
// Make newform 1008.2.cz.f in Magma, downloaded from the LMFDB on 17 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1008_cz();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1008_cz_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_1008_2_cz_f();".
// 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_1008_2_cz_f();".
// 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 := [4, 0, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 1, 0], [0, 4, 0, 1], [0, 2, 0, -1]];
    Rf_basisdens := [1, 2, 2, 2];
    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_1008_cz();"
function MakeCharacter_1008_cz()
    N := 1008;
    order := 6;
    char_gens := [127, 757, 785, 577];
    v := [3, 6, 4, 1];
    // 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_1008_cz_Hecke();"
function MakeCharacter_1008_cz_Hecke( : Kf := false)
    N := 1008;
    order := 6;
    char_gens := [127, 757, 785, 577];
    char_values := [[-1, 0, 0, 0], [1, 0, 0, 0], [-1, -1, 0, 0], [1, 1, 0, 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, -2, 0, 0], [2, 1, 0, 1], [1, 0, 1, 0], [-1, 1, 0, 0], [-1, 1, -1, 1], [2, 1, 0, 2], [1, 1, 0, 0], [-6, -3, 0, -1], [0, 3, 2, -1], [4, 0, 0, 0], [-1, -1, -1, -1], [-3, 3, -2, 2], [2, 1, 0, -2], [0, 0, -2, 4], [0, -3, -2, 1], [6, 0, 0, 0], [-4, -8, 0, 0], [0, 0, 0, 0], [-2, -4, 4, 0], [-10, -5, 0, 0], [-4, -8, -2, 0], [0, 3, -4, 2], [-1, 1, -6, 6], [-2, -1, 0, 2], [1, -1, 1, -1], [0, 1, 6, -3], [-1, 1, -4, 4], [0, -13, -2, 1], [3, 3, -2, -2], [-6, -12, -4, 0], [0, -9, 4, -2], [-9, -9, 0, 0], [-13, -13, 0, 0], [0, -15, 2, -1], [1, -1, 9, -9], [-4, -8, -2, 0], [-3, 3, 2, -2], [-3, -3, 3, 3], [0, 0, -4, 0], [10, 5, 0, 4], [2, 4, -4, 0], [-2, -4, 6, 0], [4, 0, 4, -8], [-6, 0, -2, 4], [0, -7, 6, -3], [7, -7, 2, -2], [0, -1, 2, -1], [-9, -9, 2, 2], [-26, -13, 0, -1], [21, 21, 2, 2], [-7, 7, 5, -5], [-3, 3, 2, -2], [-18, 0, -2, 4], [18, 9, 0, -4], [-1, 1, -5, 5], [-6, -3, 0, 7], [-1, -1, -3, -3], [-1, -1, -3, -3], [0, -15, 8, -4], [14, 0, -2, 4], [-3, 3, 9, -9], [-10, 0, -2, 4], [6, 0, 4, -8], [-2, -4, -2, 0], [18, 0, 0, 0], [12, 24, 4, 0], [-5, -5, 0, 0], [16, 32, 0, 0], [6, 3, 0, 1], [1, -1, 2, -2], [-5, 5, 1, -1], [19, 19, 3, 3], [0, 23, -6, 3], [-4, -8, -2, 0], [0, -15, 2, -1], [-33, -33, -1, -1], [-15, 15, -5, 5], [0, 9, -12, 6], [-10, -20, -2, 0], [-9, -9, -4, -4], [0, 1, -6, 3], [14, 7, 0, -11], [-2, -4, 4, 0], [-10, 0, -4, 8], [-8, -16, -2, 0], [-24, 0, 0, 0], [-22, 0, -4, 8], [-22, -11, 0, 3], [-9, 9, 1, -1], [15, 15, -2, -2], [-3, -3, 1, 1], [30, 15, 0, 5], [-3, 3, -4, 4], [22, 11, 0, -6], [12, 0, 0, 0], [-10, -5, 0, -7], [10, 5, 0, 0], [-1, -1, 0, 0], [23, 23, -5, -5], [-14, -7, 0, 6], [0, 3, 10, -5], [18, 0, -4, 8], [6, 0, 4, -8], [-4, -8, -16, 0], [18, 9, 0, 12], [0, -3, 8, -4], [1, -1, -6, 6], [-10, -20, -4, 0], [18, 9, 0, -4], [0, 5, -14, 7], [0, -1, -2, 1], [3, 3, 2, 2], [-19, -19, -4, -4], [-6, -12, 2, 0], [-3, -3, 8, 8], [1, 1, 6, 6], [0, 3, -6, 3], [0, -39, 2, -1], [6, 3, 0, -8], [8, 16, -12, 0], [0, 7, -8, 4], [4, 8, 6, 0], [30, 15, 0, 4], [10, 0, -8, 16], [18, 0, -4, 8], [20, 0, 2, -4], [45, 45, -1, -1], [0, 13, -14, 7], [-54, -27, 0, -1], [-42, -21, 0, -2], [-13, 13, 3, -3], [22, 11, 0, -7], [-10, 0, 8, -16], [22, 11, 0, -2], [-3, 3, 14, -14], [-18, -9, 0, -7], [-2, 0, -2, 4], [-3, 3, 5, -5], [0, -15, 8, -4], [4, 0, 8, -16], [24, 0, 6, -12], [-14, -28, 