3, -1, -5, -10, 0, 0], [-6, -4, 4, -9, 14, -4, 0, 15, -13], [9, 7, -1, 0, -16, 4, -2, -10, -1], [-10, 1, -10, 0, 8, 23, 14, 0, 8], [0, 11, 8, 7, 7, -8, 7, -10, 13], [-10, 4, -15, 17, -1, 7, 5, 17, 9], [-8, 23, 12, 8, -1, 14, -1, -6, 5], [8, 7, -15, 10, -16, -1, 7, -7, 2], [-18, 9, -11, -3, 2, 8, 1, -1, 2], [-4, 6, 12, -2, 16, 0, 1, -1, -6], [-15, 13, -9, 8, 4, 6, 10, 16, 12], [32, -12, -1, -6, -6, 2, -3, -7, -2], [-8, -1, 20, -3, 5, -10, -13, -1, -8], [10, -4, 6, 3, 10, 10, -7, -4, 12], [-6, 1, 15, -3, 0, 13, -8, 0, -5], [-4, -20, 20, -9, -2, -36, -14, -1, -9], [-19, 20, -11, 11, 8, 15, 10, 11, -1], [-8, 11, -28, 2, 7, 6, 3, -8, 4], [10, 12, -10, 9, 27, 13, 16, 7, 11], [13, 15, -7, 3, -9, 19, -2, -2, 6], [17, -4, 21, -13, -3, -26, 1, -2, -1], [11, 13, 0, -1, 3, 15, 7, 4, -10], [2, 2, 15, -13, -11, -13, -13, -7, -8], [9, -1, 10, -4, 17, -14, 7, 4, -9], [12, 3, -5, 1, -3, -9, 7, -12, 4], [-8, -4, -15, -17, 12, 14, -4, 24, -10], [-8, -5, 13, 5, -8, 1, 1, 7, -15], [12, -6, -1, -14, -5, -15, -3, -5, -2], [-2, -21, 12, 3, -4, -29, -12, 19, -5], [-3, 12, 5, -4, 11, -14, -7, 13, -2], [17, 21, -4, 0, -5, 13, -4, -1, 9], [-21, 18, -5, 4, 5, 13, 4, -1, -9], [-3, 8, -15, -10, 10, 12, 11, 7, -4], [-25, 0, -2, -2, 8, 14, 1, 17, -10], [22, -17, -9, 0, -10, -3, -8, -17, 7], [10, 13, -8, 13, 5, 11, -3, -13, 8], [-15, -10, -11, 7, -6, 15, 4, 2, 8], [-17, 0, 12, -5, 4, 11, -10, -16, 5], [-12, 12, -1, 4, 5, 3, 12, 1, 17], [-8, 18, 9, 6, 1, 8, 11, -15, 3], [-16, 21, -19, 19, 11, 6, 16, 7, 15], [-1, -17, -5, -2, -15, -13, 8, 15, 4], [3, -28, 17, 8, -2, -39, -11, 17, -16], [4, 14, -22, 27, -7, 4, 13, 8, -3], [6, 22, 21, 17, -4, -4, 1, -12, 17], [-22, -4, 0, -25, -3, -5, -10, 1, -13], [-1, -13, 26, -10, -2, -11, -6, -5, 1], [-31, 23, 14, 15, -3, 19, 3, -18, 3], [21, -17, 41, -6, -8, -32, -12, -13, -6], [-39, 0, 4, -10, 4, 5, -3, -6, -9], [-3, 16, 16, 12, 10, 7, 9, -12, 3], [-5, 17, -21, -1, 9, 23, 4, 6, 4], [-22, 19, -6, 8, 7, 7, 6, 14, -8], [-3, -12, -5, -26, -3, 1, -7, 14, -8], [7, 11, -14, 1, -3, 17, -3, -11, 15], [-3, 8, -25, -7, 5, 16, 2, 8, 8], [29, -4, 14, 4, -13, -16, -1, -2, 7], [-27, -5, 8, -17, -18, -8, -15, -8, -10], [-34, 10, 8, -23, 7, 1, 1, -4, -2], [24, -8, 1, -2, 0, 0, -13, -19, -8], [36, -6, -10, -6, -7, 5, -6, 13, -9], [-9, 10, -25, 5, -3, 14, 23, 15, 8], [-18, 0, 3, 9, 1, 6, -4, -2, -12], [-24, 2, -14, -9, 14, 15, 7, 19, -12], [5, -5, -10, -23, 2, 0, -11, 18, -2], [-11, 1, -19, -5, 9, 12, 11, 18, -9], [10, 12, -4, 21, -16, -4, 12, -8, 12], [5, -21, 6, -22, -16, -21, -16, 9, -6], [-19, 2, 2, -6, -7, -2, -5, -8, 1], [2, -19, 10, -22, -8, -20, -5, 8, -4], [6, 19, -12, 4, 7, 14, 7, 17, 2], [-13, 10, -8, 10, 21, 1, 10, -10, -1], [-1, -6, -14, 3, -15, -10, -2, 21, -1], [21, -7, 14, 