
// Make newform 2268.2.i.n in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2268_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2268_i_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_2268_2_i_n();".
// 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_2268_2_i_n();".
// 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, -4116, 0, 4606, 0, -3318, 0, 1695, 0, -282, 0, 31, 0, -9, 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], [170475459, 0, 1081706164, 0, 586034029, 0, -19347710, 0, -21569239, 0, -2800495, 0, -22952, 0, 92249, 0], [-4446117606, 0, 3126524037, 0, -2126767580, 0, 665006344, 0, -92194120, 0, 15156266, 0, -3103286, 0, 244351, 0], [-188652058, 0, -145125260, 0, -34785107, 0, 75102634, 0, -11273818, 0, 910001, 0, -431490, 0, 50804, 0], [-129671150, 0, 367714228, 0, -211337511, 0, 138865326, 0, -25351497, 0, 2336349, 0, -763661, 0, 95779, 0], [0, -6997483661, 0, 3388669823, 0, -7116987318, 0, 434366602, 0, 93415788, 0, 30604253, 0, 214456, 0, -784170], [0, -2189789518, 0, 2394515438, 0, -3169832022, 0, 810563151, 0, -90097992, 0, 20969988, 0, -3779851, 0, 202439], [401714397, 0, -4135709270, 0, 3214460858, 0, -2478376396, 0, 439930867, 0, -39713699, 0, 13906205, 0, -1700901, 0], [6024452, 0, -19057962, 0, 7818769, 0, -61209692, 0, 12291821, 0, -722548, 0, 366096, 0, -53324, 0], [-305714528, 0, 445839877, 0, -437984288, 0, 259237135, 0, -43172212, 0, 4498055, 0, -1409701, 0, 160457, 0], [0, -2003509256, 0, 504845978, 0, 875750037, 0, -878690623, 0, 152655738, 0, -13365107, 0, 5091886, 0, -635345], [0, 3200801814, 0, -571100796, 0, 2566998474, 0, 307950965, 0, -140264304, 0, -5745335, 0, -2837787, 0, 751964], [0, -6552594825, 0, 11250666413, 0, -8917693029, 0, 9658554559, 0, -1753147101, 0, 140871131, 0, -54909609, 0, 7006527], [0, -8080133320, 0, 29297334416, 0, -26134593933, 0, 13373548133, 0, -2192423802, 0, 246634027, 0, -71750653, 0, 7841464], [0, -504509782, 0, 1133172530, 0, -1221325980, 0, 2074772181, 0, -390459270, 0, 27574530, 0, -12087451, 0, 1635500], [0, 38302053342, 0, -35352763285, 0, 33437101488, 0, -19848020141, 0, 3308249367, 0, -343863724, 0, 107332095, 0, -12267456]];
    Rf_basisdens := [1, 982062165, 982062165, 140294595, 196412433, 6874435155, 1374887031, 982062165, 22838655, 42698355, 982062165, 982062165, 6874435155, 6874435155, 1374887031, 6874435155];
    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_2268_i();"
function MakeCharacter_2268_i()
    N := 2268;
    order := 3;
    char_gens := [1135, 1541, 325];
    v := [3, 2, 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_2268_i_Hecke();"
function MakeCharacter_2268_i_Hecke(Kf)
    N := 2268;
    order := 3;
    char_gens := [1135, 1541, 325];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 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, 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], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [2, 0, 0, 0, -2, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, -1, 0, 0], [2, -1, -1, -1, -2, 0, 0, 1, -1, 1, 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, 2, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0], [2, -1, 1, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, -1, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, -1, 0, 3, -1], [1, -1, 0, -2, -3, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 3, 0, -1, 0, -1, 2], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 3, 0, 0], [0, 0, 0, 0, 0, -2, -1, 0, 0, 0, 1, 0, -2, 0, -1, 0], [-3, 1, -1, 0, 1, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 0], [-1, -2, 0, -2, -2, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0, 1, 0, -2, 3], [-2, 2, -3, 2, 2, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0], [5, -1, 2, 1, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 3, -2, 0, 0, 0, 0, 3, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -3, -2, 2, -3, -3, 2], [3, -3, 3, -2, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 4, -1, -2, 4, 0, 1], [-2, 2, -1, 1, 6, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 2, 1, 0], [-2, 2, 1, 2, 5, 0, 0, -3, 3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, -3, 2, -1, -3, 0, -2], [-2, 3, -1, 2, 3, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 3, 0, 4, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -2, -1, -2, -2, -3, 1], [2, 1, 1, 1, -2, 0, 0, -1, 2, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, -5, 0, 0, 0, 0, 1, 0, -2, 0, 0], [6, 3, 3, 1, -6, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -2, 1, -1, -2, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, -2, -3, 0, 0, 0, 3, -4, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 3, 10, -1], [0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1], [0, 0, 0, 0, 0, 2, -7, 0, 0, 0, 0, 0, 0, -1, 0, 0], [-8, -3, -1, -4, -3, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -2, 0, -1, 0, 0, 3], [-3, -1, 3, 2, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 3, 0, -1, 0, -1, -5], [3, -3, -1, 0, -2, 0, 0, 4, -1, 0, 0, 0, 0, 0, 0, 0], [5, -1, -1, 2, -5, 0, 0, -2, -1, 1, 0, 0, 0, 0, 0, 0], [-3, 3, -2, 3, 10, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 3, -3, 0, -2, -3], [2, -2, 1, -3, 4, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -4, -2, 5, -4, 2, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -1, -2, -5, 0], [11, -1, -1, -4, -11, 0, 0, 4, -4, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 4, 0, 0, 0, -1, 0, -1, 0, 4, 4], [0, 0, 0, 0, 0, -5, 7, 0, 0, 0, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 4, 5, 0, 4, -1, -5], [0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0], [7, 1, 1, 3, -7, 0, 0, -3, 1, -1, 0, 0, 0, 0, 0, 0], [9, -2, -2, 2, -9, 0, 0, -2, 3, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, -1, 0, 0, 0, 0, -4, 0, 5, 0, 0], [-10, 7, -5, 2, 7, 0, 0, 3, 0, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, -1, 11, 1], [-10, 4, 1, 5, 4, 0, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 7, -8, 0, 0, 0, -6, 0, 7, 0, -8, 5], [3, -3, 4, 1, -3, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 3, -4, 0, 0, 0, -4, 0, 3, 0, -4, -1], [7, 1, -5, -4, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-6, -1, -1, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, -10, 0, 0, 0, -5, 0, 1, 0, -10, -1], [-3, 3, -5, 1, 15, 0, 0, 2, -4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -3, -1, 7, -3, 2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -1, 2, 8, 0], [-10, 2, 2, 2, 10, 0, 0, -2, -1, 1, 0, 0, 0, 0, 0, 0], [2, -2, 5, 3, 1, 0, 0, -3, 8, 0, 0, 0, 0, 0, 0, 0], [-6, 0, 2, 2, 0, 0, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 10, 0, 0, 0, 0, -3, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, -7, -5, 7, -7, -8, 5], [-3, 5, 5, 6, 3, 0, 0, -6, 3, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, -4, 0, 7, 0, 0], [-5, 