
// Make newform 2268.2.l.k in Magma, downloaded from the LMFDB on 03 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2268_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2268_l_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_l_k();".
// 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_l_k();".
// 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 := [3, -12, 19, -15, 10, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [-6, 3, -1, -5, 1, -1], [12, -21, 8, -8, 1, -1], [9, -15, 20, -11, 4, -1], [6, -21, 19, -16, 5, -2], [-9, 30, -20, 14, -4, 1]];
    Rf_basisdens := [1, 3, 3, 3, 3, 3];
    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_l();"
function MakeCharacter_2268_l()
    N := 2268;
    order := 3;
    char_gens := [1135, 1541, 325];
    v := [3, 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_2268_l_Hecke();"
function MakeCharacter_2268_l_Hecke( : Kf := false)
    N := 2268;
    order := 3;
    char_gens := [1135, 1541, 325];
    char_values := [[1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [-1, 0, 0, 0, -1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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], [1, -1, 0, 1, 0, 0], [-1, 0, 0, -1, 0, 0], [2, 0, 1, 0, 0, 1], [0, 1, 1, 0, 0, -1], [0, 1, 0, 1, -1, -1], [-2, -1, -2, -2, -2, 1], [5, -1, -1, 1, 0, -1], [2, 1, 1, 1, 2, 0], [0, 0, -1, -1, 0, 1], [1, -1, 1, 1, 1, -2], [0, -2, -1, -1, 3, 2], [4, -1, 1, 1, 4, -2], [0, -1, 1, -2, -3, 1], [0, -3, -3, 0, 0, 3], [2, -1, 0, 0, 2, -1], [0, 0, 2, -2, 3, 0], [5, 1, 0, 0, 5, 1], [2, 3, 1, -3, 0, 1], [0, -3, -2, -1, -2, 3], [0, -2, 1, -3, -3, 2], [-5, 0, 1, 1, -5, -1], [3, -2, 2, 2, 3, -4], [-4, -2, 0, 0, -4, -2], [-11, 1, -2, -1, 0, -2], [-7, 0, 0, 0, 0, 0], [-12, 1, 2, 2, -12, -1], [0, -2, 0, -2, -2, 2], [0, 4, 4, 0, 7, -4], [0, -1, -3, 1, 0, -3], [-5, -2, 1, 2, 0, 1], [10, -2, -2, 2, 0, -2], [0, -1, -1, 0, -14, 1], [6, 3, 3, -3, 0, 3], [7, 2, -1, -2, 0, -1], [7, 0, 3, 3, 7, -3], [-2, -3, -6, -6, -2, 3], [0, 3, -1, 4, 10, -3], [0, 1, -2, 3, -8, -1], [0, -7, -5, -2, 3, 7], [0, 5, 0, -5, 0, 0], [0, -5, -4, -1, 12, 5], [13, -3, 0, 0, 13, -3], [-7, 1, 0, -1, 0, 0], [0, 0, 0, 0, -7, 0], [0, 3, 0, 3, 11, -3], [-14, 0, 3, 3, -14, -3], [6, -3, 3, 3, 0, 3], [-8, 1, 3, -1, 0, 3], [-11, -2, 0, 0, -11, -2], [0, 2, 1, 1, 9, -2], [-7, 6, 0, -6, 0, 0], [3, 0, -3, 0, 0, -3], [3, -4, -2, 4, 0, -2], [12, 1, -1, -1, 0, -1], [0, -1, 0, -1, -20, 1], [-9, -1, 0, 0, -9, -1], [14, 2, 1, -2, 0, 1], [-2, -1, -4, -4, -2, 3], [-1, 3, 1, 1, -1, 2], [0, -6, -3, -3, 6, 6], [-2, 2, -1, -2, 0, -1], [-4, 0, 2, 2, -4, -2], [0, -2, -6, 4, 1, 2], [0, 1, 5, -4, 6, -1], [0, 5, 1, 4, 5, -5], [0, -5, -2, -3, 9, 5], [-12, -6, -6, -6, -12, 0], [-13, 4, 3, 3, -13, 1], [-9, 5, 4, -5, 0, 4], [-16, -1, -3, -3, -16, 2], [2, 0, -6, 0, 0, -6], [10, 4, 0, -4, 0, 0], [1, 1, 0, -1, 0, 0], [-10, 5, -1, -5, 0, -1], [-8, -1, 3, 1, 0, 3], [-9, 6, -1, -1, -9, 7], [-3, 5, -5, -5, 0, -5], [7, -7, -2, -2, 7, -5], [0, 4, 