
// Make newform 1805.2.b.i in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1805_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1805_b_Hecke();"
// 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_b_i();".
// 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_b_i();".
// 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 := [16, 0, 256, 0, 1163, 0, 2154, 0, 1862, 0, 820, 0, 190, 0, 22, 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, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3016, 2956, -22718, 6327, -38832, 1846, -27252, -3934, -9564, -3844, -1768, -1342, -164, -202, -6, -11], [3016, 2956, 22718, 6327, 38832, 1846, 27252, -3934, 9564, -3844, 1768, -1342, 164, -202, 6, -11], [-2808, 2956, -29674, 6327, -70592, 1846, -62740, -3934, -26348, -3844, -5688, -1342, -612, -202, -26, -11], [0, -10284, 0, -46879, 0, -63490, 0, -40990, 0, -14580, 0, -2918, 0, -306, 0, -13], [1276, 0, 14393, 0, 39090, 0, 41494, 0, 20776, 0, 5262, 0, 650, 0, 31, 0], [0, 3760, 0, 22015, 0, 38762, 0, 31334, 0, 13408, 0, 3110, 0, 366, 0, 17], [0, -1572, 0, -14393, 0, -39090, 0, -41494, 0, -20776, 0, -5262, 0, -650, 0, -31], [3332, 0, 36519, 0, 95610, 0, 96798, 0, 46100, 0, 11142, 0, 1322, 0, 61, 0], [0, 248, 0, 1163, 0, 2154, 0, 1862, 0, 820, 0, 190, 0, 22, 0, 1], [-2012, -2880, -8769, -27491, -9856, -66446, -856, -63992, 3492, -29754, 1704, -7126, 292, -844, 17, -39], [-924, -2880, -16835, -27491, -47100, -66446, -46284, -63992, -21128, -29754, -4900, -7126, -560, -844, -25, -39], [-924, 2880, -16835, 27491, -47100, 66446, -46284, 63992, -21128, 29754, -4900, 7126, -560, 844, -25, 39], [0, -21540, 0, -185851, 0, -442970, 0, -427886, 0, -198980, 0, -47486, 0, -5594, 0, -257]];
    Rf_basisdens := [1, 1, 1, 592, 592, 592, 592, 296, 296, 296, 296, 8, 296, 296, 296, 592];
    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_b();"
function MakeCharacter_1805_b()
    N := 1805;
    order := 2;
    char_gens := [362, 1446];
    v := [1, 2];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1805_b_Hecke();"
function MakeCharacter_1805_b_Hecke(Kf)
    N := 1805;
    order := 2;
    char_gens := [362, 1446];