1, 0, 1, 1, 0, 0, 6, 0, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -2, 2, 0, 15, 2], [-2, 2, -2, 3, -11, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, -8, 0, 0, 0, 0, 4, 0, -3, 0, 0], [-15, -4, 4, 0, -4, 0, 0, -5, 0, -1, 0, 0, 0, 0, 0, 0], [-14, 0, -5, -5, 0, 0, 0, -4, 0, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 6, 0, 0, 0, -1, 0, -2, 0, 6, -7], [0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -11, 0, 2, 0, 0, -2], [19, -7, 5, -2, -7, 0, 0, -2, 0, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, -1, 0, 0, 0, 0, -5, 0, 8, 0, 0], [-13, 1, 1, -3, 13, 0, 0, 3, 3, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -4, -1, 0, -4, -3, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 1, -5, -6, 0], [2, -2, 1, -7, 5, 0, 0, 1, -6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 4, -6, 2, 4, -14, 6], [0, 0, -4, 4, -9, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -5, 9, 0, 0, 0, 8, 0, -5, 0, 9, -3], [0, 0, 0, 0, 0, 1, 7, 0, 0, 0, 0, -2, 0, -6, 0, 0], [0, 0, 0, 0, 0, -1, -9, 0, 0, 0, 0, -7, 0, -6, 0, 0], [7, 6, 6, 0, -7, 0, 0, 0, 3, -3, 0, 0, 0, 0, 0, 0], [-13, 6, 6, 12, 13, 0, 0, -12, 5, -5, 0, 0, 0, 0, 0, 0], [-8, 8, -7, 3, 7, 0, 0, -1, -4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -7, -8, 0, 0, 0, 0, -4, 0, -7, 0, 0], [0, 0, 0, 0, 0, 1, -7, 0, 0, 0, 7, 0, 1, 0, -7, -2], [0, 0, 0, 0, 0, -3, -4, 0, 0, 0, -1, 0, -3, 0, -4, -7], [-18, 4, 1, 5, 4, 0, 0, 5, 0, 1, 0, 0, 0, 0, 0, 0], [4, -4, 0, 5, 5, 0, 0, 4, 5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -5, -8, 0, 0, 0, 0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 2, -3, 7, 2, -8, 3], [0, 0, 0, 0, 0, 4, -12, 0, 0, 0, -10, 0, 4, 0, -12, 3], [-1, 1, -8, 5, 23, 0, 0, 7, -3, 0, 0, 0, 0, 0, 0, 0], [-2, 2, -3, 9, 7, 0, 0, 1, 6, 0, 0, 0, 0, 0, 0, 0], [-2, 2, -3, -3, 16, 0, 0, 1, -6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -11, -1, 8, -11, -2, 1], [-1, 4, 4, 8, 1, 0, 0, -8, 4, -4, 0, 0, 0, 0, 0, 0], [23, -6, 5, -1, -6, 0, 0, 5, 0, 11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 1, 4, 3, -10, -1], [26, -1, -1, 4, -26, 0, 0, -4, -3, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -4, -6, 0, 0, 0, 0, -1, 0, 9, 0, 0], [0, 0, 0, 0, 0, 3, -11, 0, 0, 0, 0, 4, 0, 5, 0, 0], [0, 0, 0, 0, 0, 3, 12, 0, 0, 0, 0, 7, 0, 6, 0, 0], [-4, -5, 7, 2, -5, 0, 0, -6, 0, -1, 0, 0, 0, 0, 0, 0], [1, -1, 4, -8, -12, 0, 0, -3, -4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -8, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 3, 0, -6, 0, 0], [6, 2, 5, 7, 2, 0, 0, 5, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -5, -5, 0, 0, 0, 3, 0, -5, 0, -5, 5], [-1, 3, -7, -4, 3, 0, 0, 5, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 2, 4, -11, 0], [-3, 3, -3, -5, 3, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0], [7, -7, 4, 0, -2, 0, 0, 3, 4, 0, 0, 0, 0, 0, 0, 0], [5, -5, -3, 4, 12, 0, 0, 8, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 4, -2, 4, 4, -6, 2], [-6, 6, -6, -1, -13, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0], [-1, -7, 4, -3, -7, 0, 0, -10, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 3, 11, 0, 0, 0, 0, -1, 0, -1, 0, 0], [13, -1, -1, 8, -13, 0, 0, -8, 6, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, -1, 0, -6, 0, 0], [13, 3, 2, 5, 3, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 9, 0, 0, 0, -2, 9, -10, -2, 16, -9], [0, 0, 0, 0, 0, -1, -13, 0, 0, 0, 0, -8, 0, -6, 0, 0], [-22, 4, -4, 