10, 0], [4, 8, -6, 0], [6, 3, 0, 11], [0, -15, 2, -1], [2, 1, 0, -3], [-3, 3, 4, -4], [0, 17, -8, 4], [9, -9, -9, 9], [0, 17, 14, -7], [10, 20, -2, 0], [0, 0, -2, 0], [0, 3, 22, -11], [-5, 5, 4, -4], [-30, -15, 0, -3], [-7, 7, 3, -3], [22, 44, -4, 0], [-2, -4, 20, 0], [4, 8, -12, 0], [-22, -44, 8, 0], [36, 0, -2, 4], [9, -9, 9, -9], [21, 21, 2, 2], [0, 0, -4, 8], [21, 21, -1, -1], [-42, -21, 0, -3], [-1, 1, -3, 3], [-14, 0, 0, 0], [10, 5, 0, -1], [5, -5, 8, -8], [-9, 9, 3, -3], [-54, -27, 0, -3], [29, 29, -2, -2], [20, 0, -8, 16], [-9, 9, 8, -8], [-18, -9, 0, 2], [0, -9, -6, 3], [22, 0, -8, 16], [-4, -8, 8, 0], [-22, -44, 2, 0], [0, -9, 16, -8], [35, 35, -3, -3], [-3, 3, 12, -12], [-10, -20, -6, 0], [19, -19, -1, 1], [0, -1, 2, -1], [-1, -1, 2, 2], [0, 25, -16, 8], [0, 39, 6, -3], [-9, 9, 2, -2], [18, 9, 0, 0], [9, -9, 10, -10], [30, 60, 0, 0], [-30, -15, 0, 10], [-34, -68, 0, 0], [10, 0, -4, 8], [10, 0, -10, 20], [-14, -7, 0, 0], [-3, -3, -5, -5], [-18, 0, -8, 16], [0, -19, -10, 5], [-22, -11, 0, 1], [5, -5, 24, -24], [-18, 0, -10, 20], [22, 11, 0, 13], [43, 43, -5, -5], [14, 7, 0, -2], [0, -27, 0, 0], [-10, 0, 8, -16], [31, 31, 4, 4], [15, -15, -7, 7], [10, 5, 0, -9], [-12, 0, -4, 8], [-6, 0, -8, 16], [12, 24, 12, 0], [-14, -28, -4, 0], [-1, 1, -20, 20], [-9, 9, 11, -11], [33, 33, 5, 5], [0, -31, 6, -3], [11, 11, -5, -5], [0, 15, 4, -2], [19, -19, -11, 11], [27, 27, -4, -4], [0, 61, -2, 1], [14, 28, -16, 0], [-46, -23, 0, 9], [-8, 0, 0, 0], [0, 0, 22, 0], [-16, 0, -10, 20], [0, -37, -8, 4], [25, -25, 5, -5], [-24, 0, -6, 12], [0, -43, 4, -2], [-33, -33, -5, -5], [-18, -9, 0, 22], [27, 27, -5, -5], [-25, 25, -4, 4], [0, 0, 8, -16], [-5, 5, -24, 24], [-19, -19, -10, -10], [-52, 0, -2, 4], [-7, -7, 9, 9], [-1, 1, 10, -10], [0, 0, 2, -4], [-20, 0, 10, -20], [-6, 0, -10, 20], [-8, -16, 14, 0], [-16, -32, 18, 0], [-11, -11, 11, 11], [23, -23, 8, -8], [-10, -20, 22, 0], [-30, -15, 0, 0], [-11, 11, -3, 3], [13, -13, -6, 6], [0, -17, -6, 3], [61, 61, -2, -2], [38, 76, 4, 0], [-17, 17, 10, -10], [-9, 9, 19, -19], [46, 23, 0, -10], [12, 24, -18, 0], [-4, -8, 16, 0], [-4, -8, 8, 0], [70, 0, 2, -4], [-12, 0, 10, -20], [-2, -1, 0, 4], [-17, 17, -8, 8], [-42, 0, 4, -8], [-46, -23, 0, -5], [-30, -15, 0, -2], [-9, 9, -16, 16], [-6, -3, 0, 27], [25, -25, -8, 8], [17, 17, 5, 5], [34, 17, 0, -6], [-43, -43, -5, -5], [29, 29, 10, 10], [-18, 0, 8, -16], [-36, 0, 6, -12], [-6, -12, -8, 0], [0, -9, 2, -1], [-6, -3, 0, 9], [0, -5, 24, -12], [-11, 11, 11, -11], [0, 17, -8, 4], [-9, -9, 7, 7], [-37, -37, -1, -1], [4, 8, 28, 0], [3, 3, -3, -3], [0, 33, 0, 0], [0, 9, 28, -14], [-27, -27, -2, -2], [0, 17, 18, -9], [20, 40, 0, 0], [-4, 0, -2, 4], [6, 3, 0, -9], [39, 39, 10, 10], [0, -23, -16, 8], [38, 19, 0, -8], [-27, -27, -3, -3], [-34, -17, 0, -13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1008_cz_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1008_2_cz_f();".
// 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_1008_2_cz_f(:prec:=4)
    chi := MakeCharacter_1008_cz();
    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_1008_2_cz_f();".
// 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_1008_2_cz_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1008_cz();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![9, 18, 9, -6, 1]>,<11,R![3, 3, 1]>],Snew);
    return Vf;
end function;