12, 7, -17, 11, 11, -11], [-38, 0, 8, -6, 5, -6, 9, 1, 8], [5, -1, 25, -8, -2, -20, -10, -3, -5], [-1, -13, 28, -3, -18, -18, -3, -15, 3], [2, 3, -9, 11, 6, 8, -5, -14, 15], [9, -14, 14, 12, -9, -15, -4, 21, -7], [-11, -9, -7, -7, -8, -2, -13, -9, 1], [-16, 13, 3, 6, 9, 12, 13, -13, 10], [-19, 18, -5, -24, 19, 21, 5, 5, 5], [13, 12, -13, -8, -7, 16, 8, -1, -2], [-3, -5, 1, 9, -8, -17, 3, 15, -14], [-21, 3, 8, 13, -10, -9, -7, 15, 0], [-12, 9, -14, -8, 14, 8, -3, -4, -8], [-30, 25, -14, 3, 7, 16, 6, 3, 3], [-20, -21, -3, -11, 8, 3, 2, -6, -2], [-2, 17, 2, 12, -1, 11, 15, 0, -1], [-1, -3, 13, -5, 5, -24, -9, 14, -4], [14, -9, 6, -7, -10, -13, -3, 10, -7], [7, -5, 11, 13, -1, -23, -1, 19, 1], [-21, -9, 7, -18, 5, 12, -10, 4, 1], [-21, -10, -4, -8, 22, -13, 10, 10, 0], [-5, -1, 25, 4, 8, -9, -1, 2, 12], [11, -8, 6, 6, 5, -22, 16, 8, -8], [4, 17, -9, 25, -5, 11, 14, -5, 4], [17, -1, 26, 14, -5, -20, -8, -23, 9], [3, 0, 0, 10, -4, -20, 3, 13, -13], [-18, 0, -3, -3, 6, 1, -9, 14, -11], [-14, 4, 5, 18, -12, 20, 7, -14, 4], [22, 12, -3, -4, -5, -14, -2, -16, 9], [-9, -6, -5, -11, -4, -12, 3, 23, -6], [14, -2, -27, -1, 9, 15, -6, -23, 12], [-17, -16, 20, -12, 15, -33, -3, 17, -9], [17, 2, -20, 27, -15, 12, 17, -3, 10], [-14, 7, 4, 16, 4, -11, 16, -16, 6], [14, -10, -23, 6, 5, 11, -4, 20, -3], [0, 14, -15, 6, -6, 21, 6, -25, 4], [2, -4, 14, 4, 1, -30, -18, 5, 5], [-14, -12, -8, -8, -19, 6, -13, 6, -8], [20, -25, -8, 9, 4, 1, 5, 14, 5], [39, 4, 5, 7, -18, -10, -9, -15, 18], [-4, 20, 2, 20, -9, 13, 22, -3, 10], [4, 24, 29, -10, 14, 13, -1, -11, -2], [5, -16, 0, 11, 2, 4, -6, -14, 13], [-6, -5, 11, -1, -8, -25, 1, 9, -11], [12, -5, 0, 14, -3, -7, 5, 19, -9], [8, -20, 12, -2, -14, 7, 1, -3, 14], [-13, 4, -21, -9, -2, 3, -2, 13, -12], [6, -14, 6, 7, -1, -18, -10, 20, -2], [-3, 4, 22, -5, -4, -6, -5, 13, -9], [16, -39, -10, 1, 3, -12, 3, 29, -1], [-31, 10, -11, -2, 0, -5, 9, -3, -15], [-2, -10, 7, -17, -7, -10, -4, 2, -13], [29, -17, -16, 4, 9, -10, 0, 26, 5], [4, -9, -10, -4, 7, -7, -14, -3, -14], [1, -11, -1, -7, -3, -4, -23, 13, -20], [-8, -7, -13, -7, -17, -10, -6, 0, -11], [8, -1, -1, -5, -1, -7, -16, 18, 1], [18, 7, 15, -11, 6, 6, -4, -7, -11], [-30, -16, -10, 5, -5, -9, 2, 0, -11], [-2, -9, 0, -6, 14, 10, 4, 13, -4], [-26, 8, -4, -8, 17, -1, 9, 6, 6], [-7, 7, -14, -24, 5, 5, 6, 11, -9], [-13, -5, -3, 2, 3, 21, 6, 13, -1], [6, 3, 11, 21, -5, 9, 3, -5, 8], [-5, 4, -3, 1, 5, -3, 9, 10, -12], [-9, 8, 22, -9, 6, 4, -3, -16, -4], [-18, 16, -10, 2, 6, 0, 14, -1, -10], [13, 12, -3, -3, 4, 1, -12, 7, 7], [-29, 8, -17, 6, 4, 21, -2, -8, -13], [-3, -4, 17, 6, -7, -22, 9, -11, -4], [20, -10, -21, -11, -8, 2, 5, 26, -9], [0, -23, -2, -6, 9, -7, 2, 16, 0], [31, 24, -19, 18, -15, 32, 11, -10, 3], [10, 