
// Make newform 4410.2.b.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4410_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4410_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_4410_2_b_a();".
// 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_4410_2_b_a();".
// 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, 0, 0, 0, 0, 0, 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, 1, 0, 0, 0], [0, 0, 0, 1, 0, 1, 0, 0], [0, 0, 1, 0, 0, 0, 1, 0], [0, 1, 0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0, 0, -1], [0, 0, 1, 0, 0, 0, -1, 0], [0, 0, 0, 1, 0, -1, 0, 0]];
    Rf_basisdens := [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_4410_b();"
function MakeCharacter_4410_b()
    N := 4410;
    order := 2;
    char_gens := [3431, 2647, 1081];
    v := [1, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4410_b_Hecke();"
function MakeCharacter_4410_b_Hecke(Kf)
    N := 4410;
    order := 2;
    char_gens := [3431, 2647, 1081];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 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], [-1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 1, 0, 0, 0, 0, 0], [0, 4, 1, -1, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 3], [0, 0, 0, 1, -2, 0, 0, 0], [0, 2, -5, -1, 1, 0, 0, 0], [0, -2, 2, -2, 1, 0, 0, 0], [0, -4, -1, 2, -2, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 3, -1], [-6, 0, 0, 0, 0, 1, 0, 1], [-2, 0, 0, 0, 0, 2, -1, -5], [6, 0, 0, 0, 0, -1, 1, -2], [0, 4, -1, 2, 5, 0, 0, 0], [-2, 0, 0, 0, 0, 2, -1, -4], [0, 6, 0, 1, 2, 0, 0, 0], [0, 0, 0, 0, 0, -5, 1, 0], [0, 2, 4, 3, 0, 0, 0, 0], [0, 4, -1, 4, 5, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0], [-2, 0, 0, 0, 0, -2, 0, -2], [6, 0, 0, 0, 0, 4, -2, 4], [0, 6, 4, 0, 0, 0, 0, 0], [4, 0, 0, 0, 0, -2, -1, 0], [0, 2, 1, 6, 1, 0, 0, 0], [0, -2, -7, -2, 5, 0, 0, 0], [-2, 0, 0, 0, 0, -5, -6, 1], [0, 0, 0, 0, 6, 0, 0, 0], [4, 0, 0, 0, 0, -2, -4, 0], [4, 0, 0, 0, 0, -4, 3, -2], [0, -8, 1, 1, 7, 0, 0, 0], [0, 12, -2, -1, -6, 0, 0, 0], [0, -2, -9, 2, 6, 0, 0, 0], [-4, 0, 0, 0, 0, 0, 7, 4], [0, 4, 0, 6, -6, 0, 0, 0], [-4, 0, 0, 0, 0, -7, 5, 0], [0, 0, 0, 0, 0, 8, 1, 3], [8, 0, 0, 0, 0, -4, 6, 0], [0, -2, 0, -4, 11, 0, 0, 0], [0, 10, -6, 3, 2, 0, 0, 0], [0, 4, -10, -1, -4, 0, 0, 0], [8, 0, 0, 0, 0, -2, 4, -2], [0, 4, 1, -8, 7, 0, 0, 0], [0, 4, 3, 12, -4, 0, 0, 0], [4, 0, 0, 0, 0, 1, 0, -11], [0, 2, -1, -6, 7, 0, 0, 