
// Make newform 1008.2.cs.k in Magma, downloaded from the LMFDB on 03 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1008_cs();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1008_cs_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_cs_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_1008_2_cs_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 := [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_1008_cs();"
function MakeCharacter_1008_cs()
    N := 1008;
    order := 6;
    char_gens := [127, 757, 785, 577];
    v := [3, 6, 6, 5];
    // 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_cs_Hecke();"
function MakeCharacter_1008_cs_Hecke( : Kf := false)
    N := 1008;
    order := 6;
    char_gens := [127, 757, 785, 577];
    char_values := [[-1, 0], [1, 0], [1, 0], [0, 1]];
    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], [2, -1], [-3, 1], [3, 3], [4, -8], [-2, -2], [0, 2], [8, -4], [9, 0], [1, -1], [0, 2], [2, -4], [2, -4], [0, 0], [9, -9], [3, -3], [-8, 4], [0, 0], [-4, 8], [4, 4], [2, -1], [-15, 0], [12, -6], [5, -10], [8, 8], [0, 4], [-10, 5], [4, -4], [6, 0], [9, -18], [0, 3], [-12, 12], [-22, 0], [0, -6], [1, 1], [4, 4], [-24, 12], [-12, 0], [-16, 8], [10, 10], [-2, 4], [0, 0], [-19, 19], [-6, 0], [-16, 16], [-8, 16], [1, 0], [-9, 9], [-4, 2], [0, 18], [8, -16], [-3, -3], [-21, 0], [4, -2], [10, 10], [-1, -1], [0, -17], [-28, 28], [0, 0], [14, -14], [3, -6], [16, 0], [6, -6], [-22, 11], [0, -27], [0, 0], [23, 0], [18, 18], [-12, 24], [4, 4], [-32, 16], [23, -23], [0, -26], [-10, 20], [0, -6], [-18, 18], [24, -12], [0, 36], [-5, -5], [12, 0], [-8, 0], [-14, -14], [-4, 8], [0, -25], [26, -13], [-24, 0], [0, -17], [12, -24], [6, -12], [0, 36], [42, -42], [9, 9], [11, -22], [-20, 10], [6, 0], [22, -11], [18, 18], [0, 40], [0, -10], [-14, 28], [3, -3], [-9, 9], [0, 18], [-16, -16], [15, 15], [21, 0], [24, -12], [16, 16], [-9, 18], [0, 23], [16, -16], [-24, 0], [16, -16], [-15, 30], [24, -24], [-32, 0], [12, -12], [0, -21], [10, -20], [22, 22], [-11, 0], [-30, 15], [-5, -5], [0, -14], [27, 0], [0, -14], [0, -6], [17, 0], [-36, 18], [18, 18], [0, 0], [-14, 7], [-16, 0], [-16, 8], [3, -6], [16, 16], [-32, 32], [-5, 10], [-24, 24], [8, 0], [0, 45], [-2, -2], [21, -42], [-6, -6], [-18, 0], [4, -8], [-28, -28], [0, -10], [36, -18], [0, -8], [-32, 64], [-18, 36], [0, -12], [-32, -32], [12, -24], [-4, 2], [68, -34], [29, -58], [9, 9], [28, -14], [-6, 0], [27, -54], [0, -21], [-18, 18], [-54, 54], [-27, -27], [16, 16], [-47, 0], [-50, 25], [-7, -7], [6, -12], [-12, 6], [23, -23], [-31, 31], [-8, 16], [-18, 36], [27, -27], [-64, 0], [-32, 16], [-58, 29], [-15, 0], [4, 0], [0, 0], [-30, -30], [-23, -23], [-38, 38], [28, -28], [0, -49], [-30, 30], [6, -3], [-15, 30], [12, -6], [-24, 12], [-17, -17], [6, 6], [-46, 46], [0, 44], [6, -12], [0, -36], [21, -21], [64, 0], [-32, 16], [28, 28], [-48, 0], [-7, -7], [0, -52], [46, -23], [18, 0], [-10, 10], [0, -26], [-35, 70], [-23, 46], [0, 39], [-54, 54], [72, -36], [1, 1], [-16, -16], [8, -4], [-6, 0], [0, 40], [32, 0], [0, 36], [34, -17], [27, 0], [4, 0], [56, -28], [12, 12], [0, 73], [-76, 38], [-32, 32], [-34, 34], [13, -26], [30, -30], [22, 0], [0, 0], [6, -3], [0, -39], [-34, 68], [72, 0], [6, 6], [0, 34], [-31, 31], [0, 8], [20, -40], [0, -6], [29, 0], [-72, 72], [-8, -8], [30, -60], [32, 0], [-32, 16], [32, 32], [-12, 24], [-3, -3], [66, -33], [14, -14], [-34, 34], [-29, 58], [-34, 17], [-18, -18], [-1, 2], [-24, -24], [-2, 4], [32, 32], [0, -56], [39, 0], [-4, 8], [34, -68], [-9, 9], [-56, 28], [18, 18], [15, 15], [90, -45], [29, -58], [0, -56], [-30, 15], [20, -20], [0, -10], [0, 12], [-12, 12], [9, 9], [72, 0], [-42, 84], [0, 52], [44, -22], [-65, 65], [-45, 0], [61, -61], [28, -56], [-42, 42], [-3, 3], [0, 36], [-84, 0], [68, 0], [-21, -21], [0, -64], [-24, 48], [0, 36], [26, 0], [-8, 4], [0, -3], [-1, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1008_cs_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1008_2_cs_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_1008_2_cs_k(:prec:=2)
    chi := MakeCharacter_1008_cs();
    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_cs_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_1008_2_cs_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1008_cs();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3, -3, 1]>,<11,R![27, -9, 1]>,<13,R![48, 0, 1]>,<17,R![12, 6, 1]>,<19,R![4, -2, 1]>],Snew);
    return Vf;
end function;