
// Make newform 8820.2.d.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8820_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_8820_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_8820_2_d_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_8820_2_d_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 := [117649, -33614, -21609, 19894, -3822, -2086, 1341, -298, -78, 58, -9, -2, 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], [33614, 4802, -19894, 3822, 2086, -1341, 298, 78, -58, 9, 2, -1], [4386627, -2038449, -387933, 533659, -188559, -2980, 25114, -9433, 519, 748, -113, -10], [-4437048, 1865577, 1011164, -546644, 72072, 54744, -31616, 5319, 2088, -1423, -16, 78], [4857223, -905177, -1508514, 642194, -19922, -88269, 39066, -3369, -3538, 1648, 66, -103], [1159683, -823543, 189679, 157192, -105588, 27242, 3691, -5281, 1518, -41, -111, 24], [13815354, -10098606, 1931433, 1613717, -1300516, 312748, 37343, -60582, 18586, -561, -1177, 326], [87816575, -63271152, 7322364, 13138664, -8068508, 1464432, 512760, -431264, 104918, 6648, -9816, 2024], [-24168466, 17928267, -1920114, -3863503, 2287180, -396084, -156972, 124669, -29188, -2256, 2910, -538], [-253953770, 159827367, -15008994, -34869674, 20934158, -3581637, -1366950, 1126019, -262778, -17418, 24666, -5459], [-249415880, 165049542, -15266244, -36302189, 21672203, -3693282, -1475595, 1178894, -275603, -19923, 26256, -5624], [-69929125, 45330537, -4431609, -9970009, 6008398, -1031292, -397035, 325759, -76858, -5088, 7221, -1594]];
    Rf_basisdens := [1, 16807, 84035, 420175, 420175, 84035, 420175, 1260525, 252105, 1260525, 1260525, 180075];
    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_8820_d();"
function MakeCharacter_8820_d()
    N := 8820;
    order := 2;
    char_gens := [4411, 7841, 7057, 1081];
    v := [2, 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_8820_d_Hecke();"
function MakeCharacter_8820_d_Hecke(Kf)
    N := 8820;
    order := 2;
    char_gens := [4411, 7841, 7057, 1081];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 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, 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, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1], [0, 0, 0, 0, 0, 0, 0, -1, -1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1], [0, 0, 0, 0, -1, 1, 1, 0, 0, 0, 0, 0], [0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0], [4, 0, 0, 2, 1, -1, -1, 0, 0, 0, 0, 0], [-3, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1], [1, 0, 0, 1, 2, 0, -1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 1, -1, 1, 0], [-1, 0, 0, -1, -1, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -1, 