
// Make newform 490.2.g.c in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_490_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_490_g_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_490_2_g_c();".
// 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_490_2_g_c();".
// 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 := [2401, 0, 3038, 0, 1933, 0, 1438, 0, 852, 0, 266, 0, 61, 0, 10, 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], [-16581649, 0, 415969049, 0, 151049252, 0, -168482130, 0, -70335258, 0, -20248352, 0, -4061221, 0, -525491, 0], [53786663, 0, 34590357, 0, 51464796, 0, 30357550, 0, 9493246, 0, 2237304, 0, 364187, 0, 34297, 0], [0, 1691266857, 0, 856246593, 0, 407420164, 0, 157143590, 0, 41107094, 0, 8202336, 0, 1057753, 0, 88213], [-1076511772, 0, -843429583, 0, -445540384, 0, -373946420, 0, -125675214, 0, -30938216, 0, -5366258, 0, -623003, 0], [-8000085009, -6797535927, -220726331, -1618097653, -128527588, -3441488944, -109893770, -2888475470, -55869898, -995989274, -14263312, -238976156, -4646621, -41450603, -307671, -4174573], [-8000085009, 6797535927, -220726331, 1618097653, -128527588, 3441488944, -109893770, 2888475470, -55869898, 995989274, -14263312, 238976156, -4646621, 41450603, -307671, 4174573], [-269649107, -1945628006, 748752592, -2464268024, 207676721, -1831430982, -538978895, -1083598885, -245531944, -335301337, -68962271, -75497868, -13006098, -12140159, -1594698, -1276339], [-269649107, 1945628006, 748752592, 2464268024, 207676721, 1831430982, -538978895, 1083598885, -245531944, 335301337, -68962271, 75497868, -13006098, 12140159, -1594698, 1276339], [0, -42231189, 0, -8929484, 0, -21381008, 0, -17945290, 0, -6079906, 0, -1484692, 0, -257521, 0, -26590], [-459236869, -517014827, -190752681, -223880973, -262434718, -164843024, -158239340, -99003980, -50405418, -31585764, -10972682, -7185176, -1812391, -1257693, -151501, -124363], [459236869, -517014827, 190752681, -223880973, 262434718, -164843024, 158239340, -99003980, 50405418, -31585764, 10972682, -7185176, 1812391, -1257693, 151501, -124363], [0, 238292243, 0, 303303017, 0, 226254436, 0, 133651610, 0, 41588666, 0, 9597064, 0, 1553547, 0, 154017], [-781884621, -477019655, -614398939, -281366705, -324550940, -118396640, -272435506, -165340790, -92234954, -60556930, -22633800, -15737980, -3965353, -2868675, -472647, -357305], [781884621, -477019655, 614398939, -281366705, 324550940, -118396640, 272435506, -165340790, 92234954, -60556930, 22633800, -15737980, 3965353, -2868675, 472647, -357305], [0, 147858697, 0, 109455293, 0, 59771364, 0, 58201830, 0, 20069294, 0, 5023936, 0, 887273, 0, 104513]];
    Rf_basisdens := [1, 785969800, 34172600, 785969800, 196492450, 11003577200, 11003577200, 2750894300, 2750894300, 50016260, 239208200, 239208200, 239208200, 200065040, 200065040, 34172600];
    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_490_g();"
function MakeCharacter_490_g()
    N := 490;
    order := 4;
    char_gens := [197, 101];
    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_490_g_Hecke();"
function MakeCharacter_490_g_Hecke(Kf)
    N := 490;
    order := 4;
    char_gens := [197, 101];
