
// Make newform 1089.2.d.g in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1089_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1089_d_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_1089_2_d_g();".
// 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_1089_2_d_g();".
// 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 := [625, 0, 1000, 0, 725, 0, 360, 0, 26, 0, -72, 0, -16, 0, 2, 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], [555250, 0, -2700, 0, -18885, 0, -17472, 0, -12741, 0, -3748, 0, 906, 0, 198, 0], [0, -4371000, 0, -876975, 0, 1778925, 0, 989047, 0, -12284, 0, -61577, 0, -12231, 0, 532], [-2261250, 0, -2078950, 0, -1851615, 0, -286834, 0, 265723, 0, 73369, 0, -7568, 0, -3734, 0], [684025, 0, 599475, 0, 544476, 0, 61687, 0, -91914, 0, -21367, 0, 2574, 0, 1229, 0], [0, 7469875, 0, 2374050, 0, -1867925, 0, -922291, 0, 297652, 0, 118381, 0, -7982, 0, -6796], [0, -4720750, 0, -3942175, 0, -4205580, 0, -1255952, 0, 471569, 0, 194307, 0, -1879, 0, -7797], [0, 1047925, 0, 1381815, 0, 678048, 0, 37491, 0, -142852, 0, -30006, 0, 4062, 0, 1932], [4137375, 0, 3498700, 0, 2235885, 0, 1019404, 0, -541763, 0, -186864, 0, 8708, 0, 11794, 0], [6732875, 0, 6177175, 0, 3087765, 0, 1023533, 0, -748351, 0, -224078, 0, 14916, 0, 14403, 0], [0, 1755775, 0, 2351045, 0, 1201149, 0, 125816, 0, -209227, 0, -49056, 0, 5072, 0, 2989], [-6906250, 0, -6071300, 0, -4000385, 0, -1894396, 0, 792337, 0, 302936, 0, -6392, 0, -17146, 0], [0, 598500, 0, 412275, 0, 478015, 0, 100886, 0, -91992, 0, -24401, 0, 2572, 0, 1496], [-11238000, 0, -10624800, 0, -5540615, 0, -1983428, 0, 1116041, 0, 377023, 0, -13706, 0, -22098, 0], [0, 25694375, 0, 23823225, 0, 13611505, 0, 5518041, 0, -2819027, 0, -991506, 0, 32907, 0, 58981], [0, 41365625, 0, 38156050, 0, 21707665, 0, 8757748, 0, -4599956, 0, -1584393, 0, 59971, 0, 93868]];
    Rf_basisdens := [1, 881375, 4406875, 881375, 176275, 4406875, 4406875, 881375, 881375, 881375, 881375, 881375, 400625, 881375, 4406875, 4406875];
    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_1089_d();"
function MakeCharacter_1089_d()
    N := 1089;
    order := 2;
    char_gens := [848, 244];
    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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1089_d_Hecke();"
function MakeCharacter_1089_d_Hecke(Kf)
    N := 1089;
    order := 2;
    char_gens := [848, 244];
    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, 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], [0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 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, 0, 0, 0, 0, -1, 0, 1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, -3, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 1, 0], [0, -2, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 4, 0, 0, 0, -2, 2], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 3, 0, -2, 0, 0], [0, 0, 3, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, -2, -1, 0, 0, 0, 0, 0, -2, 0, 0, 0], [0, 0, 1, 0, 0, -1, 1, 0, 0, 0, 0, 0, -3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -2, 0, -3, 0, 0], [4, -4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, -1, 2, 0, 0, 0, 0, 0, -5, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, -4, 0, -1, 0, -3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, -3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 4, 1], [0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0], [-3, -6, 0, 3, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, -3, -5], [-8, -8, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -3, 0, 0, 0, 6, 3], [0, 0, 0, 0, 0, 0, 0, 0, 4, -3, 0, 3, 0, 2, 0, 0], [0, 0, 0, 0, 0, -3, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, -5, 0, 2, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 1, 0, 0, 0, 3, 4], [0, 0, -4, 0, 0, -1, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 0, 0, 