    char_values := [[-1, 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]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [-2, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, -1, 0, 0], [0, 2, 0, 0, 0, 0, -1, 0, -2, 1, 0, 0, 0, -1, 1, -1], [0, -2, 0, 1, 1, 0, 0, 0, 2, -1, 0, -1, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 1, 1, 0, 0, 0, 2, 1, 0, -1, 0, 0, 0, 0], [0, 0, 0, -1, 1, 0, 0, -1, 0, 0, 1, 0, 2, -1, 1, 0], [2, 0, 1, -1, 3, -2, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0], [0, 2, 0, -1, -1, 0, 1, 0, -1, 2, 0, 1, 0, -1, 1, -1], [0, 0, 1, -1, -2, 3, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 2, 0, 1, -4, 0, -1, 0, 1, -1, -1], [0, 4, 0, -2, -2, 0, 1, 0, -3, 3, 0, 1, 0, -1, 1, 1], [0, -1, 0, 0, 0, 0, 0, 0, 1, -2, 0, 1, 0, 2, -2, 1], [-1, 0, 3, 1, 0, -1, 0, 0, 0, 0, 1, 0, 0, -1, -1, 0], [0, 0, -1, -2, 2, 0, 0, 2, 0, 0, 0, 0, 1, 1, 2, 0], [0, 3, 0, -1, -1, 0, -3, 0, -1, 2, 0, 1, 0, 0, 0, 0], [0, 0, 0, 3, -1, -2, 0, 3, 0, 0, 1, 0, -1, 0, -1, 0], [0, 2, 0, -1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -1, 1, 1], [-3, 0, 2, 2, -1, -1, 0, -2, 0, 0, 0, 0, 2, 0, 2, 0], [0, 5, 0, 0, 0, 0, 0, 0, -2, 4, 0, 1, 0, -2, 2, 0], [1, 0, -1, -2, 2, 0, 0, 0, 0, 0, 1, 0, -1, 2, 1, 0], [0, -3, 0, -1, -1, 0, -2, 0, 3, 1, 0, 0, 0, 0, 0, 2], [-8, 0, 1, -1, -1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 2, 2, 0, 0, 0, 0, 3, 0, -2, 0, -1, 1, 2], [0, -2, 0, 3, 3, 0, -3, 0, 1, 0, 0, 0, 0, 1, -1, 0], [0, 0, 0, 1, -1, 0, 0, 1, 0, 0, -3, 0, 1, -1, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 1, -1, -1], [0, 1, 0, 0, 0, 0, -2, 0, -2, 4, 0, -3, 0, -1, 1, -1], [-2, 0, -1, 1, 1, -2, 0, -1, 0, 0, -5, 0, -1, 1, 0, 0], [0, -5, 0, 3, 3, 0, 5, 0, 4, -6, 0, 2, 0, 3, -3, 0], [4, 0, 1, 0, 2, -2, 0, 2, 0, 0, -1, 0, 4, -1, 3, 0], [0, 0, 2, -3, 0, 3, 0, -6, 0, 0, 0, 0, 3, -2, 1, 0], [-1, 0, 1, -1, 2, -1, 0, -4, 0, 0, -1, 0, 0, 2, 2, 0], [0, 4, 0, 1, 1, 0, -2, 0, 1, 4, 0, 0, 0, -3, 3, 1], [0, -9, 0, 3, 3, 0, 1, 0, 1, -2, 0, -1, 0, 2, -2, 0], [0, 0, 0, 0, 0, 0, 1, 0, -1, -2, 0, -2, 0, 0, 0, -2], [0, 10, 0, -3, -3, 0, 0, 0, -4, 5, 0, 1, 0, -1, 1, 3], [0, 0, -4, 0, -2, 2, 0, 1, 0, 0, -3, 0, -5, 3, -2, 0], [-4, 0, -1, 1, -1, 0, 0, -7, 0, 0, -3, 0, 0, -2, -2, 0], [-3, 0, 2, 1, 0, -1, 0, 4, 0, 0, -1, 0, -2, 0, -2, 0], [0, 6, 0, -3, -3, 0, 0, 0, -4, 1, 0, 3, 0, 1, -1, 1], [0, 1, 0, 1, 1, 0, -1, 0, 0, -3, 0, -1, 0, 0, 0, 1], [3, 0, 0, -1, 2, -1, 0, -5, 0, 0, 0, 0, 3, -1, 2, 0], [-4, 0, 1, -2, 2, 0, 0, -4, 0, 0, 1, 0, 2, 2, 4, 0], [0, 1, 0, -2, -2, 0, -1, 0, 1, 2, 0, -1, 0, 2, -2, -4], [0, 3, 0, 3, 3, 0, 0, 0, 2, 3, 0, 0, 0, -1, 1, 2], [4, 0, 0, -3, 4, -1, 0, 1, 0, 0, -2, 0, 0, 1, 1, 0], [0, -3, 0, 2, 2, 0, 5, 0, 2, -5, 0, 3, 0, 0, 0, -1], [7, 0, 1, 0, 1, -1, 0, 3, 0, 0, 1, 0, 1, 2, 3, 