1], [0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [-2, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0], [-3, 0, -1, -1, 2, -1, 0, 0, 0, 0, 0, 0], [-4, 0, -1, -1, -1, 1, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [-3, 0, 1, -1, 0, -1, 1, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, -1, -1, 2, 0], [0, -2, 0, 0, 0, 0, 0, -4, 0, -1, -1, 1], [1, 0, 1, 0, 2, 1, -1, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 4, 1, 0, -1, 2], [1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0], [-4, 0, 0, 2, 0, -2, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, -3, 2, 1, 1, 0], [0, -3, 0, 0, 0, 0, 0, 3, 1, 1, 1, 0], [0, 5, 0, 0, 0, 0, 0, -3, 2, 2, 1, -1], [-5, 0, 1, -1, 0, 0, 1, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0], [4, 0, 2, -1, -1, 1, -1, 0, 0, 0, 0, 0], [-4, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0], [-5, 0, 1, 1, 4, -2, -2, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, -2, 2, -1, 2], [0, 5, 0, 0, 0, 0, 0, -5, 1, 1, 1, -2], [0, -1, 0, 0, 0, 0, 0, -5, 0, 0, 1, -1], [4, 0, 2, 1, -2, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 4, 0, 3, 1, -1], [0, -1, 0, 0, 0, 0, 0, 0, -1, -3, 1, 0], [-2, 0, -1, -2, 2, 1, 1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 4, -1, 0, 2, 2], [-2, 0, 0, 2, -2, -1, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 3, 0, 2, 2, 2], [0, -4, 0, 0, 0, 0, 0, -4, -2, -4, -2, 0], [0, 1, 0, 0, 0, 0, 0, -3, -2, 2, 2, -3], [0, -1, 0, 0, 0, 0, 0, -6, -1, 1, 1, -2], [-3, 0, 1, 3, 2, 1, -1, 0, 0, 0, 0, 0], [-7, 0, 0, 2, 1, 1, 1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, -6, 1, 2, -1, -2], [5, 0, 0, -2, -1, 2, -3, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 1, 0, 1, 3, -1], [-4, 0, 2, 4, 3, -1, -1, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 2, 0, 2, 1, 2], [0, 1, 0, 0, 0, 0, 0, 6, 0, 3, 2, -2], [-6, 0, -1, -5, -3, 2, -1, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, -4, 1, 2, 2, 0], [-8, 0, 1, -3, -1, 3, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, -2, 0, -1, 2, 2], [0, 3, 0, 0, 0, 0, 0, -4, -1, -1, 2, -1], [-3, 0, 2, 2, 0, 4, 0, 0, 0, 0, 0, 0], [-9, 0, -2, -2, 0, -1, 1, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, -5, -2, 0, 2, -3], [0, 0, 0, 0, 0, 0, 0, -5, 1, -2, 0, -1], [-8, 0, -2, -2, -4, -1, 2, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, -8, 2, -4, 0, 3], [0, 5, 0, 0, 0, 0, 0, -6, 0, 2, 2, 0], [3, 0, 0, -3, 1, 2, 2, 0, 0, 0, 0, 0], [-3, 0, 0, 0, -4, 2, -2, 0, 0, 0, 0, 0], [-7, 0, 2, 4, 1, 2, -1, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, -4, 4, -2, -1, 3], [0, 2, 0, 0, 0, 0, 0, 2, 3, 1, 4, 0], [0, 0, 0, 0, 0, 0, 0, -4, -4, 2, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 1, -4, 0, -1], [-12, 0, 1, -2, -2, -1, 0, 0, 0, 0, 0, 0], [-5, 0, -1, 4, -2, -1, -1, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 1, 4, 4, 4, -1], [0, -1, 0, 0, 0, 0, 0, 6, 4, -2, -2, 0], [0, 3, 0, 0, 0, 0, 0, -8, 1, 1, 3, 0], [0, -2, 0, 0, 0, 0, 0, -5, -1, 2, -1, -1], [0, 2, 0, 0, 0, 0, 0, 6, -4, -2, 1, -2], [-11, 0, 2, -2, -2, -1, -3, 0, 0, 0, 0, 0], [-1, 0, 1, -3, -2, 5, 3, 0, 0, 0, 0, 0], [-1, 0, 2, -1, 3, -2, -2, 0, 0, 0, 0, 0], [-5, 0, -1, -1, 4, -4, 0, 0, 0, 0, 0, 0], [-13, 0, 1, 2, 5, -3, -1, 0, 0, 0, 0, 0], [-7, 0, 2, -5, -3, 2, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 2, 2, 1, -2], [5, 0, 2, 2, 0, 2, 2, 0, 0, 0, 0, 0], [-5, 0, -3, 1, 2, 0, 2, 0, 0, 0, 0, 0], [-13, 0, -1, 4, -3, 1, 1, 0, 0, 0, 0, 0], [11, 0, 2, 4, -3, 2, 1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, -6, 0, -4, 2, 0], [-4, 0, 0, -1, 0, -1, 4, 0, 0, 0, 0, 0], [6, 0, 0, 3, 3, -1, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, -2, 0, 2, 2], [-3, 0, -3, -1, 2, -2, 0, 0, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, 4, 4, -2, 0, 1], [-11, 0, -1, 4, 4, 1, -1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, -2, 4, -2, 0, 2], [-13, 0, 1, -5, -2, 3, 2, 0, 0, 0, 0, 0], [-3, 0, 2, -4, -5, 1, -1, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, -4, 0, -6, -1, 0], [0, -1, 0, 0, 0, 0, 0, -2, -2, 7, -1, 1], [0, 2, 0, 0, 0, 0, 0, 2, -1, -2, 4, 0], [15, 0, 0, -2, 3, 4, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 5, -3, -4, -1, 1], [0, 0, 0, 0, 0, 0, 0, -4, 1, 2, 0, 3], [-1, 0, 2, -1, 2, -3, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, -10, -2, 6, 7, -3], [0, 2, 0, 0, 0, 0, 0, 10, -1, 2, 4, -4], [7, 0, 2, 2, -1, -3, 1, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, -4, 2, 5, -3, 1], [0, -3, 0, 0, 0, 0, 0, -3, 0, -2, 0, 3], [0, -1, 0, 0, 0, 0, 0, -7, -1, -1, -3, 6], [1, 0, -2, -2, 4, 1, -3, 0, 0, 0, 0, 0], [-2, 0, 0, 0, -6, -1, 4, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 10, 2, -3, -7, 1], [0, 4, 0, 0, 0, 0, 0, 1, -2, 2, 2, 2], [0, -1, 0, 0, 0, 0, 0, 15, 2, 0, 3, -2], [-4, 0, -1, -4, -8, 3, 3, 0, 0, 0, 0, 0], [-2, 0, 2, 0, 0, -4, -2, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 2, 0, -5, 2, 0], [0, 7, 0, 0, 0, 0, 0, -2, 4, 2, 2, -4], [-3, 0, -1, 2, 2, -7, 1, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, -13, -3, 3, 0, 0], [-8, 0, 2, -3, -6, -1, 0, 0, 0, 0, 0, 0], [8, 0, 0, 1, -1, -7, -1, 0, 0, 0, 0, 0], [-2, 0, -2, 3, 3, -8, -2, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 1, 1, 1, -3, 2], [17, 0, -3, 5, 4, -7, -2, 0, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, 4, 4, -4, -2, -2], [6, 0, 1, -5, -3, 1, -1, 0, 