0], [0, 0, 0, 0, 0, 4, 2, 4], [0, 4, -6, 9, 2, 0, 0, 0], [0, 4, 0, 6, 2, 0, 0, 0], [0, -2, 2, 5, 8, 0, 0, 0], [0, 8, 0, 8, 5, 0, 0, 0], [2, 0, 0, 0, 0, -2, 2, 2], [-2, 0, 0, 0, 0, -3, 3, 4], [0, 6, 7, 1, 1, 0, 0, 0], [10, 0, 0, 0, 0, -8, -4, -12], [0, -8, 6, 2, 13, 0, 0, 0], [0, 0, 0, 0, 0, 7, -1, 2], [0, -2, 2, 11, 4, 0, 0, 0], [0, -6, -11, -3, 5, 0, 0, 0], [0, 0, 0, 0, 0, -16, 2, 6], [0, 10, 8, -2, 6, 0, 0, 0], [4, 0, 0, 0, 0, -6, -8, -2], [0, 4, 13, 0, 3, 0, 0, 0], [0, -2, -4, -14, 2, 0, 0, 0], [-12, 0, 0, 0, 0, -7, 4, -1], [-6, 0, 0, 0, 0, -2, -4, -12], [0, 0, 4, 12, 8, 0, 0, 0], [0, -2, -6, -3, -14, 0, 0, 0], [0, 0, 0, 0, 0, 16, 1, 1], [0, 8, 4, 11, -8, 0, 0, 0], [0, 4, 0, 0, 4, 0, 0, 0], [-14, 0, 0, 0, 0, 10, 11, 3], [-28, 0, 0, 0, 0, -2, -2, 4], [0, 0, 0, 0, 0, 6, -7, 5], [0, 6, 7, -6, 0, 0, 0, 0], [0, 4, -2, 14, -12, 0, 0, 0], [0, 2, 2, -1, 6, 0, 0, 0], [0, 4, 8, -16, 3, 0, 0, 0], [-2, 0, 0, 0, 0, 8, -8, -12], [10, 0, 0, 0, 0, -15, -10, -3], [0, 0, 10, 13, 2, 0, 0, 0], [0, -6, -4, 14, 2, 0, 0, 0], [0, -12, -14, -2, 3, 0, 0, 0], [0, 4, -8, -4, -10, 0, 0, 0], [0, 10, -16, -7, -14, 0, 0, 0], [-10, 0, 0, 0, 0, 4, 12, -2], [6, 0, 0, 0, 0, 4, -6, 4], [10, 0, 0, 0, 0, 6, 8, -8], [2, 0, 0, 0, 0, -14, 2, -2], [0, 0, 0, 0, 0, -8, 0, -12], [-6, 0, 0, 0, 0, 14, 12, 2], [0, 14, -10, -4, 3, 0, 0, 0], [0, 0, 0, 0, 0, 4, -16, 2], [6, 0, 0, 0, 0, -15, 11, 12], [-2, 0, 0, 0, 0, 4, -15, 2], [-6, 0, 0, 0, 0, 4, -14, 6], [0, -10, 13, -3, -3, 0, 0, 0], [6, 0, 0, 0, 0, -7, -12, 5], [10, 0, 0, 0, 0, 0, -5, -5], [0, -18, 0, -2, 2, 0, 0, 0], [-12, 0, 0, 0, 0, 2, 0, 8], [0, 6, -4, 3, -8, 0, 0, 0], [8, 0, 0, 0, 0, 2, -6, -8], [0, 8, 3, -2, -17, 0, 0, 0], [2, 0, 0, 0, 0, 4, -8, -12], [-26, 0, 0, 0, 0, 1, 3, 2], [0, -12, 0, -25, -2, 0, 0, 0], [0, -12, 1, 4, 14, 0, 0, 0], [0, -26, -5, -10, 7, 0, 0, 0], [24, 0, 0, 0, 0, 11, 7, -10], [0, -4, -9, 11, -19, 0, 0, 0], [0, 2, 4, 9, 2, 0, 0, 0], [-6, 0, 0, 0, 0, 2, 9, 6], [0, 8, 2, 13, -14, 0, 0, 0], [0, 14, -1, -15, 3, 0, 0, 0], [-6, 0, 0, 0, 0, -9, 3, 10], [0, 6, 8, 2, 6, 0, 0, 0], [0, 6, 9, -16, 14, 0, 0, 0], [0, 2, -6, -19, 4, 0, 0, 0], [12, 0, 0, 0, 0, -8, -2, -2], [-18, 0, 0, 0, 0, 0, -6, -6], [0, 18, 9, 4, -9, 0, 0, 0], [0, 2, -18, -5, 2, 0, 0, 0], [0, 2, -17, 4, -6, 0, 0, 0], [6, 0, 0, 0, 0, 2, -10, -14], [-16, 0, 0, 0, 0, 19, 10, 13], [0, 2, -19, 2, 3, 0, 0, 0], [0, -10, 18, 8, 2, 0, 0, 0], [-20, 0, 0, 0, 0, 0, -4, 20], [0, -12, -10, -2, 2, 