
// Make newform 168.2.q.c in Magma, downloaded from the LMFDB on 07 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_168_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_168_q_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_168_2_q_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_168_2_q_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 := [25, -5, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-5, -4, 4, 1], [5, 4, 0, -1]];
    Rf_basisdens := [1, 1, 20, 4];
    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_168_q();"
function MakeCharacter_168_q()
    N := 168;
    order := 3;
    char_gens := [127, 85, 113, 73];
    v := [3, 3, 3, 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_168_q_Hecke();"
function MakeCharacter_168_q_Hecke( : Kf := false)
    N := 168;
    order := 3;
    char_gens := [127, 85, 113, 73];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 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], [1, 0, -1, 0], [-1, 2, 1, -1], [-2, 1, 0, -1], [1, -1, 0, 2], [3, -1, -1, -1], [4, 0, -4, 0], [1, -2, 2, 1], [0, 0, -4, 0], [2, -1, -1, -1], [1, 0, -1, 0], [1, -2, -2, 1], [2, 2, 2, 2], [-3, -1, -1, -1], [0, 0, 6, 0], [-5, -1, 6, 2], [-3, 1, 2, -2], [0, 0, -10, 0], [-4, 1, 3, -2], [2, 0, 0, 0], [0, 1, -1, -2], [-2, 4, -3, -2], [4, -1, -1, -1], [2, -4, 2, 2], [12, 1, 1, 1], [6, 0, -6, 0], [-3, 6, 2, -3], [3, -6, 3, 3], [-4, -3, 7, 6], [2, 2, 2, 2], [-13, -2, -2, -2], [-1, 2, -3, -1], [-10, 2, 8, -4], [5, -1, -1, -1], [2, -4, 6, 2], [-1, -5, 6, 10], [-8, 2, 6, -4], [0, 0, 12, 0], [-6, 0, 0, 0], [2, -4, 14, 2], [8, 2, -10, -4], [-9, 3, 3, 3], [-2, 4, 4, -2], [-9, 2, 7, -4], [-8, -2, -2, -2], [-12, 4, 8, -8], [-8, 6, 6, 6], [2, 1, 1, 1], [7, 1, -8, -2], [-1, 2, -14, -1], [2, -4, -8, 2], [22, 2, 2, 2], [3, -3, 0, 6], [-2, 5, 5, 5], [-4, 8, 6, -4], [18, -2, -16, 4], [-25, 1, 24, -2], [-1, 2, 3, -1], [14, 3, -17, -6], [-20, 2, 2, 2], [-4, 7, -3, -14], [-18, 3, 3, 3], [-5, -5, -5, -5], [-4, 2, 2, -4], [2, -4, 11, 2], [3, -6, 15, 3], [3, -6, -14, 3], [-11, 4, 4, 4], [4, -8, 4, 16], [-6, 0, 0, 0], [12, 2, -14, -4], [6, -12, -6, 6], [-5, -4, 9, 8], [1, -2, 2, 1], [23, 3, 3, 3], [-2, 4, -2, -2], [2, 0, -2, 0], [5, -10, -18, 5], [0, 0, -24, 0], [-19, 2, 17, -4], [22, -2, -2, -2], [-1, -3, -3, -3], [-8, 6, 2, -12], [13, 3, 3, 3], [-1, 2, -13, -1], [5, -10, -15, 5], [14, -2, -2, -2], [-6, 12, -11, -6], [16, 6, 6, 6], [23, 3, 3, 3], [-2, 4, -32, -2], [14, -2, -12, 4], [-1, 0, 1, 0], [14, -7, -7, -7], [-1, 2, -28, -1], [-18, 6, 6, 6], [1, -2, 7, 1], [6, 2, -8, -4], [-1, 2, 22, -1], [1, -2, 4, 1], [-28, 0, 0, 0], [17, 3, -20, -6], [-11, -1, 12, 2], [2, -4, -20, 2], [-4, -1, 5, 2], [25, 0, -25, 0], [6, -1, -1, -1], [2, -4, -8, 2], [-14, -2, 16, 4], [-9, -4, -4, -4], [0, 0, 7, 0], [0, 2, -2, -4], [-6, 0, 0, 0], [-16, -3, 19, 6], [10, -3, -3, -3], [-22, 6, 16, -12], [-11, -5, -5, -5], [34, 0, -34, 0], [7, -14, -23, 7], [-16, 8, 8, 8], [-20, 3, 17, -6], [35, -2, -2, -2], [1, -2, 9, 1], [-25, 3, 22, -6], [3, -6, -2, 3], [10, 7, 7, 7], [8, -16, -6, 8], [-8, 16, 6, -8], [25, -2, -2, -2], [-7, 14, 24, -7], [20, -5, -15, 10], [-18, 8, 8, 8], [-2, 4, 19, -2], [-22, 0, 0, 0], [4, -8, 8, 4], [-13, 6, 6, 6], [14, -4, -10, 8], [6, -2, -4, 4], [-2, 9, 9, 9], [4, 6, -10, -12], [-36, 2, 2, 2], [1, -2, -17, 1], [0, 8, -8, -16], [-8, 11, 11, 11], [-26, -1, 27, 2], [-4, 8, 8, 8], [-3, -5, -5, -5], [22, -2, -20, 4], [6, -12, 30, 6], [-4, 8, -2, -4], [-10, 20, -4, -10], [16, 2, 2, 2], [-9, -5, -5, -5], [-6, 12, -8, -6], [34, -1, -33, 2], [8, -14, -14, -14], [3, -6, -10, 3], [8, -16, 6, 8], [7, 4, 4, 4], [-39, -3, 42, 6], [-2, 4, 28, -2], [36, -6, -6, -6], [-47, 2, 2, 2], [-13, 26, -1, -13], [-26, 4, 22, -8], [12, -12, 0, 24], [-3, 4, -1, -8], [32, 3, -35, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_168_q_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_168_2_q_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_168_2_q_c(:prec:=4)
    chi := MakeCharacter_168_q();
    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_168_2_q_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_168_2_q_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_168_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![196, 14, 15, -1, 1]>],Snew);
    return Vf;
end function;