    char_values := [[0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 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, -1, 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, -1, 0, 0, -1, 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], [1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, -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, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, -2], [1, 0, 1, -1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, -2, 0, 0, -2, -2, 0, 0, 0, 0, -1, 1, 0], [0, 0, 1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0], [-1, 0, -1, 1, 0, 0, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0], [0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, -2, 0, -1], [4, 0, 1, -1, 0, 0, 2, 0, 0, 4, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, -2], [0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0], [-4, 0, 0, 0, 0, -5, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0], [2, 0, 2, 0, 0, -4, -4, 0, 0, 0, 1, -1, 0, 0, 0, 0], [0, 1, 0, 0, 1, 0, 0, 0, 6, 0, 0, 0, 0, 0, -4, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, -2, 0, 0, 0, 0], [0, 2, 0, 0, -2, 0, 0, 0, 3, 0, 0, 0, 4, 0, -5, -2], [0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 2, 1, 1, -1], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, -4, 0, 1], [0, 7, 0, 0, 1, 0, 0, 3, 3, 0, 0, 0, 0, 2, -2, 0], [0, -4, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, -3, 0, -5, -1], [-3, 0, -1, -1, 0, 1, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 4, -4, 0, 0, 2, 2, 2, 0, 0, 0, 0], [1, 0, 3, -3, 0, 0, 4, 0, 0, 1, 0, 4, 0, 0, 0, 0], [-6, 0, -1, -1, 0, -10, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, -3, 3, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 2, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, -2, 5, 5, 0], [0, 0, 0, -3, 0, 2, -2, 0, 0, 6, 3, 3, 0, 0, 0, 0], [0, 0, 0, 0, 0, -5, -5, 0, 0, 0, 3, -3, 0, 0, 0, 0], [0, -2, 0, 0, -2, 0, 0, 5, 0, 0, 0, 0, 0, 6, 0, 2], [-8, 0, -2, 2, 0, 0, 6, 0, 0, -8, 0, 2, 0, 0, 0, 0], [0, -1, 0, 0, -2, 0, 0, -2, 0, 0, 0, 0, -1, -5, 0, 2], [0, 1, 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 0, 0, -2, 1], [0, 0, 0, -1, 0, -2, 2, 0, 0, 1, -1, -1, 0, 0, 0, 0], [0, -3, 0, 0, 1, 0, 0, 3, 3, 0, 0, 0, 0, 2, -2, 0], [2, 0, 4, 0, 0, 5, 5, 0, 0, 0, -1, 1, 0, 0, 0, 0], [-5, 0, 0, 0, 0, 0, 8, 0, 0, -5, 0, -2, 0, 0, 0, 0], [4, 0, -2, -2, 0, 0, 0, 0, 0, -4, 5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 6, 0, 0, -6], [5, 0, -3, 0, 0, -5, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, -2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 4, -2], [0, -5, 0, 0, -2, 0, 0, -8, 0, 0, 0, 0, 3, -4, 0, 2], [0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 4, -4, -4, 0], [-9, 0, 0, 0, 0, 0, -2, 0, 0, -9, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 1, 0, 0, -14, 2, 2, 0, 0, 0, 0], [0, -1, 0, 0, 6, 0, 0, -2, -2, 0, 0, 0, 0, -5, 5, 0], [0, -3, 0, 0, -2, 0, 0, -5, -5, 0, 0, 0, 0, -4, 4, 0], [0, -3, 0, 0, -4, 0, 0, 6, 0, 0, 0, 0, -1, 2, 0, 4], [3, 0, 0, 0, 0, 0, -5, 0, 0, 3, 0, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 8, 2, 2, -1], [0, 2, 0, 0, -6, 0, 0, -4, -4, 0, 0, 0, 0, -4, 4, 0], [6, 0, -3, -3, 0, 2, 0, 0, 0, -6, 2, 0, 0, 0, 0, 0], [-11, 0, -1, 0, 0, 7, 7, 0, 0, 0, 2, -2, 0, 0, 0, 0], [0, -2, 0, 0, 4, 0, 0, 0, -4, 0, 0, 0, -6, 0, -2, 4], [0, 7, 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 6, 0, -2, 