0, 0, 0], [2, 0, 0, 0, 0, -2, 5, 10], [-2, 0, 0, 0, 0, -2, 5, 9], [-14, 0, 0, 0, 0, 2, -20, -16], [0, 4, -13, -10, -14, 0, 0, 0], [-2, 0, 0, 0, 0, 12, -14, 4], [0, -18, 4, -6, 6, 0, 0, 0], [-2, 0, 0, 0, 0, 12, 6, 10], [0, 12, -6, -9, 0, 0, 0, 0], [0, -10, 12, 23, 4, 0, 0, 0], [0, 18, 2, 14, -1, 0, 0, 0], [20, 0, 0, 0, 0, 8, 20, -16], [0, 28, -2, 10, -4, 0, 0, 0], [0, 12, 18, 3, -4, 0, 0, 0], [36, 0, 0, 0, 0, 3, -10, 1], [0, -2, 0, 2, 24, 0, 0, 0], [-10, 0, 0, 0, 0, -7, 13, 6], [0, -18, -2, -11, -10, 0, 0, 0], [0, -4, 18, -16, -6, 0, 0, 0], [6, 0, 0, 0, 0, 22, -9, -11], [6, 0, 0, 0, 0, 7, 8, -13], [24, 0, 0, 0, 0, 17, -7, 6], [-20, 0, 0, 0, 0, -22, 9, -1], [-4, 0, 0, 0, 0, -3, -1, 12], [0, 6, 6, 15, 0, 0, 0, 0], [18, 0, 0, 0, 0, -24, -9, -4], [-10, 0, 0, 0, 0, 0, -10, 6], [0, -10, 10, 12, 10, 0, 0, 0], [4, 0, 0, 0, 0, -4, 23, 14], [0, -6, -13, 6, -9, 0, 0, 0], [0, 0, -19, -1, 11, 0, 0, 0], [16, 0, 0, 0, 0, 18, -4, -2], [-14, 0, 0, 0, 0, -2, -18, -14], [0, -2, -22, 4, -10, 0, 0, 0], [-20, 0, 0, 0, 0, 2, 9, 15], [10, 0, 0, 0, 0, 8, 3, 24], [0, 18, 0, 2, 8, 0, 0, 0], [-6, 0, 0, 0, 0, -20, 11, 0], [-14, 0, 0, 0, 0, -12, -8, 6], [0, -14, -5, 0, -28, 0, 0, 0], [0, -8, 16, 17, 4, 0, 0, 0], [0, 18, 2, -19, 4, 0, 0, 0], [12, 0, 0, 0, 0, 6, 4, -6], [0, -16, 16, -14, -15, 0, 0, 0], [-18, 0, 0, 0, 0, 6, -6, -6], [4, 0, 0, 0, 0, 19, -12, 11], [0, -18, -11, 6, 8, 0, 0, 0], [0, -10, -3, 0, 21, 0, 0, 0], [0, 6, -28, -9, -2, 0, 0, 0], [-8, 0, 0, 0, 0, -6, -4, -8], [-2, 0, 0, 0, 0, 18, -19, 0], [12, 0, 0, 0, 0, 13, 1, -16], [-4, 0, 0, 0, 0, 4, -15, -3], [0, 10, -7, 9, -13, 0, 0, 0], [-24, 0, 0, 0, 0, 0, -6, 4], [8, 0, 0, 0, 0, -9, 11, -2], [0, -10, -15, 19, -7, 0, 0, 0], [18, 0, 0, 0, 0, 24, -2, -12], [12, 0, 0, 0, 0, 1, 6, 7], [0, 42, -6, -12, -4, 0, 0, 0], [0, -30, 9, 12, 5, 0, 0, 0], [-8, 0, 0, 0, 0, -8, -2, -24], [4, 0, 0, 0, 0, -10, -4, 2], [0, 30, -11, 4, -7, 0, 0, 0], [18, 0, 0, 0, 0, 3, 9, 10], [2, 0, 0, 0, 0, -12, 9, -24], [-4, 0, 0, 0, 0, 3, 17, -10], [-2, 0, 0, 0, 0, -1, -13, 10], [-8, 0, 0, 0, 0, -10, 11, -9], [0, -18, -15, -6, -6, 0, 0, 0], [0, 12, 5, -12, 12, 0, 0, 0], [0, 4, 7, 15, 7, 0, 0, 0], [0, 12, 15, -2, -6, 0, 0, 0], [22, 0, 0, 0, 0, 6, -2, 10], [-16, 0, 0, 0, 0, -4, -8, -18], [0, 20, -7, -2, -2, 0, 0, 0], [0, 0, -14, 20, 6, 0, 0, 0], [0, 12, 8, -5, -4, 0, 0, 0], [0, -40, -6, -5, -2, 0, 0, 0], [28, 0, 0, 0, 0, -10, -12, -4], [-24, 0, 0, 0, 0, -6, -12, -10], [8, 0, 