0], [-10, 0, 3, 1, -3, 2, 0, -1, 0, 0, -2, 0, 2, -3, -1, 0], [-13, 0, 0, 1, -4, 3, 0, -2, 0, 0, 3, 0, -4, 0, -4, 0], [0, -1, 0, 0, 0, 0, -2, 0, 0, -7, 0, 0, 0, 0, 0, -3], [0, 0, 0, -1, -1, 0, -1, 0, 4, -1, 0, -4, 0, 3, -3, -2], [7, 0, -3, -4, 3, 1, 0, -8, 0, 0, 3, 0, 0, 1, 1, 0], [-14, 0, -3, 0, 2, -2, 0, -4, 0, 0, -2, 0, 0, -1, -1, 0], [0, 7, 0, 1, 1, 0, 2, 0, -2, 2, 0, 3, 0, -1, 1, 2], [-10, 0, -3, -1, -3, 4, 0, 1, 0, 0, 2, 0, -5, 2, -3, 0], [0, 10, 0, 0, 0, 0, -9, 0, -5, 7, 0, -4, 0, -3, 3, 5], [0, 2, 0, -2, -2, 0, 2, 0, -7, 2, 0, 2, 0, -2, 2, -3], [0, 0, 0, 1, 1, 0, -5, 0, -4, -3, 0, -3, 0, -2, 2, -2], [-10, 0, 0, -1, -3, 4, 0, 3, 0, 0, 2, 0, 1, -4, -3, 0], [0, -4, 0, -1, -1, 0, 3, 0, -5, -3, 0, 6, 0, 3, -3, 0], [0, -1, 0, 2, 2, 0, 4, 0, 6, -4, 0, 3, 0, 1, -1, -2], [7, 0, 4, -3, 6, -3, 0, 2, 0, 0, 2, 0, 4, -2, 2, 0], [0, 5, 0, -2, -2, 0, -2, 0, -4, 2, 0, 3, 0, -1, 1, 2], [0, -9, 0, 3, 3, 0, -2, 0, -1, -4, 0, 0, 0, 2, -2, 4], [-2, 0, -2, -1, 3, -2, 0, -8, 0, 0, 0, 0, 3, -3, 0, 0], [0, -2, 0, 0, 0, 0, -3, 0, 1, 5, 0, 0, 0, -2, 2, -2], [6, 0, 3, 1, 3, -4, 0, -1, 0, 0, 4, 0, -1, 2, 1, 0], [0, -6, 0, -1, -1, 0, 8, 0, 0, -3, 0, 5, 0, 1, -1, -4], [0, 9, 0, 2, 2, 0, -9, 0, -4, 5, 0, 1, 0, -6, 6, 5], [0, 0, 2, -1, 3, -2, 0, 8, 0, 0, -4, 0, -4, 4, 0, 0], [0, -4, 0, 5, 5, 0, 1, 0, 5, 0, 0, -1, 0, 0, 0, -1], [3, 0, 2, 4, 5, -9, 0, -3, 0, 0, 2, 0, 4, -3, 1, 0], [0, -13, 0, 0, 0, 0, 9, 0, 12, -13, 0, 1, 0, 9, -9, 1], [-8, 0, -5, 3, 0, -3, 0, 2, 0, 0, -4, 0, 1, 0, 1, 0], [1, 0, 5, 1, 3, -4, 0, 7, 0, 0, 0, 0, 2, -1, 1, 0], [13, 0, 0, -2, 5, -3, 0, 12, 0, 0, 0, 0, 2, 2, 4, 0], [-1, 0, 7, 0, -1, 1, 0, 0, 0, 0, 2, 0, 3, -5, -2, 0], [6, 0, -6, 3, -1, -2, 0, 3, 0, 0, 3, 0, -7, 1, -6, 0], [0, -2, 0, 2, 2, 0, -5, 0, 1, 3, 0, 0, 0, 0, 0, 0], [6, 0, -4, 0, 2, -2, 0, 13, 0, 0, -5, 0, 1, 5, 6, 0], [0, -4, 0, 5, 5, 0, -2, 0, 2, -7, 0, 1, 0, -2, 2, 0], [-14, 0, 3, 0, -3, 3, 0, -11, 0, 0, 5, 0, -2, 0, -2, 0], [0, -16, 0, 8, 8, 0, -8, 0, 6, -8, 0, -2, 0, -1, 1, 2], [-4, 0, -3, -3, 3, 0, 0, -7, 0, 0, 5, 0, -4, 5, 1, 0], [0, -6, 0, 4, 4, 0, 6, 0, 9, -6, 0, 2, 0, 3, -3, 2], [0, 5, 0, 0, 0, 0, -8, 0, 2, 1, 0, -8, 0, -2, 2, -2], [12, 0, -1, 2, 0, -2, 0, 4, 0, 0, -5, 0, 0, -1, -1, 0], [0, 8, 0, 1, 1, 0, 2, 0, -5, 0, 0, -1, 0, -1, 1, 0], [-2, 0, 3, 6, 0, -6, 0, 7, 0, 0, 6, 0, -1, 0, -1, 0], [11, 0, 0, -3, 2, 1, 0, -4, 0, 0, 4, 0, 0, -3, -3, 0], [0, 9, 0, -2, -2, 0, -3, 0, -13, 2, 0, -7, 0, -4, 4, -4], [-4, 0, -2, -5, -4, 9, 0, -12, 0, 0, 5, 0, 