0, 4, 0, 0, -5, 0, -9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 7, -13, 0, 0, 0, 1, 0, 7, 0, -13, 7], [-7, -2, 2, 0, -2, 0, 0, 4, 0, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, -6, 0, 0, 0, -13, 0, -2, 0, -6, -3], [-4, 4, 3, -3, 7, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, -19, 0, 0, 0, 0, -4, 0, -8, 0, 0], [2, -2, 0, -1, 17, 0, 0, 2, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 1, -4, 3, 15, -1], [-5, 5, 2, 1, 33, 0, 0, -7, 3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, -9, -4, -1, -9, 9, 4], [-5, 5, 0, -1, -1, 0, 0, -5, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 7, 0, 0, 0, 3, 0, 1, 0, 7, 4], [0, 2, -2, 0, 2, 0, 0, -2, 0, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -9, 7, 0, 0, 0, 0, -5, 0, -16, 0, 0], [17, 6, 6, 2, -17, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 7, 5, 0, 0, 0, 0, 0, 0, -8, 0, 0], [-10, -2, -2, 3, 10, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 8, 0, -8, 0, 0, -3], [-21, 5, -5, 0, 5, 0, 0, 7, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5, -4, 0, 0, 0, 3, 0, 5, 0, -4, 15], [0, 0, 0, 0, 0, -6, -4, 0, 0, 0, 5, 0, -6, 0, -4, 3], [0, 0, 0, 0, 0, 8, -6, 0, 0, 0, -8, 0, 8, 0, -6, 2], [7, 0, 0, 10, -7, 0, 0, -10, 5, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 1, -4, -3, 1, 2, 4], [0, 0, 0, 0, 0, -7, -3, 0, 0, 0, 4, 0, -7, 0, -3, -12], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, -15, -3, -3, -15, -11, 3], [-13, 13, -10, 8, 23, 0, 0, -3, -2, 0, 0, 0, 0, 0, 0, 0], [14, -1, -1, 0, -14, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [-21, 20, -17, 3, 20, 0, 0, 13, 0, -7, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 7, 14, 0, 0, 0, 0, 15, 0, 7, 0, 0], [0, 0, 0, 0, 0, 0, -9, 0, 0, 0, -1, -9, -1, -1, 5, 9], [9, 3, 3, 1, -9, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0], [-3, 2, 2, -3, 3, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], [10, -3, -2, -5, -3, 0, 0, -1, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 6, 0, 0, 0, -2, 6, 6, -2, 11, -6], [6, -6, -4, 2, 14, 0, 0, 10, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 10, -5, 0, 0, 0, 0, 9, 0, 2, 0, 0], [-8, 4, 0, 4, 4, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0], [9, -1, 2, 1, -1, 0, 0, -7, 0, -6, 0, 0, 0, 0, 0, 0], [21, 1, -1, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 8, -6, 0, 0, 0, 0, 10, 0, 8, 0, 0], [-11, 2, 2, 6, 11, 0, 0, -6, -1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -5, -2, 2, -5, -11, 2], [0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 2, 0, -8, 0, 0, -2], [0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 18, 7, -10, 18, 9, -7], [6, -6, 0, 2, -1, 0, 0, 6, 2, 0, 0, 0, 0, 0, 0, 0], [-17, 9, 9, -3, 17, 0, 0, 3, -3, 3, 0, 0, 0, 0, 0, 0], [-4, 4, 2, 4, 15, 0, 0, -6, 6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5, 2, 0, 0, 0, 0, 3, 0, 5, 0, 0], [-11, -4, -4, -1, 11, 0, 0, 1, 3, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -4, -2, 0, 0, 0, 0, -2, 0, -8, 0, 0], [-5, -10, -10, -1, 5, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 6, -8, 0, 0, 0, 13, 0, 6, 0, -8, 6], [0, 0, 0, 0, 0, -4, -1, 0, 0, 0, 0, 0, 0, 14, 0, 0], [0, 0, 0, 0, 0, -3, 13, 0, 0, 0, -2, 0, -3, 0, 13, 0], [7, -11, 5, -6, -11, 0, 0, -13, 0, -2, 0, 0, 0, 0, 0, 0], [-5, 2, -2, 0, 2, 0, 0, 6, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5, 13, 0, 0, 0, 0, 10, 0, -5, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, -2, 