0, 0, 0, 0, 12, 20], [20, 0, 0, 0, 0, 16, -2, 2], [-8, 0, 0, 0, 0, -9, 18, -15], [0, 16, 2, 14, -3, 0, 0, 0], [-18, 0, 0, 0, 0, 4, -12, -4], [6, 0, 0, 0, 0, -25, -22, 1], [0, 14, 11, -1, -7, 0, 0, 0], [0, -6, 2, 6, -22, 0, 0, 0], [-14, 0, 0, 0, 0, 6, -4, 6], [0, -24, 16, -16, -27, 0, 0, 0], [0, -32, 8, 1, 22, 0, 0, 0], [-30, 0, 0, 0, 0, 10, 24, 2], [-14, 0, 0, 0, 0, 2, 8, -10], [-22, 0, 0, 0, 0, -2, -2, 10], [4, 0, 0, 0, 0, -2, -19, 9], [0, -30, 20, -1, -10, 0, 0, 0], [0, -20, 2, -6, 4, 0, 0, 0], [0, -14, -2, 4, -13, 0, 0, 0], [22, 0, 0, 0, 0, -28, -9, -7], [0, -32, 26, 5, -8, 0, 0, 0], [-20, 0, 0, 0, 0, 6, -22, -14], [0, -16, -10, 31, -10, 0, 0, 0], [0, 14, 6, 20, -28, 0, 0, 0], [-14, 0, 0, 0, 0, 13, 13, -12], [0, -24, -21, 0, -12, 0, 0, 0], [0, -8, -3, -2, -3, 0, 0, 0], [0, 14, -1, -14, 18, 0, 0, 0], [-16, 0, 0, 0, 0, 5, 20, -7], [0, 14, 1, -8, -17, 0, 0, 0], [0, -16, 4, -13, -10, 0, 0, 0], [0, -28, 2, -20, -12, 0, 0, 0], [6, 0, 0, 0, 0, 25, 0, 3], [-10, 0, 0, 0, 0, -13, 17, -12], [24, 0, 0, 0, 0, 9, -10, 21], [0, -26, -1, -4, 7, 0, 0, 0], [-20, 0, 0, 0, 0, -4, -21, 18], [-32, 0, 0, 0, 0, -21, 12, -17], [0, -30, 20, -16, -26, 0, 0, 0], [-20, 0, 0, 0, 0, 23, 1, -4], [-2, 0, 0, 0, 0, -15, 18, -17], [0, 14, 26, -8, 12, 0, 0, 0], [0, -8, 2, 26, 3, 0, 0, 0], [12, 0, 0, 0, 0, 14, 8, 24], [0, -6, -12, -8, -7, 0, 0, 0], [18, 0, 0, 0, 0, 1, 14, -9], [0, -22, 15, 21, 3, 0, 0, 0], [6, 0, 0, 0, 0, 4, 20, 16], [0, 2, 8, 0, 26, 0, 0, 0], [22, 0, 0, 0, 0, -12, 32, 22], [0, 0, 26, -6, 12, 0, 0, 0], [0, -18, 14, 7, -8, 0, 0, 0], [0, -44, -25, 3, 7, 0, 0, 0], [12, 0, 0, 0, 0, -36, -1, 7], [0, 26, 2, 1, 6, 0, 0, 0], [0, 38, -2, -16, -11, 0, 0, 0], [10, 0, 0, 0, 0, -10, 20, 28], [28, 0, 0, 0, 0, 17, 9, 6], [0, 20, -3, 10, -5, 0, 0, 0], [0, 2, -28, -1, 16, 0, 0, 0], [-2, 0, 0, 0, 0, -10, 1, 17], [0, -24, 7, -22, 17, 0, 0, 0], [26, 0, 0, 0, 0, -6, 8, 34], [0, 36, -23, 0, -7, 0, 0, 0], [0, -16, -8, -6, -30, 0, 0, 0], [0, -2, -17, 18, 13, 0, 0, 0], [-30, 0, 0, 0, 0, 6, 6, 14], [-18, 0, 0, 0, 0, 30, 3, -10], [-2, 0, 0, 0, 0, -38, 7, 4], [2, 0, 0, 0, 0, 11, 1, 14], [2, 0, 0, 0, 0, -12, 4, 24], [4, 0, 0, 0, 0, -16, 7, -41], [0, 6, -16, 3, 26, 0, 0, 0], [0, 2, 1, 29, 21, 0, 0, 0], [0, 14, 2, -5, -10, 0, 0, 0], [-8, 0, 0, 0, 0, 10, -36, 10], [0, -20, 9, 30, 1, 0, 0, 0], [0, 28, -15, -30, -8, 0, 0, 0], [-2, 0, 0, 0, 0, 24, -2, -2], [0, 6, -14, 0, 7, 0, 0, 0], [40, 0, 0, 0, 0, 2, 12, -18], [0, 