1, -2, -1, 0], [12, 0, -1, 4, 0, -4, 0, 17, 0, 0, 0, 0, 1, 3, 4, 0], [0, 1, 0, -2, -2, 0, 2, 0, 4, 1, 0, -4, 0, -2, 2, 0], [9, 0, -6, -1, 6, -5, 0, 9, 0, 0, -1, 0, -3, 4, 1, 0], [0, -9, 0, -2, -2, 0, 4, 0, -3, -16, 0, 5, 0, 5, -5, 0], [0, -9, 0, 0, 0, 0, 11, 0, 6, 3, 0, -1, 0, 2, -2, -1], [0, -11, 0, -1, -1, 0, 3, 0, 4, -2, 0, -6, 0, 8, -8, -5], [-1, 0, 6, 5, -4, -1, 0, -2, 0, 0, 2, 0, -1, -2, -3, 0], [-3, 0, -1, 1, 2, -3, 0, 0, 0, 0, 4, 0, 6, -2, 4, 0], [0, -8, 0, 4, 4, 0, 5, 0, 4, 1, 0, -4, 0, 7, -7, 3], [0, -1, 0, 5, 5, 0, 2, 0, -4, -3, 0, -2, 0, -4, 4, -2], [0, 6, 0, -1, -1, 0, -1, 0, -10, -6, 0, -1, 0, 0, 0, -8], [3, 0, -6, 0, -3, 3, 0, 4, 0, 0, 5, 0, -2, 2, 0, 0], [22, 0, -2, -2, 10, -8, 0, 12, 0, 0, -10, 0, 2, 5, 7, 0], [0, -21, 0, 8, 8, 0, 0, 0, 11, -6, 0, -3, 0, -1, 1, 0], [0, 6, 0, 3, 3, 0, -3, 0, 0, 6, 0, -9, 0, -8, 8, 2], [0, 11, 0, -2, -2, 0, -12, 0, -5, 9, 0, 2, 0, -1, 1, 2], [2, 0, 2, 3, 1, -4, 0, 16, 0, 0, 1, 0, -2, 0, -2, 0], [-13, 0, 0, -4, -3, 7, 0, -7, 0, 0, 1, 0, 3, 0, 3, 0], [-7, 0, -2, 3, -1, -2, 0, -10, 0, 0, 2, 0, 1, -2, -1, 0], [0, -2, 0, 2, 2, 0, -4, 0, 5, 0, 0, 1, 0, 1, -1, 6], [0, 10, 0, -3, -3, 0, -2, 0, 4, -1, 0, -5, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, 0, 11, 2, 0, 0, 0, -3, 3, -5], [-14, 0, 0, 1, 3, -4, 0, -3, 0, 0, -6, 0, -1, 1, 0, 0], [-3, 0, -3, 0, 6, -6, 0, 0, 0, 0, -4, 0, 1, -1, 0, 0], [0, -10, 0, 0, 0, 0, 3, 0, 2, -8, 0, 5, 0, 6, -6, -6], [0, 2, 0, 0, 0, 0, 9, 0, 3, -5, 0, 4, 0, -2, 2, 4], [0, 0, 0, 9, 9, 0, 2, 0, 10, 5, 0, -3, 0, -3, 3, 10], [-3, 0, 4, -2, 7, -5, 0, 1, 0, 0, 3, 0, 7, -1, 6, 0], [-4, 0, 2, 5, 5, -10, 0, 2, 0, 0, -4, 0, 5, -2, 3, 0], [8, 0, 3, 4, -3, -1, 0, 6, 0, 0, -2, 0, -1, -3, -4, 0], [22, 0, -5, 0, -8, 8, 0, 0, 0, 0, 0, 0, -6, 4, -2, 0], [0, 1, 0, -3, -3, 0, -3, 0, -6, 4, 0, 2, 0, 4, -4, -1], [0, -1, 0, 0, 0, 0, 3, 0, -2, -12, 0, 2, 0, 7, -7, -2], [-4, 0, 0, -3, -5, 8, 0, -5, 0, 0, 3, 0, -3, 1, -2, 0], [0, -19, 0, 3, 3, 0, 5, 0, 4, -12, 0, 2, 0, 5, -5, -2], [16, 0, -7, 11, -3, -8, 0, 4, 0, 0, 2, 0, -4, -1, -5, 0], [0, 10, 0, 1, 1, 0, 7, 0, -5, 12, 0, -3, 0, -2, 2, -5], [-1, 0, 3, 2, 3, -5, 0, -8, 0, 0, 0, 0, 3, 0, 3, 0], [4, 0, -6, -7, -1, 8, 0, -3, 0, 0, -1, 0, -4, 3, -1, 0], [0, -14, 0, 6, 6, 0, 8, 0, 1, -15, 0, -1, 0, 2, -2, -12], [0, -10, 0, -1, -1, 0, 7, 0, -4, -2, 0, -1, 0, 8, -8, -3], [0, 9, 0, -10, -10, 0, 0, 0, -8, 2, 0, 5, 0, 2, -2, 4], [-1, 0, -6, 4, -10, 6, 0, 1, 0, 0, -3, 0, -6, -5, -11, 0], [2, 0, -7, -2, -6, 8, 0, -6, 0, 0, 