
// Make newform 2268.2.x.h in Magma, downloaded from the LMFDB on 26 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2268_x();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2268_x_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_x_h();".
// 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_x_h();".
// 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 := [1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_x();"
function MakeCharacter_2268_x()
    N := 2268;
    order := 6;
    char_gens := [1135, 1541, 325];
    v := [6, 5, 3];
    // 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_x_Hecke();"
function MakeCharacter_2268_x_Hecke(Kf)
    N := 2268;
    order := 6;
    char_gens := [1135, 1541, 325];
    char_values := [[1, 0], [0, 1], [-1, 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, 3], [-1, 3], [-3, -3], [-4, 2], [6, 0], [1, -2], [-6, 3], [6, 6], [-6, 3], [1, 0], [0, -3], [-10, 10], [-6, 6], [0, 0], [0, 6], [-8, -8], [0, -2], [3, -6], [-2, 4], [-14, 14], [6, -6], [-9, 0], [-4, -4], [18, -18], [10, -5], [6, -12], [-11, 0], [0, 0], [2, 0], [0, -18], [6, 6], [-20, 10], [0, 0], [-4, 4], [12, -6], [-10, 0], [0, 12], [-9, 9], [-6, 12], [-4, 8], [3, 3], [0, 2], [-12, 24], [-1, 2], [0, 20], [15, 15], [0, 0], [24, -12], [-12, 24], [12, -6], [-2, -2], [24, 0], [0, 9], [3, 3], [21, 0], [2, -4], [19, -19], [-6, -6], [12, -6], [0, 6], [9, -18], [0, -6], [2, 2], [18, 18], [-4, 4], [0, 29], [6, -3], [16, 16], [-15, 15], [-6, 12], [-1, -1], [0, -29], [10, 0], [0, -12], [-12, -12], [-12, 24], [12, -6], [-20, 10], [0, -24], [-5, 5], [-21, 42], [-6, 12], [18, 18], [9, 9], [12, -24], [17, -17], [15, -15], [0, 16], [18, 0], [24, -24], [-26, 0], [-18, 9], [0, 14], [-6, 0], [0, 42], [-33, 0], [-7, 14], [17, 0], [40, -40], [-12, 24], [0, -18], [-24, -24], [0, -20], [-20, 40], [-36, 36], [39, 0], [-30, 15], [-6, -6], [20, -10], [-19, 0], [-36, 18], [-19, -19], [26, 0], [-12, -12], [54, -27], [18, 0], [-12, 6], [21, 21], [-28, 14], [-22, 22], [27, -27], [9, -18], [-6, -6], [-18, 36], [23, -23], [-6, 0], [-18, -18], [-12, 6], [-16, 0], [42, -21], [0, 2], [-26, 0], [0, 6], [4, -2], [-39, 0], [44, -22], [0, -45], [-18, 36], [-3, 6], [18, 18], [0, -20], [-27, 54], [26, -52], [-42, 42], [-28, -28], [21, -21], [-34, 17], [-6, 12], [0, -2], [27, 0], [28, 0], [0, -42], [32, -32], [6, 6], [16, 0], [30, -30], [-32, 64], [0, -21], [-15, -15], [18, -36], [0, 16], [-6, 0], [48, -24], [42, -42], [10, 0], [0, 0], [35, 0], [0, -33], [-6, -6], [36, -18], [36, -18], [-59, 59], [-22, 11], [0, 21], [-44, 44], [-12, 24], [-1, 2], [24, 24], [-32, 32], [60, -60], [0, -5], [-6, 0], [-6, 3], [15, -15], [-26, 0], [-2, -2], [0, -47], [24, 0], [26, -52], [-30, -30], [28, 0], [-6, 6], [21, -42], [0, 21], [0, 65], [-14, 14], [9, -9], [12, 0], [-36, 18], [-5, -5], [76, -38], [-28, -28], [6, 0], [30, 0], [10, -20], [6, -3], [-6, -6], [-20, 10], [-37, 0], [0, -9], [14, -14], [6, -6], [0, -12], [-8, -8], [0, -22], [-27, 27], [3, -6], [-24, 12], [0, 38], [-36, 18], [-36, 18], [-2, 0], [0, -12], [-13, 13], [-15, 15], [-33, 66], [7, 7], [21, 21], [0, -25], [-3, 6], [0, 14], [24, -12], [-7, -7], [54, -54], [-32, 16], [30, -60], [12, -6], [-6, 0], [-27, -27], [-7, 14], [10, -5], [1, 0], [0, 9], [-6, 6], [-1, 2], [0, -6], [0, -14], [-21, 42], [0, -22], [-18, 0], [60, -30], [-8, -8], [39, -39], [21, -42], [-55, 0], [21, 21], [-18, 0], [6, -12], [58, -58], [45, 45], [44, -22], [14, -28], [0, -63], [21, 21], [30, -60], [-54, 54], [0, 4], [-36, 18], [52, -26], [28, 0], [6, 6], [0, 2], [84, -42], [-29, 58], [-42, 21], [-46, 46], [-2, 0], [30, -30], [0, 66], [0, 4], [-48, 48], [-10, -10], [46, -23], [39, -78], [-17, 0], [-60, 30], [-17, -17], [-57, 0], [0, 0], [30, 0], [-12, 6], [0, -60], [-17, 17], [0, 3], [23, 23], [-30, 30], [42, 0], [41, 41], [-40, 20], [-30, 60], [56, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2268_x_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2268_2_x_h();".
// 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_x_h(:prec:=2)
    chi := MakeCharacter_2268_x();
    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_x_h();".
// 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_x_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2268_x();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![9, -3, 1]>,<11,R![27, 9, 1]>,<13,R![12, 6, 1]>],Snew);
    return Vf;
end function;