0, 6, 12, 12, 0, 0, 0], [28, 0, 0, 0, 0, -8, -19, -10], [12, 0, 0, 0, 0, -21, 17, 6], [30, 0, 0, 0, 0, 22, -11, -4], [0, -36, 12, -2, -7, 0, 0, 0], [-20, 0, 0, 0, 0, -24, 22, 4], [34, 0, 0, 0, 0, -12, -2, 4], [0, 8, -22, 12, 18, 0, 0, 0], [0, 14, 12, 24, -14, 0, 0, 0], [0, 10, 4, 12, -2, 0, 0, 0], [4, 0, 0, 0, 0, 14, -2, 26], [0, 2, 9, -2, 27, 0, 0, 0], [20, 0, 0, 0, 0, 30, -4, -8], [-10, 0, 0, 0, 0, -34, 20, -6], [0, 26, -3, -2, 17, 0, 0, 0], [0, -40, 20, 11, 6, 0, 0, 0], [0, 8, -16, -17, -28, 0, 0, 0], [-18, 0, 0, 0, 0, -10, 19, -9], [14, 0, 0, 0, 0, 7, 33, 14], [0, 14, 8, 10, 21, 0, 0, 0], [0, 0, -12, 3, 8, 0, 0, 0], [34, 0, 0, 0, 0, -26, -5, -3], [0, 4, -6, 18, -26, 0, 0, 0], [0, -12, -9, 22, 12, 0, 0, 0], [-38, 0, 0, 0, 0, -4, 12, 16], [0, -4, -12, 3, 36, 0, 0, 0], [0, -6, 20, -14, -6, 0, 0, 0], [0, 10, 12, -15, 2, 0, 0, 0], [0, 0, -8, 19, 4, 0, 0, 0], [-8, 0, 0, 0, 0, 16, -8, 14], [10, 0, 0, 0, 0, 22, 9, -4], [-2, 0, 0, 0, 0, -32, 8, 2], [0, 20, -15, 9, 19, 0, 0, 0], [0, 40, 2, 0, 27, 0, 0, 0], [20, 0, 0, 0, 0, 8, 6, 4], [0, 28, 4, 18, 20, 0, 0, 0], [0, 34, 24, -6, 10, 0, 0, 0], [0, -16, -1, -2, -3, 0, 0, 0], [-2, 0, 0, 0, 0, 20, 6, 2], [0, -2, -26, 2, -10, 0, 0, 0], [0, -16, 10, -20, -9, 0, 0, 0], [28, 0, 0, 0, 0, 2, -12, -2], [4, 0, 0, 0, 0, -7, 38, 17], [-8, 0, 0, 0, 0, -4, 40, -12], [-6, 0, 0, 0, 0, -11, -22, 27], [24, 0, 0, 0, 0, -4, -1, 33], [0, -32, -4, -6, 15, 0, 0, 0], [0, 16, -12, -30, -8, 0, 0, 0], [14, 0, 0, 0, 0, -20, 19, -3], [0, -40, 3, 3, -15, 0, 0, 0], [30, 0, 0, 0, 0, 12, 15, 14], [-20, 0, 0, 0, 0, -22, -26, 26], [0, -48, -19, -4, 13, 0, 0, 0], [0, 18, 29, -6, 4, 0, 0, 0], [0, -44, 0, 1, -24, 0, 0, 0], [12, 0, 0, 0, 0, -9, 13, 8], [-24, 0, 0, 0, 0, 29, 20, 23], [-26, 0, 0, 0, 0, 24, 28, 8], [0, 8, -6, -21, 10, 0, 0, 0], [-20, 0, 0, 0, 0, -10, 18, 6], [0, -6, 24, -26, 5, 0, 0, 0], [0, 12, 24, 10, 8, 0, 0, 0], [-30, 0, 0, 0, 0, 13, 20, 11], [-16, 0, 0, 0, 0, 14, 21, -9], [-12, 0, 0, 0, 0, -17, 16, -5], [20, 0, 0, 0, 0, -10, 26, 14], [0, 18, -28, 10, 0, 0, 0, 0], [0, 6, 3, -33, 17, 0, 0, 0], [26, 0, 0, 0, 0, -18, -25, 11], [22, 0, 0, 0, 0, -35, -12, -3], [0, 22, 7, 33, -27, 0, 0, 0], [0, -30, -22, -18, 13, 0, 0, 0], [0, -24, -6, 36, -6, 0, 0, 0], [-16, 0, 0, 0, 0, 12, -6, 16], [-42, 0, 0, 0, 0, 18, -6, -16], [12, 0, 0, 0, 0, 34, -14, 32], [-12, 0, 0, 0, 0, -8, 8, 12], [0, 10, -2, -2, -25, 0, 0, 0], [0, 2, 0, -15, 8, 0, 0, 0], [0, -42, -11, 1, -21, 