1], [0, -3, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 3, 1, 0, 0], [0, -2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 4, -4, 0], [0, -1, 0, 0, 4, 0, 0, 0, -10, 0, 0, 0, -5, 0, 2, 4], [2, 0, 1, 1, 0, -2, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0], [-9, 0, -9, 0, 0, -8, -8, 0, 0, 0, 5, -5, 0, 0, 0, 0], [2, 0, 0, 0, 0, 14, 0, 0, 0, -2, -4, 0, 0, 0, 0, 0], [-5, 0, -1, -1, 0, -5, 0, 0, 0, 5, 5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 10, -10, 0, 0, 0, -2, 4, 4, -11], [0, 6, 0, 0, 6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 6], [0, 0, 0, 4, 0, 5, -5, 0, 0, -14, -4, -4, 0, 0, 0, 0], [0, 1, 0, 0, -4, 0, 0, -4, 0, 0, 0, 0, -5, -7, 0, 4], [-2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -4, 0, 0, 0, 0], [0, 0, 0, -1, 0, -7, 7, 0, 0, 1, -2, -2, 0, 0, 0, 0], [0, 5, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 3, 2], [0, 0, 0, 2, 0, -2, 2, 0, 0, -2, 3, 3, 0, 0, 0, 0], [0, 7, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0, -4, 16, 0, -3], [-3, 0, 0, 0, 0, 12, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6, -6, 0, 0, 0, -12, 3, 3, -1], [0, 0, 0, 0, 0, 0, 0, 6, -6, 0, 0, 0, -5, -3, -3, 12], [8, 0, 11, 0, 0, 4, 4, 0, 0, 0, -4, 4, 0, 0, 0, 0], [8, 0, 4, 0, 0, 3, 3, 0, 0, 0, -4, 4, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, 0, 0, 2, 0, 0, 0, -6, 0, -2, 6], [0, 0, 0, 0, 0, 0, 0, 9, -9, 0, 0, 0, 0, -4, -4, -10], [8, 0, 2, -2, 0, 0, -3, 0, 0, 8, 0, 1, 0, 0, 0, 0], [0, 0, 0, 6, 0, -8, 8, 0, 0, -11, 0, 0, 0, 0, 0, 0], [10, 0, 0, 0, 0, 12, 0, 0, 0, -10, 2, 0, 0, 0, 0, 0], [0, -10, 0, 0, -2, 0, 0, -1, -1, 0, 0, 0, 0, 4, -4, 0], [7, 0, 3, -3, 0, 0, -1, 0, 0, 7, 0, 4, 0, 0, 0, 0], [0, -5, 0, 0, -6, 0, 0, -11, 0, 0, 0, 0, -1, -3, 0, 6], [0, 0, 0, 0, 0, 0, 0, 3, -3, 0, 0, 0, -10, 5, 5, 4], [-2, 0, 4, 4, 0, 10, 0, 0, 0, 2, -6, 0, 0, 0, 0, 0], [16, 0, 6, 0, 0, 7, 7, 0, 0, 0, -3, 3, 0, 0, 0, 0], [0, 0, 0, 0, 0, -15, 15, 0, 0, -4, 3, 3, 0, 0, 0, 0], [0, 2, 0, 0, -9, 0, 0, 0, -1, 0, 0, 0, 11, 0, -1, -9], [0, 0, 0, 0, 0, 0, 0, 5, -5, 0, 0, 0, -4, 0, 0, 3], [0, -1, 0, 0, -14, 0, 0, -4, -4, 0, 0, 0, 0, -1, 1, 0], [0, 8, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 8, 0, -14, 0], [-8, 0, 5, 0, 0, -14, -14, 0, 0, 0, -5, 5, 0, 0, 0, 0], [-10, 0, 2, 2, 0, 21, 0, 0, 0, 10, -1, 0, 0, 0, 0, 0], [-12, 0, -2, -2, 0, 12, 0, 0, 0, 12, -3, 0, 0, 0, 0, 0], [0, 1, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, -1, 0, 7, 2], [0, 0, 0, -3, 0, 5, -5, 0, 0, 21, -2, -2, 0, 0, 0, 0], [6, 0, 0, 0, 0, -2, -2, 0, 0, 0, -2, 2, 0, 0, 0, 0], [0, 1, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 4, 6, 0, -5], [0, 1, 0, 0, 10, 0, 0, 4, 0, 0, 0, 0, 9, 8, 0, -10], [0, -9, 0, 0, -4, 0, 0, 0, -4, 0, 0, 0, -5, 0, 12, -4], [0, 0, 0, -2, 0, 2, -2, 0, 0, -12, -3, -3, 0, 0, 0, 0], [0, 4, 0, 0, 6, 0, 0, -12, -12, 0, 0, 0, 0, -2, 2, 0], [0, 9, 0, 0, 12, 0, 0, 4, 0, 0, 0, 0, 3, 3, 0, -12], [-4, 0, -4, 4, 0, 0, -22, 0, 0, -4, 0, -3, 0, 0, 0, 0], [8, 0, -4, -4, 0, -8, 0, 0, 0, -8, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -11, 11, 0, 0, 0, 15, -8, -8, 4], [-4, 0, -8, 0, 0, 9, 9, 0, 0, 0, 2, -2, 0, 0, 0, 0], [3, 0, -4, 0, 0, -2, -2, 0, 0, 0, 4, -4, 0, 0, 0, 0], [0, -6, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -4, 0, 2, -2], [0, -5, 0, 0, -2, 0, 0, 2, 0, 0, 0, 0, 3, -15, 0, 2], [6, 0, -4, 4, 0, 0, -2, 0, 0, 6, 0, -1, 