18, -13, 0, -1, 30, 18, -20, 5], [3, -24, -9, -13, -18, -5, -7, 6, -5], [32, -21, 0, -3, -15, 6, -1, -4, -7], [-24, 21, 6, 18, 6, 1, 9, 6, 8], [-7, -16, 8, -27, 8, -18, -11, 31, -16], [30, -9, -27, -8, 2, 5, -5, -2, 6], [0, -10, -39, -6, 0, 19, 1, 3, 7], [-4, 11, -33, 1, 8, 21, 0, 1, 11], [7, 8, 15, 14, 6, -5, 14, 10, -12], [9, -18, -13, 5, -2, 4, 8, 22, -6], [17, 7, -3, -3, -11, 18, -11, -12, 20], [16, -3, 21, -13, 4, 1, -5, -12, 7], [18, 26, -2, 7, 8, 0, 2, -6, -15], [8, 8, -10, -4, -7, 18, -16, -17, -1], [6, 4, 22, -5, -13, -17, -12, -8, 3], [-5, -9, 38, -12, -12, -22, -8, -12, -1], [25, -12, -7, 22, -1, 12, 10, 1, 20], [18, 0, 11, 5, -8, -23, -5, 3, -11], [9, -9, -11, -5, 8, 16, -13, 10, -6], [-9, 21, 1, 4, 18, -2, 5, 11, -7], [-10, 14, 7, 0, 15, 18, -5, -8, -2], [-10, 6, 19, -5, -10, -19, 0, -7, -18], [17, -5, 19, -22, 8, 1, -8, -24, 1], [-31, 8, 16, 4, -6, 14, -3, -9, 10], [29, -9, 7, 10, 2, -3, -2, -5, 0], [-16, 19, 4, -2, 9, 12, -9, -20, 17], [0, 2, 3, 10, -6, 0, -1, -7, -11], [30, 8, -2, 10, 11, 18, 9, 2, 4], [2, -13, 1, 11, -5, 8, 4, -6, 6], [-32, 19, -2, -15, 21, 24, 12, 4, -10], [16, 0, -25, -4, 2, 16, -6, -18, 8], [-4, 8, -19, 4, 2, 22, -12, 0, -7], [-26, -1, -22, -20, 19, 19, -7, 18, -18], [26, 15, -6, -1, -17, -3, -2, -20, 10], [-14, 2, -8, -18, -5, -12, -5, 14, -12], [11, -1, 0, 13, -9, -1, -1, -18, -6], [28, -1, 5, 19, -8, -6, -3, -22, 20], [6, -17, 32, -13, -5, -42, -22, -3, -5], [-13, -17, 36, -29, 3, -26, -17, 6, -16], [-9, 3, -15, 3, 6, 13, -3, -1, -15], [9, -8, 17, 15, 9, -21, -4, 0, -1], [28, -25, 4, -21, 8, -20, -19, 23, -23], [-14, 13, 9, -3, -6, 36, -9, -24, -3], [27, -10, -7, 1, -1, -2, -1, -22, 14], [23, -29, -23, -20, 1, -8, -10, 26, -21], [10, -1, -14, 18, 2, 37, 2, -8, 15], [-14, -14, 12, -8, 8, -36, 15, 11, 0], [23, -1, -2, -14, 1, -22, -13, 14, 4], [31, 16, -1, 14, 3, 22, 18, -16, 15], [-8, -17, -25, -17, 6, 6, 0, 4, -1], [11, -20, -13, 28, -7, -13, 0, -5, 4], [61, -25, -5, 12, -15, -17, -25, -7, -4], [-1, 3, -17, 12, -25, 4, 10, -8, 4], [-23, 13, -20, 18, 17, 23, 17, 13, 9], [1, -9, 29, -32, 15, -41, -20, 21, -16], [16, -5, 38, -8, -6, -14, -19, -8, -8], [36, -29, 12, 10, -17, -20, -4, 14, -1], [-31, 14, 14, -13, 14, -11, 5, -1, -4], [-8, 1, 9, -17, -5, -11, -21, 28, -19]];
    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_v();".
// 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_v(: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_v();".
// 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_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1805_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -3, -15, 30, 21, -60, 26, 6, -6, 1]>,<3,R![1, -9, -48, 44, 81, -87, 1, 24, -9, 1]>],Snew);
    return Vf;
end function;