0, 0, 0], [0, -18, 3, -20, 18, 0, 0, 0], [0, -52, 8, 10, 9, 0, 0, 0], [34, 0, 0, 0, 0, 12, -25, -5], [12, 0, 0, 0, 0, -34, -7, 28], [0, 60, 4, -5, 4, 0, 0, 0], [0, 30, 28, 2, -16, 0, 0, 0], [-2, 0, 0, 0, 0, -26, -6, 18], [0, 14, -4, -22, 16, 0, 0, 0], [2, 0, 0, 0, 0, -18, 0, -22], [0, -4, 4, -18, -22, 0, 0, 0], [-12, 0, 0, 0, 0, -22, 2, 16], [0, -66, -8, -6, 16, 0, 0, 0], [0, 42, 16, 1, 12, 0, 0, 0], [56, 0, 0, 0, 0, -8, 9, 15], [-12, 0, 0, 0, 0, -8, -32, 16], [0, -6, -20, -6, -4, 0, 0, 0], [28, 0, 0, 0, 0, -11, -13, -12], [26, 0, 0, 0, 0, -17, -25, -12], [24, 0, 0, 0, 0, -10, -9, 18], [-30, 0, 0, 0, 0, 14, -18, -22], [0, -6, -2, -18, -3, 0, 0, 0], [0, 2, -22, -12, -8, 0, 0, 0], [0, 2, 10, 13, 34, 0, 0, 0], [14, 0, 0, 0, 0, -8, 2, -8], [0, -24, -24, 8, -21, 0, 0, 0], [2, 0, 0, 0, 0, -3, -18, -35], [20, 0, 0, 0, 0, 40, -2, 2], [0, -14, 36, 8, -3, 0, 0, 0], [0, 28, 0, -7, -26, 0, 0, 0], [0, -24, -23, 5, -21, 0, 0, 0], [-54, 0, 0, 0, 0, 4, -8, 8], [62, 0, 0, 0, 0, -1, 7, 6], [-12, 0, 0, 0, 0, 10, 15, 16], [0, 8, -7, 32, 2, 0, 0, 0], [18, 0, 0, 0, 0, -24, -13, -5], [0, 26, 0, 25, 6, 0, 0, 0], [0, 22, 9, 3, 5, 0, 0, 0], [42, 0, 0, 0, 0, 16, -17, 2], [0, -22, -30, 18, -16, 0, 0, 0], [0, 30, -8, -22, -8, 0, 0, 0], [0, -32, -4, 22, 2, 0, 0, 0], [28, 0, 0, 0, 0, -31, -4, -13], [-32, 0, 0, 0, 0, -10, 10, -14], [30, 0, 0, 0, 0, 0, -7, -14], [0, 14, 18, -29, 4, 0, 0, 0], [0, 36, -34, -10, 16, 0, 0, 0], [-22, 0, 0, 0, 0, -7, 17, 24], [-50, 0, 0, 0, 0, 1, 25, 4], [0, 2, -25, -6, 16, 0, 0, 0], [0, 4, 33, -3, 13, 0, 0, 0], [0, -14, 0, -58, -2, 0, 0, 0], [-20, 0, 0, 0, 0, 6, -18, 34], [0, 16, -27, 26, -5, 0, 0, 0], [-28, 0, 0, 0, 0, -40, 6, -10], [0, 40, -30, 10, 22, 0, 0, 0], [0, 34, 8, 11, 30, 0, 0, 0], [0, 12, 10, -5, -32, 0, 0, 0], [-16, 0, 0, 0, 0, 3, -10, 23]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4410_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4410_2_b_a();".
// 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_4410_2_b_a(:prec:=8)
    chi := MakeCharacter_4410_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(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_4410_2_b_a();".
// 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_4410_2_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4410_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![4, 0, 176, 0, 148, 0, 24, 0, 1]>,<17,R![94, -72, -40, 0, 1]>],Snew);
    return Vf;
end function;