5, 0, 2, -1, 1, 0], [13, 0, -2, 1, 9, -10, 0, 18, 0, 0, -8, 0, -1, 1, 0, 0], [0, 9, 0, -2, -2, 0, 0, 0, 5, 0, 0, 0, 0, 2, -2, 8], [0, -10, 0, 1, 1, 0, 8, 0, 0, -18, 0, -2, 0, 5, -5, -9], [-7, 0, -8, 0, 1, -1, 0, 3, 0, 0, -3, 0, -8, 8, 0, 0], [11, 0, 5, 7, -4, -3, 0, 7, 0, 0, -1, 0, -5, 5, 0, 0], [0, -15, 0, 3, 3, 0, 1, 0, 8, -2, 0, -6, 0, 6, -6, -4], [0, -2, 0, -5, -5, 0, -4, 0, -7, -4, 0, 2, 0, 8, -8, -5], [-2, 0, -6, 3, -3, 0, 0, 6, 0, 0, -2, 0, -10, 9, -1, 0], [0, -5, 0, 4, 4, 0, -1, 0, 0, -8, 0, -2, 0, -1, 1, 3], [0, 9, 0, -4, -4, 0, 2, 0, -2, -4, 0, -5, 0, -2, 2, 4], [-6, 0, 6, -2, 0, 2, 0, 1, 0, 0, 1, 0, 4, 2, 6, 0], [0, 9, 0, -4, -4, 0, 4, 0, -3, 13, 0, 2, 0, -3, 3, -1], [0, -6, 0, 1, 1, 0, 6, 0, 1, -1, 0, 10, 0, 9, -9, -1], [0, 4, 0, 3, 3, 0, 4, 0, 2, 1, 0, -5, 0, -5, 5, 3], [-10, 0, -5, 0, -4, 4, 0, 4, 0, 0, -9, 0, -4, 4, 0, 0], [-9, 0, -2, -5, 0, 5, 0, -11, 0, 0, -5, 0, -1, 4, 3, 0], [16, 0, -2, -1, 0, 1, 0, 4, 0, 0, -8, 0, -5, 4, -1, 0], [0, -7, 0, 0, 0, 0, -7, 0, 5, -3, 0, -9, 0, 0, 0, 4], [-10, 0, 4, -6, 3, 3, 0, 1, 0, 0, -1, 0, -6, 4, -2, 0], [0, -17, 0, 0, 0, 0, 11, 0, 18, -9, 0, 3, 0, 9, -9, 0], [0, -13, 0, 5, 5, 0, 3, 0, 8, -5, 0, 1, 0, 1, -1, 3], [0, 8, 0, -6, -6, 0, -2, 0, -5, -4, 0, 1, 0, -3, 3, -6], [1, 0, -9, -3, 0, 3, 0, -15, 0, 0, 4, 0, 1, -5, -4, 0], [0, 15, 0, -5, -5, 0, -5, 0, -8, 4, 0, 2, 0, -3, 3, -4], [0, -10, 0, 0, 0, 0, -8, 0, 2, -11, 0, -12, 0, 1, -1, -2], [-15, 0, -1, 9, -10, 1, 0, -10, 0, 0, 3, 0, -4, -1, -5, 0], [0, -10, 0, 3, 3, 0, -12, 0, -10, 2, 0, -8, 0, -3, 3, -6], [-1, 0, -3, 4, -3, -1, 0, 11, 0, 0, 3, 0, -4, 3, -1, 0], [0, 14, 0, -7, -7, 0, -14, 0, -11, 9, 0, 7, 0, -1, 1, 0], [-11, 0, 4, 8, 1, -9, 0, -18, 0, 0, -4, 0, 8, -5, 3, 0], [-19, 0, 3, -3, 6, -3, 0, -15, 0, 0, 4, 0, 6, -1, 5, 0], [0, 0, 0, 6, -4, -2, 0, -2, 0, 0, 7, 0, -4, -3, -7, 0], [0, -4, 0, -7, -7, 0, 4, 0, -2, -6, 0, -2, 0, 4, -4, -6], [-1, 0, 8, 9, 4, -13, 0, 0, 0, 0, 0, 0, 4, -1, 3, 0], [-1, 0, 4, 2, 5, -7, 0, 2, 0, 0, 5, 0, 5, -4, 1, 0], [-5, 0, 2, 7, -8, 1, 0, -19, 0, 0, 5, 0, -7, -2, -9, 0], [-8, 0, 1, 2, 3, -5, 0, 2, 0, 0, -4, 0, -5, 9, 4, 0], [0, 26, 0, -1, -1, 0, -16, 0, -16, 17, 0, -2, 0, -12, 12, 4], [0, 0, 5, 8, -1, -7, 0, 5, 0, 0, -1, 0, -4, 7, 3, 0], [0, 4, 0, -7, -7, 0, 3, 0, -15, -7, 0, -1, 0, 4, -4, -9], [-19, 0, 6, 10, -1, -9, 0, -9, 0, 0, 0, 0, 3, -5, -2, 0], [0, -10, 0, 2, 2, 0, 3, 0, -4, -11, 0, 6, 0, 2, -2, 3], [0, 0, 0, 5, 5, 0, 0, 0, 15, 5, 0, -5, 0, -3, 3, 2], [0, -27, 0, 0, 0, 0, 