8, -4, 15, -4], [-3, 3, -4, -4, -3, 7], [0, 5, 4, 1, 9, -5], [14, -1, 7, 1, 0, 7], [0, 6, 2, 4, 15, -6], [0, 5, 2, 3, 20, -5], [-20, 2, -3, -2, 0, -3], [0, 2, -1, 3, -23, -2], [-7, -1, -6, -6, -7, 5], [0, -4, -4, 0, -8, 4], [2, 1, -5, -5, 2, 6], [2, -1, -4, 1, 0, -4], [0, 5, 4, 1, 5, -5], [0, -1, 4, -5, 18, 1], [-5, 8, -1, -8, 0, -1], [0, 2, 1, -2, 0, 1], [-6, 2, 1, -2, 0, 1], [0, 1, 2, -1, -6, -1], [13, 3, 0, 0, 13, 3], [3, 3, 5, 5, 3, -2], [6, -1, -6, -6, 6, 5], [0, 8, 9, -1, -2, -8], [1, 4, 9, 9, 1, -5], [0, 4, -1, 5, -18, -4], [3, -3, -13, -13, 3, 10], [0, -3, 1, -4, 13, 3], [22, 4, 6, 6, 22, -2], [-24, -2, -7, -7, -24, 5], [9, 8, 10, 10, 9, -2], [-20, 4, -1, -1, -20, 5], [-15, 2, 3, -2, 0, 3], [0, 12, 5, 7, 18, -12], [0, -4, -10, 6, -16, 4], [25, 5, 2, -5, 0, 2], [-1, 3, 0, -3, 0, 0], [14, -5, 0, 5, 0, 0], [0, -3, -3, 0, -28, 3], [0, 1, 5, -4, 27, -1], [-18, 4, 5, -4, 0, 5], [-14, 7, -6, -6, -14, 13], [-1, 10, 3, 3, -1, 7], [-8, 2, 4, 4, -8, -2], [0, -10, -2, -8, -9, 10], [0, 3, 6, -3, -6, -3], [0, -1, -5, 4, 8, 1], [-2, -7, 6, 7, 0, 6], [0, -3, 0, -3, -16, 3], [-12, -10, -14, -14, -12, 4], [-25, -2, 3, 3, -25, -5], [-15, -7, 3, 7, 0, 3], [0, -8, -5, -3, 12, 8], [0, 0, -3, 3, 27, 0], [6, 2, 0, -2, 0, 0], [9, -2, 4, 2, 0, 4], [8, 9, 1, -9, 0, 1], [0, -4, -1, -3, -20, 4], [0, -6, -7, 1, 13, 6], [27, -4, 6, 6, 27, -10], [0, 8, -2, 10, 6, -8], [0, 1, 7, -6, -5, -1], [-8, 5, -4, -5, 0, -4], [0, 9, 10, -1, -16, -9], [17, 4, 3, 3, 17, 1], [3, 11, -2, -11, 0, -2], [0, 11, 5, 6, 19, -11], [1, -2, -9, -9, 1, 7], [1, -3, 0, 0, 1, -3], [-12, -2, -1, 2, 0, -1], [-1, 3, 3, -3, 0, 3], [14, 0, -1, -1, 14, 1], [-2, 8, 8, -8, 0, 8], [-24, 1, 2, -1, 0, 2], [5, -7, -5, 7, 0, -5], [0, 4, -4, -4, 0, -4], [18, -10, -6, 10, 0, -6], [-12, 1, 8, 8, -12, -7], [41, -2, 0, 0, 41, -2], [0, -1, -2, 1, 45, 1], [17, 5, -2, -5, 0, -2], [-22, 1, 1, 1, -22, 0], [0, -2, 1, -3, 1, 2], [17, -9, -2, 9, 0, -2], [0, -3, 3, -6, 11, 3], [-18, 1, -4, -4, -18, 5], [-15, -11, -13, -13, -15, 2], [30, 0, 3, 0, 0, 3], [0, -9, -9, 0, -19, 9], [11, -6, 9, 6, 0, 9], [-7, -9, -3, 9, 0, -3], [6, -5, -1, 5, 0, -1], [-20, 4, 4, -4, 0, 4], [0, 9, -2, 11, 1, -9], [18, -11, 2, 11, 0, 2], [33, 5, 0, -5, 0, 0], [6, 3, 11, 11, 6, -8], [0, -8, 1, -9, -23, 8], [18, 3, 8, 8, 18, -5], [0, -3, -5, 2, -31, 3], [-1, 0, 3, 0, 0, 3], [0, -4, 4, -8, -16, 4], [0, -14, -5, -9, 6, 14], [22, 3, 7, 7, 22, -4], [0, 9, 7, 2, 34, -9], [-13, 7, 1, 1, -13, 6], [2, 1, -2, -2, 2, 3], [-21, 13, 2, -13, 0, 2], [0, -15, -10, -5, -21, 15], [20, 4, -4, -4, 0, -4], [30, -10, -3, 10, 0, -3], [0, 8, 1, 7, 18, -8], [-7, 0, 4, 4, -7, -4], [-21, 0, 0, 0, -21, 0], [20, -2, 9, 9, 20, -11], [0, 8, 2, 6, -1, -8], [0, 2, -8, 10, 6, -2], [-5, 3, -5, -5, -5, 8], [7, 5, -5, -5, 7, 10], [0, 3, 10, -7, -11, -3], [21, 0, -3, -3, 21, 3], [9, 4, 11, 11, 9, -7], [51, 1, 5, 5, 51, -4], [-15, 3, 11, 11, -15, -8], [-1, -6, 3, 6, 0, 3], [-24, 1, -5, -1, 0, -5], [-9, -8, 5, 8, 0, 5], [0, 6, 12, -6, 12, -6], [22, -3, 9, 