2, 0, -2, -10, -2], [0, 0, 0, 0, 0, -10, -12, 0, 0, 0, -2, 0, -10, 0, -12, 6], [7, -7, -3, -6, -20, 0, 0, 10, -9, 0, 0, 0, 0, 0, 0, 0], [-7, 7, -13, 8, 29, 0, 0, 6, -5, 0, 0, 0, 0, 0, 0, 0], [32, 0, 0, -8, -32, 0, 0, 8, -8, 8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, -2, 0, 0, 0, -6, 0, 1, 0, -2, -2], [0, 0, 0, 0, 0, -3, -10, 0, 0, 0, 0, 6, 0, 0, 0, 0], [-3, -5, -5, -11, 3, 0, 0, 11, -2, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, 5, 0, 0, 0, 0, 3, 0, -18, 0, 0], [0, 0, 0, 0, 0, -5, 2, 0, 0, 0, 0, 9, 0, -8, 0, 0], [-1, -6, 7, 1, -6, 0, 0, -5, 0, 1, 0, 0, 0, 0, 0, 0], [-10, 2, 2, -14, 10, 0, 0, 14, -6, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 14, -8, -8, 14, -5, 8], [-9, 9, -10, 14, 11, 0, 0, 1, 4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 6, 6, 0, 0, 0, 0, 0, 6, 0, 6, 4], [0, 0, 0, 0, 0, -2, -6, 0, 0, 0, 14, 0, -2, 0, -6, -6], [-15, -4, 8, 4, -4, 0, 0, -7, 0, -3, 0, 0, 0, 0, 0, 0], [-8, -7, 8, 1, -7, 0, 0, -8, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 0, 0, 0, -3, 5, -3, -3, 16, -5], [0, 0, 0, 0, 0, 0, -7, 0, 0, 0, -8, -7, 14, -8, -2, 7], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, -8, -4, -7, -8, 4, 4], [10, -10, 7, 2, 14, 0, 0, 3, 9, 0, 0, 0, 0, 0, 0, 0], [13, -4, -4, 8, -13, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -7, -7, 0, 0, 0, 0, -3, 0, -1, 0, 0], [-1, -2, -2, 8, 1, 0, 0, -8, 4, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 9, 0, 0, 0, -11, 9, -7, -11, 19, -9], [-3, 3, -9, 6, 1, 0, 0, 6, -3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, -12, 0, 0, 0, -3, 11], [0, 0, 0, 0, 0, -6, -3, 0, 0, 0, 0, -9, 0, 2, 0, 0], [45, -11, 7, -4, -11, 0, 0, -7, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 3, 8, 0, 0, 0, -7, 0, 3, 0, 8, -7], [17, 2, -3, -1, 2, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0], [-2, 2, -5, 5, 20, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], [-19, 13, 13, 10, 19, 0, 0, -10, 5, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 10, -11, 0, 0, 0, 3, 0, 10, 0, -11, 1], [1, -1, 3, 1, 6, 0, 0, -2, 4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, -4, -5, -3, -4, -3, 5], [7, -7, 0, -4, -16, 0, 0, 7, -4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 3, -11, 2, 7, -3], [0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 7, 7, 1, 7, 0, -7], [0, 0, 0, 0, 0, 1, -4, 0, 0, 0, -8, 0, 1, 0, -4, -3], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 2, 1, 2, 4, -2], [-5, -6, -6, -5, 5, 0, 0, 5, -13, 13, 0, 0, 0, 0, 0, 0], [25, 4, 1, 5, 4, 0, 0, 7, 0, 3, 0, 0, 0, 0, 0, 0], [36, -4, -4, -7, -36, 0, 0, 7, -8, 8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 9, 8, -15, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 5, 0, -2, 0, 0, -12], [25, -6, -2, -8, -6, 0, 0, 2, 0, 8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 8, -12, 0, 0, 0, -11, 0, 8, 0, -12, 11], [-27, 17, -16, 1, 17, 0, 0, 11, 0, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 19, 0, 0, 0, -5, 0, 0, 0, 19, -2], [-21, -6, -6, -5, 21, 0, 0, 5, 6, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, -2, -9, 2, -3, 2], [0, 0, 0, 0, 0, 1, 17, 0, 0, 0, 7, 0, 1, 0, 17, -7], [-14, 14, -5, -1, 2, 0, 0, -9, -6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 3, -8, 1, 3, 11, 8], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -11, 1, 9, -11, 5, -1], [-3, 3, 15, -9, -23, 0, 0, -18, 6, 0, 0, 0, 0, 0, 0, 0], [-25, -3, -3, 8, 25, 