3, 0, 16, -14, 0, 6, 0, 10, -10, 8], [0, 0, 9, -2, -5, 7, 0, -23, 0, 0, 9, 0, 2, -4, -2, 0], [0, -7, 0, 2, 2, 0, 9, 0, 8, -22, 0, 2, 0, 8, -8, -10], [0, 11, 0, 8, 8, 0, -7, 0, -6, 7, 0, 3, 0, -8, 8, 9], [21, 0, -2, 1, 2, -3, 0, 12, 0, 0, -3, 0, 0, -2, -2, 0], [-6, 0, 3, 8, -6, -2, 0, 15, 0, 0, -5, 0, -1, 1, 0, 0], [0, -12, 0, 0, 0, 0, 0, 0, 11, -5, 0, -5, 0, 2, -2, 10], [0, 10, 0, -4, -4, 0, 2, 0, 8, 0, 0, 9, 0, 3, -3, 1], [-11, 0, -2, -2, 1, 1, 0, 16, 0, 0, 3, 0, -8, 3, -5, 0], [4, 0, -8, 3, 5, -8, 0, 18, 0, 0, -1, 0, 1, 0, 1, 0], [0, -1, 0, -4, -4, 0, 1, 0, -4, 3, 0, -3, 0, -2, 2, -2], [0, -6, 0, 7, 7, 0, -4, 0, 3, 8, 0, -18, 0, -4, 4, -1], [20, 0, 0, 1, -3, 2, 0, 6, 0, 0, 4, 0, -3, 3, 0, 0], [0, -7, 0, 5, 5, 0, -7, 0, -2, -9, 0, -3, 0, 1, -1, -9], [0, -4, 0, 1, 1, 0, 3, 0, 6, -7, 0, 2, 0, 10, -10, -1], [0, 17, 0, -9, -9, 0, -8, 0, -8, 11, 0, -8, 0, -4, 4, 3], [-28, 0, -6, 2, 4, -6, 0, -6, 0, 0, 4, 0, 5, 0, 5, 0], [-11, 0, 8, -7, -8, 15, 0, -9, 0, 0, 4, 0, -3, 0, -3, 0], [0, 5, 0, -4, -4, 0, -19, 0, -14, 10, 0, -4, 0, -7, 7, -6], [-17, 0, 12, 0, 0, 0, 0, 5, 0, 0, -7, 0, -2, 4, 2, 0], [0, 0, -4, 10, -6, -4, 0, -10, 0, 0, -2, 0, 0, -8, -8, 0], [0, -2, 0, -4, -4, 0, -5, 0, -7, 3, 0, -4, 0, -1, 1, -2], [8, 0, -6, 2, -4, 2, 0, 18, 0, 0, -5, 0, -4, 0, -4, 0], [0, -4, 0, -5, -5, 0, -3, 0, -2, 5, 0, -8, 0, -3, 3, 3], [28, 0, -1, 8, 4, -12, 0, -1, 0, 0, -2, 0, -6, 3, -3, 0], [-9, 0, -1, -9, -2, 11, 0, -3, 0, 0, 16, 0, -5, 3, -2, 0], [0, 7, 0, 5, 5, 0, 7, 0, 9, 9, 0, 1, 0, -5, 5, 4], [-14, 0, 9, -6, 3, 3, 0, -4, 0, 0, -2, 0, -7, 5, -2, 0], [0, 27, 0, 2, 2, 0, -16, 0, -7, 12, 0, -2, 0, -16, 16, 0], [0, -2, 0, -1, -1, 0, -15, 0, -13, 10, 0, -1, 0, -2, 2, 2], [12, 0, -7, -7, 13, -6, 0, 26, 0, 0, -4, 0, 4, 5, 9, 0], [6, 0, 3, -9, 10, -1, 0, 9, 0, 0, 8, 0, 6, 1, 7, 0], [0, 29, 0, -10, -10, 0, -16, 0, -5, 21, 0, 5, 0, 0, 0, 1], [5, 0, 4, 5, -10, 5, 0, 0, 0, 0, 13, 0, -3, 2, -1, 0], [0, 22, 0, -3, -3, 0, 5, 0, -6, -2, 0, -6, 0, 0, 0, 1], [0, -6, 0, 4, 4, 0, 5, 0, 2, 11, 0, -2, 0, 1, -1, 9], [4, 0, 4, 9, -10, 1, 0, 15, 0, 0, 3, 0, -2, 3, 1, 0], [-4, 0, 9, -1, 3, -2, 0, 7, 0, 0, 3, 0, 1, 3, 4, 0], [17, 0, 5, -7, 6, 1, 0, -10, 0, 0, -1, 0, 3, -1, 2, 0], [0, -29, 0, 6, 6, 0, 13, 0, 15, -8, 0, 4, 0, 7, -7, 5], [0, 1, 0, 12, 12, 0, 1, 0, 17, 1, 0, -3, 0, -4, 4, 11], [-19, 0, 0, -1, -5, 6, 0, -15, 0, 0, 1, 0, -4, 1, -3, 0], [0, 9, 0, -3, -3, 0, -4, 0, -6, 2, 0, -3, 0, -4, 4, 2], [23, 0, 1, 9, 4, -13, 0, 18, 0, 0, -2, 0, 6, -1, 5, 