0, -5, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 5, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -3, 0, -1, 0, 2, 0, 0], [-9, -1, 0, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -7, 0, 0, 0, 7, 4], [0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 3, 0, 0, 0, -2, -2], [0, 0, -4, 0, 0, 1, -2, 0, 0, 0, 0, 0, 3, 0, 0, 0], [-5, 9, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 1, -11, 0, 0, 0, 0, 0, 6, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 4, 0, 8, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 1, 0, 0, 0, -4, -6], [-6, -8, 0, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, -3, 0, 0, 0, 0], [-6, -5, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, 0, 0, -4, 0, 0, 0, 0, 8], [11, 7, 0, -2, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 0, -1, 0, 0, 0, -2, -8], [0, 0, 0, 0, 0, 0, 0, 8, 0, 0, -1, 0, 0, 0, -1, -8], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 3, 0, 1, 0, 0], [0, 0, -1, 0, 0, 6, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 9, -2, 0, 0, 0, 0, 0, 6, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 4, 0, 0, 0, -7, 3], [0, 0, 4, 0, 0, -2, 5, 0, 0, 0, 0, 0, 4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, -1, 0, 5, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, -5, 0, 4, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 2, 0, 0, 0, -5, 2], [0, 0, 0, 0, 0, 0, 0, 0, -4, -1, 0, -3, 0, -10, 0, 0], [0, 0, 0, 0, 0, 0, 0, 3, 0, 0, -10, 0, 0, 0, 4, -6], [0, 0, 0, 0, 0, 0, 0, 0, 3, 5, 0, 7, 0, 1, 0, 0], [0, 0, -2, 0, 0, -3, 4, 0, 0, 0, 0, 0, -3, 0, 0, 0], [-10, -8, 0, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -6, 0, 0, -1, -4, 0, 0, 0, 0, 0, 4, 0, 0, 0], [-17, -10, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -10, 2, 0, 8, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -9, 0, 0, 0, 5, 7], [0, 0, 0, 0, 0, 0, 0, 0, -3, 4, 0, 7, 0, 2, 0, 0], [0, 0, -2, 0, 0, 0, 6, 0, 0, 0, 0, 0, -5, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, -8, -3], [10, -1, 0, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -3, 0, -9, 0, -1, 0, 0], [20, 12, 0, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 3, -10, 0, 0, 0, 0, 0, 7, 0, 0, 0], [0, 0, -3, 0, 0, -6, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 3, 0, -5, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 8, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 3, 0, -6, 0, -7, 0, 0], [0, 0, 3, 0, 0, 11, -6, 0, 0, 0, 0, 0, 10, 0, 0, 0], [9, 2, 0, -1, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 2, 0, 0, 0, 3, -3], [9, 20, 0, -1, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, 7, 0, 2, 0, 2, 0, 0], [0, 0, -2, 0, 0, 8, -11, 0, 0, 0, 0, 0, -3, 0, 0, 0], [0, 0, 0, 0, 0, 6, -7, 0, 0, 0, 0, 0, 7, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, -5, 0, -4, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, -6, 0, 0, -12, 0, 0, 0, 6, 3], [-11, -6, 0, -4, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 5, 2, 0, 0, 0, 0, 0, -14, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 7, 1], [10, 4, 0, 1, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 12, 0, 0, 0, -10, 5], [-4, -7, 0, -5, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 11, 0, 0, 0, 6, -1], [0, 0, 5, 0, 0, 2, -5, 0, 0, 0, 0, 0, 12, 0, 0, 0], [0, 0, 2, 0, 0, -4, 13, 0, 0, 0, 0, 0, 3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -11, 2, 0, -8, 0, -13, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 0, -6, 0, -6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 10, 0, -8, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 9, 0, 0, -3, 0, 0, 0, 11, -4], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 13, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 1, 0, 0, 0, -10, -9], [0, 0, 0, 0, 0, 0, 0, 0, 9, -10, 0, -8, 0, -8, 0, 0], [0, 10, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -6, 0, 0, 11, -2, 0, 0, 0, 0, 0, 18, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 9, 0, 0, 0, 7, -8], [0, 0, 0, 0, 0, 10, -13, 0, 0, 0, 0, 0, 7, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, -4, 0, -7, 0, -12, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -20, 1, 0, 2, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 6, 5, 0, -3, 0, -2, 0, 0], [0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [19, -1, 0, -3, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-16, -5, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -9, 0, 0, 1, -5, 0, 0, 0, 0, 0, 18, 0, 0, 0], [-3, -10, 0, -2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, -6, -7, 0, 0, 0, 0, 0, 7, 0, 0, 0], [0, 0, -3, 0, 0, 10, -2, 0, 0, 0, 0, 0, 9, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 10, 0, 0, 0, -5, -10], [9, -8, 0, 6, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 8, 4, 0, 14, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 0, -6, 0, 0, 0, -10, -8], [0, 0, -10, 0, 0, -3, -6, 0, 0, 0, 0, 0, 16, 0, 0, 0], [-13, 6, 0, 5, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -6, 0, 0, -3, 0, 0, 0, 4, 4], [-1, 3, 0, -1, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 4, 0, 0, 3, -14, 0, 0, 0, 0, 0, 15, 0, 0, 0], [18, 15, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, -6, 0, -11, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -3, 0, -6, 0, 8, 0, 0], [0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 17, 0, 0, 0, -6, 7], [-3, 17, 0, 8, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [9, 12, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, -11, 0, 0, 0, -7, 7], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, -6, 0, 15, 0, 0], [0, 0, 0, 0, 0, 1, 13, 0, 0, 0, 0, 0, -18, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 8, 0, -17, 0, 1, 0, 0], [0, 0, -8, 0, 0, -10, 8, 0, 0, 0, 0, 0, 4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -13, 0, 0, -4, 0, 0, 0, -11, 5], [0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, -10, 0, 3, 0, 0], [0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 2, 0, 0, 0, 1, 0], [16, 1, 0, 6, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 7, 0, 0, 0, -11, -9], [11, 14, 0, 4, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -6, 0, 0, -15, 16, 0, 0, 0, 0, 0, -9, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 8, 0, 10, 0, 0], [0, 0, 0, 0, 0, 0, 0, -8, 0, 0, -14, 0, 0, 0, -7, -6], [-10, 5, 0, -2, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 8, 0, 0, -16, -10, 0, 0, 0, 0, 0, -17, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 8, -11, 0, 1, 0, -3, 0, 0], [0, 0, -7, 0, 0, -4, 24, 0, 0, 0, 0, 0, -20, 0, 0, 0], [19, 9, 0, -4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 19, 0, 0, 0, -11, -1], [-6, 7, 0, -7, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, 3, 14, 0, 0, 0, 0, 0, -11, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 14, -7, 0, 1, 0, 1, 0, 0], [0, 0, -6, 0, 0, -15, -4, 0, 0, 0, 0, 0, 7, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 5, 0, -11, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -9, 0, 0, -2, 0, 0, 0, 4, 1], [0, 0, 7, 0, 0, 1, -6, 0, 0, 0, 0, 0, 22, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -11, 0, 0, -17, 0, 0, 0, 13, 17], [0, 0, 0, 0, 0, 0, 0, 0, -5, -9, 0, -3, 0, 8, 0, 0], [0, 0, 6, 0, 0, 8, 19, 0, 0, 0, 0, 0, -18, 0, 0, 0], [0, 0, -6, 0, 0, 9, -7, 0, 0, 0, 0, 0, 10, 0, 0, 0], [0, 0, -2, 0, 0, 16, -11, 0, 0, 0, 0, 0, -3, 0, 0, 0], [8, 0, 0, 6, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, 1, 0, -9, 0, -8, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -13, 2, 0, 0, 0, -1, 0, 0], [0, 0, 6, 0, 0, -10, 21, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -17, 0, 0, -16, 0, 0, 0, -7, 13], [-13, -1, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, -15, 0, 0, 0, -3, -12], [0, 0, 0, 0, 0, 0, 0, 0, -2, -10, 0, -2, 0, -12, 0, 0], [-10, 1, 0, -5, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -12, 0, 0, 