9, 22, -12], [15, 0, -3, 0, 0, -3], [9, 1, -7, -7, 9, 8], [-8, 3, 3, 3, -8, 0], [-11, -4, 1, 1, -11, -5], [0, -2, -5, 3, 13, 2], [12, -4, -6, -6, 12, 2], [0, -12, -4, -8, 4, 12], [30, -1, 4, 4, 30, -5], [0, 9, 3, 6, 47, -9], [-4, -8, 3, 3, -4, -11], [23, -7, 8, 7, 0, 8], [4, 9, 13, 13, 4, -4], [0, -1, 5, -6, 2, 1], [-14, 15, -2, -15, 0, -2], [0, 9, 16, -7, 7, -9], [-27, -7, 4, 7, 0, 4], [14, -5, -12, -12, 14, 7], [0, -7, -11, 4, -15, 7], [-35, -3, -6, -6, -35, 3], [0, 9, -5, 14, 8, -9], [0, -2, -5, 3, -8, 2], [0, 10, 4, 6, -9, -10], [0, 13, 11, 2, -3, -13], [-12, 6, -10, -10, -12, 16], [0, -9, 1, -10, -41, 9], [0, 17, 5, 12, 25, -17], [-36, 10, -1, -1, -36, 11], [-17, 2, -5, -5, -17, 7], [3, 17, -2, -17, 0, -2], [1, -11, -3, 11, 0, -3], [-9, -8, -4, -4, -9, -4], [0, 9, 0, 9, -3, -9], [22, -2, -2, 2, 0, -2], [32, -15, -5, 15, 0, -5], [13, 3, -9, -9, 13, 12], [37, -6, 0, 0, 37, -6], [-15, -1, 12, 12, -15, -13], [0, 5, 8, -3, -34, -5], [0, 2, -5, 7, 36, -2], [18, -4, 9, 4, 0, 9], [-8, 5, -1, -1, -8, 6], [4, -6, -12, -12, 4, 6], [30, 8, 1, -8, 0, 1], [0, 15, 16, -1, -8, -15], [-3, 2, 19, 19, -3, -17], [-21, 0, -12, -12, -21, 12], [5, -8, -15, -15, 5, 7], [-7, 7, -9, -7, 0, -9], [25, 2, 14, 14, 25, -12], [0, -9, -9, 0, -37, 9], [3, -1, -9, 1, 0, -9], [-8, -3, -4, 3, 0, -4], [16, -6, -12, -12, 16, 6], [-14, 13, -9, -13, 0, -9], [-22, -11, -20, -20, -22, 9], [-28, 7, 6, 6, -28, 1], [-5, 4, -6, -4, 0, -6], [0, 0, 0, 0, 0, 0], [0, -8, 0, -8, 19, 8], [-48, -3, -3, 3, 0, -3], [-4, -9, -13, -13, -4, 4], [0, 0, 10, -10, 7, 0], [21, 0, -15, -15, 21, 15], [0, 17, 0, -17, 0, 0], [0, 3, 6, -3, 11, -3], [-12, 2, -6, -2, 0, -6], [38, -3, 3, 3, 0, 3], [0, -12, -8, -4, -8, 12], [-33, 8, 3, 3, -33, 5], [-25, 4, -9, -4, 0, -9], [3, -16, -3, 16, 0, -3], [-7, 9, 4, 4, -7, 5], [0, -1, 1, -2, -3, 1], [-42, 2, 1, 1, -42, 1], [-27, 3, -13, -13, -27, 16], [12, 12, 12, 12, 12, 0], [16, 15, 15, 15, 16, 0], [15, -14, 4, 14, 0, 4], [-11, -2, -9, -9, -11, 7], [0, -2, 1, -3, -18, 2], [0, -8, -3, -5, -31, 8], [8, 0, -3, 0, 0, -3], [33, -1, -2, 1, 0, -2], [0, 1, -7, -1, 0, -7], [0, -8, -13, 5, -36, 8], [54, -3, 6, 3, 0, 6], [0, -4, -9, 5, -23, 4], [13, 6, 3, 3, 13, 3], [6, 5, 10, 10, 6, -5], [0, 1, -4, 5, -25, -1], [-27, 10, 2, 2, -27, 8], [-15, -7, -11, -11, -15, 4], [-6, -3, -1, -1, -6, -2], [28, -7, -7, 7, 0, -7], [6, 3, 9, 9, 6, -6], [0, -9, -8, -1, -5, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2268_l_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2268_2_l_k();".
// 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_l_k(:prec:=6)
    chi := MakeCharacter_2268_l();
    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_l_k();".
// 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_l_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2268_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-9, -12, -1, 1]>,<13,R![441, -231, 163, -20, 15, -2, 1]>,<19,R![2401, 1764, 1443, -10, 45, 3, 1]>],Snew);
    return Vf;
end function;