0], [0, -30, 0, 0, 0, 0, 5, 0, 12, -10, 0, 7, 0, 11, -11, 4], [-18, 0, -5, -8, -2, 10, 0, -2, 0, 0, -2, 0, -8, 5, -3, 0], [0, -1, 0, -2, -2, 0, 8, 0, 13, -2, 0, -1, 0, 3, -3, 9], [10, 0, 0, 5, -7, 2, 0, 17, 0, 0, 7, 0, -13, 4, -9, 0], [-10, 0, -11, 2, 8, -10, 0, -8, 0, 0, -7, 0, 0, -3, -3, 0], [4, 0, -8, 16, 0, -16, 0, 11, 0, 0, 3, 0, 0, -2, -2, 0], [0, 24, 0, -6, -6, 0, -15, 0, -14, 14, 0, -3, 0, -7, 7, -6], [0, 8, 0, -6, -6, 0, -7, 0, -5, 4, 0, 8, 0, -3, 3, -6], [0, 0, 0, 10, 6, -16, 0, 17, 0, 0, -2, 0, 1, -2, -1, 0], [0, -9, 0, 3, 3, 0, 3, 0, -5, 4, 0, 2, 0, 4, -4, -9], [8, 0, -3, -9, 7, 2, 0, -6, 0, 0, -8, 0, 4, -1, 3, 0], [5, 0, 5, 4, 1, -5, 0, 17, 0, 0, -7, 0, 7, -4, 3, 0], [0, 1, 0, 1, 1, 0, 1, 0, 1, -1, 0, 5, 0, 3, -3, 7], [9, 0, 12, -7, 0, 7, 0, -11, 0, 0, 9, 0, 7, 0, 7, 0], [0, -28, 0, -2, -2, 0, 3, 0, 11, -10, 0, -13, 0, 5, -5, -5], [-21, 0, 5, -4, -7, 11, 0, -15, 0, 0, 3, 0, 0, -5, -5, 0], [0, 5, 0, 7, 7, 0, -4, 0, 10, 16, 0, -5, 0, -1, 1, 6], [34, 0, -1, -4, -8, 12, 0, 20, 0, 0, 10, 0, -4, -2, -6, 0], [0, 5, 0, -3, -3, 0, 12, 0, -9, 11, 0, 4, 0, -8, 8, 2], [8, 0, -3, 4, -6, 2, 0, -1, 0, 0, 3, 0, 3, -11, -8, 0], [-3, 0, -2, -14, 13, 1, 0, 22, 0, 0, -4, 0, 4, 2, 6, 0], [0, 8, 0, -8, -8, 0, 1, 0, 6, 10, 0, 2, 0, -2, 2, 3], [0, -18, 0, 7, 7, 0, -1, 0, -5, -16, 0, -5, 0, -2, 2, -1], [13, 0, 2, 9, -4, -5, 0, 24, 0, 0, -2, 0, -5, 1, -4, 0], [0, 13, 0, -3, -3, 0, 5, 0, 14, 1, 0, 5, 0, -6, 6, -4], [-17, 0, 7, 9, -5, -4, 0, -19, 0, 0, -1, 0, -2, 3, 1, 0], [0, 9, 0, 0, 0, 0, 7, 0, -3, 7, 0, -7, 0, -6, 6, 0], [-11, 0, 11, 1, -10, 9, 0, 7, 0, 0, 8, 0, 1, -6, -5, 0], [40, 0, -1, 1, 11, -12, 0, 29, 0, 0, -9, 0, 4, 1, 5, 0], [0, -21, 0, 4, 4, 0, -11, 0, 2, -9, 0, -1, 0, 4, -4, 9], [0, 1, 0, -1, -1, 0, 19, 0, -2, 5, 0, 5, 0, 5, -5, -7], [0, 2, 0, -5, -5, 0, 3, 0, 17, -5, 0, 8, 0, 7, -7, 4], [0, 3, 0, -9, -9, 0, 8, 0, -9, -1, 0, 2, 0, 4, -4, -10], [0, -7, 0, 2, 2, 0, 13, 0, 1, -9, 0, 6, 0, -2, 2, -3], [-9, 0, 8, -7, 7, 0, 0, 12, 0, 0, 3, 0, 11, -5, 6, 0], [0, 0, 0, -8, -8, 0, 11, 0, 8, 4, 0, 7, 0, 4, -4, -8], [0, 1, 0, -3, -3, 0, -3, 0, 6, 8, 0, 12, 0, 6, -6, 4], [12, 0, 10, 5, -3, -2, 0, 14, 0, 0, -3, 0, 4, -7, -3, 0], [33, 0, 3, 12, -2, -10, 0, -1, 0, 0, -2, 0, -4, -1, -5, 0], [-10, 0, 5, -9, 0, 9, 0, -5, 0, 0, -1, 0, -8, 7, -1, 0], [0, 1, 0, -13, -13, 0, 0, 0, -3, -11, 0, 10, 0, 2, -2, -11], [0, 15, 0, -3, -3, 0, -7, 0, -16, 18, 0, 2, 0, -10, 10, 10], [-6, 0, 3, -3, -9, 12, 0, -11, 0, 0, -6, 0, 6, -4, 2, 0], [0, 13, 