0, 0, 0, 0], [0, 7, 0, 0, 0, 0, 0, 1, -4, 8, 2, -3], [0, -7, 0, 0, 0, 0, 0, -7, -2, -3, 3, 1], [0, -4, 0, 0, 0, 0, 0, 12, 2, -4, -2, 3], [21, 0, 2, -2, 1, -2, -2, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 4, 0, -2, -2, 0], [0, 5, 0, 0, 0, 0, 0, 0, 0, 5, -1, -1], [-14, 0, -2, -1, 3, -2, 2, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 8, -2, 1, -4, -2], [-11, 0, 3, -3, -6, 3, -2, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 10, 5, -5, -1, 0], [0, 4, 0, 0, 0, 0, 0, -6, 4, 2, 2, -2], [-4, 0, -2, 7, -1, -1, 1, 0, 0, 0, 0, 0], [13, 0, 1, -2, 3, 3, 3, 0, 0, 0, 0, 0], [0, 0, 0, -6, -3, 3, 5, 0, 0, 0, 0, 0], [-6, 0, -2, -2, 6, 3, -4, 0, 0, 0, 0, 0], [-2, 0, -2, 3, -2, 0, -2, 0, 0, 0, 0, 0], [0, -7, 0, 0, 0, 0, 0, -9, -4, -2, -1, 0], [-6, 0, 0, 1, 4, 1, -4, 0, 0, 0, 0, 0], [6, 0, -2, -3, 7, -2, -2, 0, 0, 0, 0, 0], [0, -7, 0, 0, 0, 0, 0, 8, 2, -3, -4, 6], [5, 0, 4, 7, 4, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, -1, -3, -6, 6], [0, 5, 0, 0, 0, 0, 0, 5, -2, 0, -2, -1], [-6, 0, 2, 3, -8, 6, 2, 0, 0, 0, 0, 0], [3, 0, -3, -2, -3, 3, 5, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 1, 8, -3, -1, 4], [-19, 0, -3, -5, 4, -1, -4, 0, 0, 0, 0, 0], [-17, 0, 1, 6, 6, -3, -3, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, -8, 0, 7, 0, -2], [-6, 0, -2, 1, 4, 1, -2, 0, 0, 0, 0, 0], [1, 0, -1, 1, -6, 5, 6, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 1, -2, -2, 4, -5], [0, 1, 0, 0, 0, 0, 0, -9, -2, -3, -5, 1], [0, 4, 0, 0, 0, 0, 0, 2, 0, -6, 5, 1], [23, 0, 0, 0, 2, -1, 5, 0, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, -7, -2, 4, 2, 0], [12, 0, -1, 0, 2, 3, 2, 0, 0, 0, 0, 0], [-14, 0, 2, -3, -10, 1, -2, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 16, 0, -4, 5, -4], [0, 2, 0, 0, 0, 0, 0, 4, 6, 0, 4, 0], [0, -8, 0, 0, 0, 0, 0, 1, 0, 0, -2, 0], [-2, 0, -2, -5, -4, 6, 0, 0, 0, 0, 0, 0], [-6, 0, 4, 5, 1, 4, -4, 0, 0, 0, 0, 0], [7, 0, 0, -5, -9, 4, -2, 0, 0, 0, 0, 0], [3, 0, 2, -8, -5, 0, -3, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 1, -1, 4, -7, -1], [-27, 0, -2, -4, -5, 4, 5, 0, 0, 0, 0, 0], [4, 0, 0, 6, 2, -2, 2, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 2, -6, -3, -2, 0], [12, 0, -1, -1, 6, -6, -1, 0, 0, 0, 0, 0], [6, 0, 4, -5, 1, 6, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, -2, -2, 4, -4, 0], [0, -1, 0, 0, 0, 0, 0, -15, -4, -3, -3, -4], [-4, 0, 0, -2, -4, 2, -4, 0, 0, 0, 0, 0], [2, 0, -2, -4, 6, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 17, 1, 1, 4, -4], [-1, 0, -6, -2, 1, -5, -1, 0, 0, 0, 0, 0], [-14, 0, 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0, 0, 0, 0, 0], [13, 0, -1, 11, 6, 7, 2, 0, 0, 0, 0, 0], [-8, 0, 2, -1, -16, -3, 