0, 0, -8, 15, -15, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 11, 0, 0, 0, -3, 0, -1, 0, 11, 0], [-11, -6, -6, -8, 11, 0, 0, 8, 2, -2, 0, 0, 0, 0, 0, 0], [0, 6, 6, -3, 0, 0, 0, 3, 10, -10, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 3, 0, -6, 0, 0], [-37, 4, -2, 2, 4, 0, 0, 3, 0, -1, 0, 0, 0, 0, 0, 0], [-24, 9, 0, 9, 9, 0, 0, -1, 0, -10, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -8, 17, 0, 0, 0, -5, 0, -8, 0, 17, -16], [-44, 8, -6, 2, 8, 0, 0, -4, 0, -12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 5, 0, 0, 0, -11, 0, 1, 0, 5, 11], [0, 0, 0, 0, 0, -6, 6, 0, 0, 0, 0, -9, 0, -7, 0, 0], [23, -5, -5, -11, -23, 0, 0, 11, -1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, -9, 0, 0, 0, 15, 0, -3, 0, -9, 0], [6, -6, 13, -1, -7, 0, 0, -7, 12, 0, 0, 0, 0, 0, 0, 0], [3, -3, 10, -3, 13, 0, 0, -7, 7, 0, 0, 0, 0, 0, 0, 0], [12, 5, 5, 8, -12, 0, 0, -8, 4, -4, 0, 0, 0, 0, 0, 0], [-2, 2, 3, 1, 12, 0, 0, -5, 4, 0, 0, 0, 0, 0, 0, 0], [-8, -13, -13, -12, 8, 0, 0, 12, 2, -2, 0, 0, 0, 0, 0, 0], [15, -7, -7, -7, -15, 0, 0, 7, -8, 8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -12, 19, 0, 0, 0, 0, -9, 0, 1, 0, 0], [0, -4, -4, -8, 0, 0, 0, 8, -7, 7, 0, 0, 0, 0, 0, 0], [5, -5, -5, -2, -5, 0, 0, 2, 5, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 10, 12, 0, 0, 0, 2, 0, 10, 0, 12, 3], [0, 0, 0, 0, 0, -3, -8, 0, 0, 0, -14, 0, -3, 0, -8, 4], [23, -7, 12, 5, -7, 0, 0, 1, 0, 8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -13, -2, 0, 0, 0, 0, -11, 0, -8, 0, 0], [6, -6, 2, 6, 13, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 0], [-5, 5, -8, 15, -17, 0, 0, 3, 7, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 11, 13, -10, 11, 11, -13], [2, -2, 7, 7, -18, 0, 0, -5, 14, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 11, 2, -2, 11, 7, -2], [-30, -11, -11, -8, 30, 0, 0, 8, -8, 8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -8, -12, 0, 0, 0, -19, 0, -8, 0, -12, 6], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 4, -1, -2, 0], [0, 0, 0, 0, 0, -14, -1, 0, 0, 0, 0, -15, 0, -14, 0, 0], [0, 0, 0, 0, 0, -1, -5, 0, 0, 0, 0, -8, 0, 4, 0, 0], [0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 11, -10, 6, 11, -18, 10], [-12, 12, -1, -3, -17, 0, 0, -11, -4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 14, 0, 0, 0, 18, 0, 2, 0, 14, -7], [-18, 17, -19, -2, 17, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 17, 0, 0, 0, 0, 3, 0, 7, 0, 0], [0, 0, 0, 0, 0, 0, 5, 0, 0, 0, -6, 5, 16, -6, 23, -5], [7, -7, 7, -3, -26, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0], [-3, 3, -9, 10, 30, 0, 0, 6, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 7, 13, -2, 7, -15, -13], [4, -4, 1, 8, -10, 0, 0, 3, 9, 0, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2268_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2268_2_i_n();".
// 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_2268_2_i_n(:prec:=16)
    chi := MakeCharacter_2268_i();
    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_2268_2_i_n();".
// 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_2268_2_i_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2268_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![81, 0, 783, 0, 6723, 0, 7818, 0, 7087, 0, 1706, 0, 306, 0, 20, 0, 1]>,<13,R![21609, -18522, 12936, -3990, 1177, -152, 45, -5, 1]>,<19,R![49, -49, 217, 224, 541, 110, 40, -4, 1]>],Snew);
    return Vf;
end function;