0, 0, 0, 0], [0, 0, 0, -4, 0, -6, 6, 0, 0, -10, -2, -2, 0, 0, 0, 0], [0, -5, 0, 0, -7, 0, 0, -3, -3, 0, 0, 0, 0, -10, 10, 0], [-3, 0, 3, -3, 0, 0, -14, 0, 0, -3, 0, 4, 0, 0, 0, 0], [0, -2, 0, 0, -10, 0, 0, -7, 0, 0, 0, 0, -8, -8, 0, 10], [3, 0, 1, -1, 0, 0, 23, 0, 0, 3, 0, 3, 0, 0, 0, 0], [0, -18, 0, 0, -4, 0, 0, -11, -11, 0, 0, 0, 0, -13, 13, 0], [-14, 0, -7, 0, 0, -8, -8, 0, 0, 0, 5, -5, 0, 0, 0, 0], [0, 0, 0, -7, 0, -10, 10, 0, 0, -4, -2, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 8, -5, -5, 4], [0, -15, 0, 0, -4, 0, 0, -8, 0, 0, 0, 0, 11, -1, 0, 4], [0, -10, 0, 0, -10, 0, 0, 0, -9, 0, 0, 0, 0, 0, -2, -10], [0, 0, 0, -5, 0, -6, 6, 0, 0, 11, 1, 1, 0, 0, 0, 0], [25, 0, -1, 1, 0, 0, 9, 0, 0, 25, 0, -2, 0, 0, 0, 0], [0, 0, 6, 0, 0, -22, -22, 0, 0, 0, -1, 1, 0, 0, 0, 0], [20, 0, -1, -1, 0, 10, 0, 0, 0, -20, 2, 0, 0, 0, 0, 0], [0, 1, 0, 0, 2, 0, 0, -8, -8, 0, 0, 0, 0, -1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1, 1, 8], [0, -12, 0, 0, -10, 0, 0, 0, -5, 0, 0, 0, -2, 0, 8, -10], [0, -7, 0, 0, -6, 0, 0, -11, 0, 0, 0, 0, 1, -3, 0, 6], [0, -12, 0, 0, -12, 0, 0, -16, 0, 0, 0, 0, 0, -8, 0, 12], [0, 0, 0, -2, 0, 12, -12, 0, 0, -11, 2, 2, 0, 0, 0, 0], [0, -9, 0, 0, -22, 0, 0, 1, 1, 0, 0, 0, 0, 2, -2, 0], [18, 0, 4, 0, 0, 8, 8, 0, 0, 0, -4, 4, 0, 0, 0, 0], [9, 0, -2, 2, 0, 0, -17, 0, 0, 9, 0, 1, 0, 0, 0, 0], [-3, 0, 5, 5, 0, -5, 0, 0, 0, 3, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, -10, -4, -4, 4], [0, 0, 0, 0, 0, 0, 0, 11, -11, 0, 0, 0, -24, 11, 11, 10], [0, 3, 0, 0, 9, 0, 0, 0, 5, 0, 0, 0, -6, 0, 0, 9], [0, -2, 0, 0, 3, 0, 0, -4, 0, 0, 0, 0, 5, 12, 0, -3], [0, 0, 0, 0, 0, 0, 0, -10, 10, 0, 0, 0, -8, -6, -6, 16], [9, 0, -3, 3, 0, 0, -19, 0, 0, 9, 0, -4, 0, 0, 0, 0], [-2, 0, -1, -1, 0, 22, 0, 0, 0, 2, 5, 0, 0, 0, 0, 0], [0, -10, 0, 0, -7, 0, 0, -8, -8, 0, 0, 0, 0, -14, 14, 0], [0, 0, 4, -4, 0, 0, 20, 0, 0, 0, 0, 12, 0, 0, 0, 0], [0, 6, 0, 0, 5, 0, 0, 23, 0, 0, 0, 0, -1, 9, 0, -5], [-2, 0, -2, -2, 0, 17, 0, 0, 0, 2, -3, 0, 0, 0, 0, 0], [4, 0, -6, 0, 0, 5, 5, 0, 0, 0, -3, 3, 0, 0, 0, 0], [0, 0, 0, -4, 0, -4, 4, 0, 0, 24, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6, -6, 0, 0, 0, -9, 14, 14, 3], [0, 16, 0, 0, 2, 0, 0, 14, 0, 0, 0, 0, -14, 12, 0, -2], [0, 8, 0, 0, 8, 0, 0, -4, -4, 0, 0, 0, 0, 4, -4, 0], [-4, 0, -8, -8, 0, -23, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0], [-4, 0, -8, 8, 0, 0, -4, 0, 0, -4, 0, 0, 0, 0, 0, 0], [-17, 0, -6, -6, 0, -25, 0, 0, 0, 17, 11, 0, 0, 0, 0, 0], [0, 3, 0, 0, -8, 0, 0, 12, 12, 0, 0, 0, 0, 9, -9, 0], [6, 0, 7, 7, 0, 10, 0, 0, 0, -6, 2, 0, 0, 0, 0, 0], [0, 15, 0, 0, -2, 0, 0, 0, 14, 0, 0, 0, 17, 0, -11, -2], [-22, 0, 6, 0, 0, 1, 1, 0, 0, 0, -9, 9, 0, 0, 0, 0], [0, -2, 0, 0, -12, 0, 0, 2, 0, 0, 0, 0, -10, 2, 0, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_490_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_490_2_g_c();".
// 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_490_2_g_c(:prec:=16)
    chi := MakeCharacter_490_g();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_490_2_g_c();".
// 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_490_2_g_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_490_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 0, 0, 0, 2280, 0, 0, 0, 1361, 0, 0, 0, 90, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;