2, 0, 0, 0, 0, 0], [28, 0, -2, 2, 8, 3, -2, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 26, 10, -4, -1, 3], [0, -10, 0, 0, 0, 0, 0, 16, -5, -5, -10, 8], [0, 0, 0, 0, 0, 0, 0, 6, 0, -4, -6, -1], [17, 0, 0, 18, 9, -4, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -12, 0, -10, 6, -4], [-9, 0, 2, 6, -11, 4, 7, 0, 0, 0, 0, 0], [-6, 0, 1, -1, 8, 2, -11, 0, 0, 0, 0, 0], [0, 5, 0, 0, 0, 0, 0, -9, 2, 14, 9, -5], [0, 3, 0, 0, 0, 0, 0, 23, 2, 9, 5, -1], [0, -6, 0, 0, 0, 0, 0, 8, -4, 7, -3, -9], [7, 0, 0, 14, 3, -4, 0, 0, 0, 0, 0, 0], [-33, 0, 2, 0, -3, -1, 3, 0, 0, 0, 0, 0], [17, 0, 1, -17, 4, -5, -1, 0, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, -9, -4, -6, -4, 8], [-22, 0, 4, 0, -10, 0, 8, 0, 0, 0, 0, 0], [0, 5, 0, 0, 0, 0, 0, -5, 0, 14, 1, -1], [0, -8, 0, 0, 0, 0, 0, 2, 1, -4, -10, 2], [-13, 0, 1, -8, -5, 13, 5, 0, 0, 0, 0, 0], [0, -17, 0, 0, 0, 0, 0, 4, -1, -2, -4, 6], [0, -7, 0, 0, 0, 0, 0, 9, 2, 0, 6, 1], [0, -8, 0, 0, 0, 0, 0, 26, 0, 3, -9, 5], [21, 0, 4, 7, 16, -5, -10, 0, 0, 0, 0, 0], [-45, 0, -2, 5, -1, -4, -2, 0, 0, 0, 0, 0], [6, 0, -4, 7, 9, -6, 2, 0, 0, 0, 0, 0], [0, 11, 0, 0, 0, 0, 0, -11, 6, 8, 7, -4], [0, 4, 0, 0, 0, 0, 0, 8, 1, 10, 6, 0], [-21, 0, -2, -10, -11, 4, 5, 0, 0, 0, 0, 0], [33, 0, 5, -1, 8, 2, -6, 0, 0, 0, 0, 0], [0, -16, 0, 0, 0, 0, 0, -4, -2, -4, -1, 10], [0, -12, 0, 0, 0, 0, 0, 14, -3, 3, -6, -2], [0, -7, 0, 0, 0, 0, 0, -12, -7, 1, -4, -3], [16, 0, -2, 6, 12, 0, 4, 0, 0, 0, 0, 0], [0, 13, 0, 0, 0, 0, 0, 0, 1, 4, 2, -8], [-8, 0, 5, 7, 5, -7, -1, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, -7, -1, -2, 3, -3], [0, 0, 0, 0, 0, 0, 0, 10, 4, -6, 3, -8], [0, 2, 0, 0, 0, 0, 0, 2, 4, 0, -6, 6], [-14, 0, 1, 5, 3, 3, -4, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 11, 10, -4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 18, 12, 4, 7, 3], [-29, 0, 2, 6, 4, 2, 0, 0, 0, 0, 0, 0], [-12, 0, -3, -17, -7, 1, 7, 0, 0, 0, 0, 0], [0, 9, 0, 0, 0, 0, 0, -22, 0, 10, 2, -4], [50, 0, 0, 6, -4, 6, 4, 0, 0, 0, 0, 0], [11, 0, 3, -14, 2, 3, -1, 0, 0, 0, 0, 0], [12, 0, 0, 7, 6, -11, -6, 0, 0, 0, 0, 0], [34, 0, -2, 4, 5, -5, 3, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, -7, 6, 1, -1, -5], [-14, 0, -2, -8, -14, 9, 4, 0, 0, 0, 0, 0], [0, 11, 0, 0, 0, 0, 0, 21, 8, 4, -1, 5], [16, 0, -4, -13, -6, 5, 6, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, -10, -14, 10, -1, 8], [0, -1, 0, 0, 0, 0, 0, -22, -5, 5, -1, 2], [0, 11, 0, 0, 0, 0, 0, -9, 2, 4, 2, 3], [0, 0, 0, 0, 0, 0, 0, 24, 0, -9, 0, 4], [-11, 0, -2, -6, 13, -12, -7, 0, 0, 0, 0, 0], [0, 13, 0, 0, 0, 0, 0, -5, 14, 3, 5, 1], [0, 11, 0, 0, 0, 0, 0, 1, 5, 9, 7, -10], [22, 0, 2, 11, 6, 2, -6, 0, 0, 0, 0, 0], [-44, 