0, -6, -6, 0, -2, 0, -22, 7, 0, 6, 0, -3, 3, -7], [0, -22, 0, -6, -6, 0, 5, 0, -8, -25, 0, 4, 0, 4, -4, -13], [7, 0, -9, -7, 4, 3, 0, 12, 0, 0, 0, 0, -10, 16, 6, 0], [0, 22, 0, 1, 1, 0, 2, 0, -8, 13, 0, -7, 0, -4, 4, -9], [0, -16, 0, -9, -9, 0, 11, 0, 8, -15, 0, 11, 0, 17, -17, -2], [0, 7, 0, 3, 3, 0, 13, 0, 12, 5, 0, 9, 0, -2, 2, 9], [17, 0, 2, 3, 5, -8, 0, -6, 0, 0, 11, 0, 3, -5, -2, 0], [-6, 0, 5, 6, -10, 4, 0, -30, 0, 0, 1, 0, -7, 0, -7, 0], [-5, 0, 9, -17, 10, 7, 0, -23, 0, 0, 5, 0, 9, -7, 2, 0], [0, -7, 0, -5, -5, 0, 19, 0, 10, -11, 0, 14, 0, 5, -5, -2], [17, 0, -6, 11, -6, -5, 0, 33, 0, 0, -3, 0, -9, 4, -5, 0], [0, 4, 0, -5, -5, 0, -6, 0, -13, 8, 0, -6, 0, 4, -4, -2], [9, 0, -2, -1, -6, 7, 0, 9, 0, 0, 13, 0, -9, -1, -10, 0], [0, 9, 0, 0, 0, 0, -9, 0, -6, 23, 0, -4, 0, -2, 2, 8], [-2, 0, -12, -7, 9, -2, 0, -1, 0, 0, -2, 0, 11, 2, 13, 0], [0, -7, 0, -1, -1, 0, -12, 0, 10, -6, 0, 3, 0, -5, 5, 8], [0, 18, 0, -2, -2, 0, 11, 0, -6, 13, 0, 12, 0, -11, 11, -9], [-2, 0, 10, 7, -7, 0, 0, 4, 0, 0, -7, 0, -6, 4, -2, 0], [9, 0, -4, 4, 9, -13, 0, -5, 0, 0, -7, 0, -4, 5, 1, 0], [5, 0, 0, 0, 5, -5, 0, -16, 0, 0, 0, 0, 9, -2, 7, 0], [0, -39, 0, 16, 16, 0, 11, 0, 28, -20, 0, -15, 0, 5, -5, -7], [0, 3, 0, 3, 3, 0, -25, 0, -5, 0, 0, -6, 0, -11, 11, -1], [-21, 0, 8, 4, 5, -9, 0, 7, 0, 0, 1, 0, 9, -7, 2, 0], [0, 9, 0, 1, 1, 0, 5, 0, -16, 9, 0, 7, 0, -1, 1, 1], [3, 0, -7, 10, -1, -9, 0, 4, 0, 0, -6, 0, -8, 8, 0, 0], [-28, 0, 17, -3, -7, 10, 0, -19, 0, 0, 0, 0, 9, -10, -1, 0], [0, 2, 0, 8, 8, 0, -4, 0, -6, 2, 0, -2, 0, 3, -3, 8], [35, 0, -4, 2, 1, -3, 0, 18, 0, 0, -1, 0, -6, 1, -5, 0], [0, -24, 0, 5, 5, 0, 2, 0, 9, -9, 0, -2, 0, 5, -5, -1], [0, 29, 0, -13, -13, 0, -6, 0, -16, 18, 0, 9, 0, -9, 9, 2], [0, -4, 0, -1, -1, 0, -5, 0, 0, 1, 0, 6, 0, 9, -9, -5], [0, 0, -11, -3, -13, 16, 0, -20, 0, 0, 8, 0, 2, -6, -4, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1805_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1805_2_b_i();".
// 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_b_i(:prec:=16)
    chi := MakeCharacter_1805_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_b_i();".
// 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_b_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1805_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![16, 0, 256, 0, 1163, 0, 2154, 0, 1862, 0, 820, 0, 190, 0, 22, 0, 1]>,<29,R![-27484, -16054, 16029, 14309, 2570, -381, -116, 1, 1]>],Snew);
    return Vf;
end function;