12, 1, 0, 0, 0, 0, 0, 18, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -10, 0, 16, 0, -16, 0, 0], [0, 0, -2, 0, 0, -2, 23, 0, 0, 0, 0, 0, -14, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -20, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 13, -2, 0, 11, 0, -1, 0, 0], [-4, -3, 0, -4, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -9, 0, 0, -13, 0, 0, 0, 8, 11], [0, -1, 0, -4, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5, 0, 0, -4, 0, 0, 0, -8, 9], [0, 0, 7, 0, 0, -1, 7, 0, 0, 0, 0, 0, -12, 0, 0, 0], [0, 0, -5, 0, 0, -7, 20, 0, 0, 0, 0, 0, -26, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 11, 0, 1, 0, 4, 0, 0], [-20, -3, 0, 8, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, -2, 0, 4, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, 0, 0, 5, 0, 0, 0, 12, 8], [7, -3, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, -19, 0, 0, 0, 1, -3], [10, -1, 0, 8, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -12, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 0, -7, 0, 0, 0, 16, 12], [0, 0, 3, 0, 0, 6, -1, 0, 0, 0, 0, 0, 7, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, -12, 0, -9, 0, 3, 0, 0], [4, -28, 0, 9, -13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 20, 0, 0, 7, 0, 0, 0, -4, -15], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 9, 0, 0, 0, 7, 2], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -2, 0, 0, 0, 1, 8], [0, 0, 0, 0, 0, 0, 0, 0, -5, 7, 0, -34, 0, 1, 0, 0], [17, 24, 0, -1, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, -11, 0, 2, 0, -10, 0, 0], [0, 0, -8, 0, 0, 9, 8, 0, 0, 0, 0, 0, 22, 0, 0, 0], [0, 0, 1, 0, 0, 16, -6, 0, 0, 0, 0, 0, 10, 0, 0, 0], [-20, -20, 0, 4, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, -2, -11], [0, 0, 0, 0, 0, 0, 0, 3, 0, 0, -6, 0, 0, 0, 6, 10], [8, 33, 0, 3, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -12, -15, 0, 6, 0, 9, 0, 0], [0, 0, -10, 0, 0, 1, -11, 0, 0, 0, 0, 0, 30, 0, 0, 0], [-34, -31, 0, -9, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 6, 2, 0, 0, 0, 0, 0, 22, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -12, 0, 0, 0, 4, -14], [-8, -7, 0, -12, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -11, 10, 0, 3, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 7, 0, 0, 0, -2, -16], [0, 0, -2, 0, 0, -13, -2, 0, 0, 0, 0, 0, -12, 0, 0, 0], [0, 0, 8, 0, 0, -14, -7, 0, 0, 0, 0, 0, -14, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, -23, 0, -10, 0, -6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 0, -9, 0, 3, 0, 0], [0, 0, 10, 0, 0, -21, 6, 0, 0, 0, 0, 0, -11, 0, 0, 0], [-26, -28, 0, 6, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -19, 0, 0, 8, 0, 0, 0, -19, 1], [0, 0, 0, 0, 0, 0, 0, 0, -13, -1, 0, 16, 0, 2, 0, 0], [0, 0, 9, 0, 0, 5, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 6, 0, 0, 7, -10, 0, 0, 0, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7, 17, 0, -2, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 13, 0, 0, 0, -5, 20], [31, 24, 0, -1, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -7, -2, 0, 0, 0, -3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 5, 0, 9, 0, 0], [0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 7, 0, 0, 0, -2, 5], [24, -5, 0, -7, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 2, 0, 0, 0, 18, 12], [7, 5, 0, 9, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3, 0, 0, 0, 6, 9], [0, 0, 0, 0, 0, 4, -5, 0, 0, 0, 0, 0, -18, 0, 0, 0], [0, 0, -9, 0, 0, 0, -11, 0, 0, 0, 0, 0, 18, 0, 0, 0], [0, 0, -2, 0, 0, -12, 2, 0, 0, 0, 0, 0, 13, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -19, 2, 0, 10, 0, -6, 0, 0], [-1, 1, 0, -13, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [22, 11, 0, -1, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -13, 0, 0, 16, 0, 0, 0, -15, -7], [0, 0, 0, 0, 0, 0, 0, -3, 0, 0, -10, 0, 0, 0, 24, 17], [53, 3, 0, -4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 19, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -18, -13, 0, -5, 0, 3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 22, 0, 0, 11, 0, 0, 0, -6, -13], [0, 0, 0, 0, 0, 0, 0, 0, -6, 1, 0, 2, 0, -6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 12, 0, 0, -1, 0, 0, 0, 20, 6], [0, 0, 2, 0, 0, 4, -9, 0, 0, 0, 0, 0, -23, 0, 0, 0], [-4, 13, 0, -1, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6, 0, 0, -16, 0, 0, 0, -14, -15], [0, 0, 0, 0, 0, 0, 0, -3, 0, 0, -6, 0, 0, 0, 12, -2], [-38, -1, 0, -6, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 17, 0, 8, 0, 0], [0, 0, 8, 0, 0, 3, -22, 0, 0, 0, 0, 0, 10, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -7, 12, 0, 16, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 30, 0, 9, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 16, 0, 0, -16, 0, 0, 0, 9, -14], [0, 0, 0, 0, 0, 0, 0, 0, 31, -9, 0, -7, 0, -4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, -6, 0, -3, 0, -4, 0, 0], [0, 0, -1, 0, 0, 3, -17, 0, 0, 0, 0, 0, -6, 0, 0, 0], [-13, -14, 0, 4, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -6, 0, 0, 21, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 6, 0, 0, -10, 26, 0, 0, 0, 0, 0, -20, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 12, 6, 0, -8, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 5, 0, 0, 0, -21, -20], [19, -12, 0, -1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-16, -6, 0, -10, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [48, 0, 0, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 12, 11, 0, -2, 0, -5, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, -8, 0, -16, 0, 17, 0, 0], [43, -7, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 12, 0, 0, 3, 8, 0, 0, 0, 0, 0, -11, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -20, -12, 0, 16, 0, -7, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -14, -14, 0, 1, 0, 13, 0, 0], [0, 0, 0, 0, 0, 0, 0, -13, 0, 0, -11, 0, 0, 0, 8, -3], [0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 16, 0, 0, 0, 17, 9], [24, 9, 0, 15, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -9, 0, 0, -21, 0, 0, 0, -6, 9], [0, 0, 0, 0, 0, 0, 0, 0, -14, -9, 0, -6, 0, 24, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 22, 9, 0, -7, 0, 14, 0, 0], [0, 0, 15, 0, 0, 8, 3, 0, 0, 0, 0, 0, -23, 0, 0, 0], [-11, 16, 0, -14, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 28, 0, 0, 0, 3, 19], [22, -6, 0, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 16, 0, 0, 0, 11, 6], [0, 0, 3, 0, 0, 1, 8, 0, 0, 0, 0, 0, -41, 0, 0, 0], [0, 0, 2, 0, 0, -14, 34, 0, 0, 0, 0, 0, -24, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 25, 0, 0, 0, 0, -11], [0, 0, 0, 0, 0, 0, 0, -5, 0, 0, -8, 0, 0, 0, -4, -28], [0, 0, 0, 0, 0, 0, 0, 0, -21, -1, 0, -9, 0, 25, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5, 0, 0, -1, 0, 0, 0, -12, 6], [19, 1, 0, -17, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -8, 0, 0, -9, 1, 0, 0, 0, 0, 0, 24, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -25, 0, 0, 4, 0, 0, 0, -9, 31], [0, 0, 0, 0, 0, 0, 0, 0, -12, -10, 0, -19, 0, 7, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -12, 7, 0, 5, 0, 21, 0, 0], [0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 2, 0, 0, 0, 3, 4], [0, 0, 0, 0, 0, 0, 0, 0, 7, 2, 0, 0, 0, -15, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1089_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1089_2_d_g();".
// 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_1089_2_d_g(:prec:=16)
    chi := MakeCharacter_1089_d();
    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_1089_2_d_g();".
// 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_1089_2_d_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1089_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 0, -18, 0, 39, 0, -12, 0, 1]>,<17,R![25, 0, -4300, 0, 815, 0, -50, 0, 1]>],Snew);
    return Vf;
end function;