0, -1, -2, -2, 1, -2, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 0, 12, -10, 10, 2, -10], [-7, 0, 0, 6, 13, -2, -9, 0, 0, 0, 0, 0], [-25, 0, -2, -5, 7, 8, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 14, 0, 10, 1, 1], [20, 0, 3, 1, -8, 4, 3, 0, 0, 0, 0, 0], [7, 0, -1, 7, -2, 3, 7, 0, 0, 0, 0, 0], [0, 10, 0, 0, 0, 0, 0, -2, -4, -2, 6, -8], [0, -7, 0, 0, 0, 0, 0, -6, 5, 12, 7, 0], [-14, 0, -1, -8, 2, 7, 1, 0, 0, 0, 0, 0], [7, 0, -1, -4, -2, -3, 7, 0, 0, 0, 0, 0], [0, 17, 0, 0, 0, 0, 0, -1, 4, 4, 7, -8], [-13, 0, 0, -2, -2, -4, -2, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 10, 6, 3, -4, 6], [-2, 0, 2, 3, -17, 3, -5, 0, 0, 0, 0, 0], [-11, 0, -3, -2, 2, 7, 7, 0, 0, 0, 0, 0], [-19, 0, 4, 6, -1, -3, -11, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 31, -10, -12, -4, 4], [0, 17, 0, 0, 0, 0, 0, -20, -8, 3, 3, -5], [29, 0, 2, -5, -7, 14, 4, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 33, 0, 0, 4, -1], [-11, 0, 2, -6, -3, -6, -5, 0, 0, 0, 0, 0], [4, 0, 1, 5, 12, -6, 1, 0, 0, 0, 0, 0], [0, 13, 0, 0, 0, 0, 0, -1, 0, 6, 5, -5], [0, 15, 0, 0, 0, 0, 0, -12, -3, 6, -2, 0], [0, -12, 0, 0, 0, 0, 0, 16, -2, -8, 2, 2], [0, -8, 0, 0, 0, 0, 0, -4, 5, -2, 4, 1], [1, 0, -5, 3, -4, 1, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, -4, 11, 4, -1, 2], [0, -13, 0, 0, 0, 0, 0, 12, 6, 1, -10, 18], [-34, 0, -9, 1, 11, -1, 6, 0, 0, 0, 0, 0], [0, 12, 0, 0, 0, 0, 0, 8, -9, 7, 4, 2], [-36, 0, 0, -5, 1, 6, -6, 0, 0, 0, 0, 0], [0, 9, 0, 0, 0, 0, 0, -8, 1, 10, 14, -10], [0, 8, 0, 0, 0, 0, 0, -20, 0, -1, -9, -5], [4, 0, 3, 1, -4, 4, 1, 0, 0, 0, 0, 0], [4, 0, 7, 2, 4, 3, -12, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, -6, -4, 2, 0, -14], [-7, 0, 3, 5, -2, 0, 5, 0, 0, 0, 0, 0], [0, 15, 0, 0, 0, 0, 0, -12, 9, 4, 4, 4], [10, 0, 3, 5, 11, 0, -11, 0, 0, 0, 0, 0], [-3, 0, 1, 10, 6, -17, -5, 0, 0, 0, 0, 0], [-54, 0, 2, -12, -16, -2, -4, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 1, 2, 6, -3, 0], [0, -12, 0, 0, 0, 0, 0, 6, -5, -8, -4, 3], [0, 6, 0, 0, 0, 0, 0, 8, 9, -7, 8, -4], [0, 4, 0, 0, 0, 0, 0, -1, 3, -5, 6, 0], [0, -1, 0, 0, 0, 0, 0, -10, 0, -5, 11, 3], [-3, 0, -6, 12, -2, -6, -4, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 10, 0, 4, 6, 4], [0, 9, 0, 0, 0, 0, 0, -18, -2, -3, 8, -8], [0, -2, 0, 0, 0, 0, 0, -13, -9, 11, -2, -4], [21, 0, -4, -12, -6, 7, 5, 0, 0, 0, 0, 0], [10, 0, -4, -6, 4, 6, 4, 0, 0, 0, 0, 0], [-4, 0, 7, -7, -11, 4, 1, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 0, -27, -7, -5, 2, -10], [0, -1, 0, 0, 0, 0, 0, -7, -3, 3, 17, -4], [-21, 0, -2, -5, 5, -2, 2, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 0, -4, -2, 0, -8, 2], [-2, 0, 3, -6, -4, 3, 6, 0, 0, 0, 0, 0], [18, 0, 1, 1, 1, 5, -6, 0, 0, 0, 0, 0], [0, 12, 0, 0, 0, 0, 0, -12, -1, 1, 12, -4], [0, -10, 0, 0, 0, 0, 0, -16, 4, -8, 12, 6], [-23, 0, 3, 3, -2, 6, 3, 0, 0, 0, 0, 0], [21, 0, 2, 21, 9, -8, -4, 0, 0, 0, 0, 0], [-8, 0, -2, -7, -9, -6, 2, 0, 0, 0, 0, 0], [0, -14, 0, 0, 0, 0, 0, -6, 3, -12, 0, 13], [0, 14, 0, 0, 0, 0, 0, -2, 6, 10, 10, -4], [0, -3, 0, 0, 0, 0, 0, 16, 0, -7, 0, 8], [38, 0, 2, 1, 8, -2, -2, 0, 0, 0, 0, 0], [21, 0, -1, 13, 6, -14, 1, 0, 0, 0, 0, 0], [0, 5, 0, 0, 0, 0, 0, -17, -6, 4, -2, -4], [0, -7, 0, 0, 0, 0, 0, -3, -10, 5, -7, -4], [0, -10, 0, 0, 0, 0, 0, 12, 3, -8, 0, 5], [13, 0, 2, 7, -6, 4, 3, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, -12, -4, -8, -6, -6], [39, 0, 8, 12, -9, 4, 3, 0, 0, 0, 0, 0], [0, -12, 0, 0, 0, 0, 0, 2, -4, 6, 4, 8], [4, 0, 4, 3, 5, 2, 6, 0, 0, 0, 0, 0], [2, 0, 4, 8, 2, 0, -14, 0, 0, 0, 0, 0], [0, 20, 0, 0, 0, 0, 0, -1, 1, 4, -5, 3], [8, 0, 8, 2, -7, -11, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 34, 4, 0, 11, -4], [0, 7, 0, 0, 0, 0, 0, -6, 7, 2, 16, -2], [31, 0, 2, 22, 7, 0, 1, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 4, 5, -7, 6, 4], [2, 0, -1, -4, -2, 5, 10, 0, 0, 0, 0, 0], [-8, 0, 1, 1, -6, -12, -1, 0, 0, 0, 0, 0], [34, 0, -8, 4, -5, -7, -1, 0, 0, 0, 0, 0], [-10, 0, -4, -9, -1, 10, -2, 0, 0, 0, 0, 0], [0, 7, 0, 0, 0, 0, 0, -20, 5, 0, -3, 4], [0, -18, 0, 0, 0, 0, 0, 40, -3, -4, -4, 5], [50, 0, 5, -5, -7, -2, -3, 0, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, -8, 4, 0, -17, 8], [-8, 0, -3, -18, 4, 3, 3, 0, 0, 0, 0, 0], [-21, 0, 2, -13, -7, 6, -4, 0, 0, 0, 0, 0], [-28, 0, 3, -17, -11, 3, 5, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 20, 8, 7, 0, 8], [17, 0, -2, 0, 15, 8, -1, 0, 0, 0, 0, 0], [0, -14, 0, 0, 0, 0, 0, 2, 12, -12, 0, 1], [0, 24, 0, 0, 0, 0, 0, -8, 4, 18, 10, -10], [-62, 0, -4, -2, -8, 0, 4, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, -22, 8, 5, 3, -7], [0, -1, 0, 0, 0, 0, 0, 5, 4, 0, -3, -6], [0, 12, 0, 0, 0, 0, 0, -18, -5, -2, 2, -7], [-22, 0, 6, -11, -11, 9, 1, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8820_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8820_2_d_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_8820_2_d_a(:prec:=12)
    chi := MakeCharacter_8820_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(4027) - 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_8820_2_d_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_8820_2_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8820_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![272484, 0, 256320, 0, 93880, 0, 16988, 0, 1572, 0, 68, 0, 1]>,<17,R![-450, 372, 262, -108, -52, 0, 1]>],Snew);
    return Vf;
end function;