
// Make newform 196.6.e.d in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_196_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_196_e_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_196_6_e_d();".
// 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_196_6_e_d();".
// 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_196_e();"
function MakeCharacter_196_e()
    N := 196;
    order := 3;
    char_gens := [99, 101];
    v := [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_196_e_Hecke();"
function MakeCharacter_196_e_Hecke(Kf)
    N := 196;
    order := 3;
    char_gens := [99, 101];
    char_values := [[1, 0], [0, -1]];
    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 := 6;
    raw_aps := [[0, 0], [-12, 12], [0, 54], [0, 0], [-540, 540], [418, 0], [594, -594], [0, 836], [0, 4104], [-594, 0], [4256, -4256], [0, 298], [-17226, 0], [-12100, 0], [0, -1296], [-19494, 19494], [-7668, 7668], [0, -34738], [-21812, 21812], [-46872, 0], [67562, -67562], [0, 76912], [-67716, 0], [0, 29754], [122398, 0], [11286, -11286], [0, -27256], [0, -122364], [-99902, 99902], [-29646, 0], [336512, 0], [0, 100980], [317142, -317142], [148324, 0], [0, -196614], [-74360, 74360], [120878, -120878], [0, 111340], [491832, 0], [0, 707454], [-493668, 493668], [559450, 0], [0, 724032], [-7106, 7106], [-530442, 0], [56168, -56168], [-339196, 0], [-779360, 0], [-744876, 744876], [0, -272746], [0, 153846], [1154736, 0], [657074, -657074], [-1341900, 0], [0, 132354], [-943272, 943272], [967518, -967518], [0, -518320], [-2222726, 2222726], [-196614, 0], [-1552276, 1552276], [1072170, 0], [-1585892, 0], [-730728, 730728], [0, 584858], [0, 2482866], [0, -377948], [639122, 0], [2904660, -2904660], [3991570, 0], [1429218, -1429218], [0, -1161864], [-1089232, 1089232], [0, -3505766], [4043852, 0], [0, 5187456], [950346, -950346], [0, -520738], [0, -764370], [2640506, -2640506], [4980204, 0], [-237994, 0], [3882384, -3882384], [66958, 0], [0, -6501352], [0, 4607604], [3776706, 0], [0, 3180694], [-6685470, 0], [-4351216, 0], [0, 7079940], [3221856, -3221856], [-2297096, 2297096], [2821500, 0], [0, 4136284], [-8332632, 0], [0, 4341006], [-6741846, 6741846], [0, -7721956], [0, 682066], [2157716, 0], [2675970, -2675970], [-3553308, 3553308], [0, 12922470], [6083572, -6083572], [-15824062, 15824062], [-4602204, 0], [0, 8611218], [7982280, -7982280], [-10174010, 0], [0, -9958432], [-4195862, 4195862], [9125514, 0], [6457340, -6457340], [-14051368, 0], [-8471682, 8471682], [-488564, 0], [2481192, -2481192], [0, 5291298], [4724676, 0], [-6174202, 6174202], [-9409246, 0], [0, 15008598], [12970692, -12970692], [0, 22655588], [19016910, 0], [0, -15131126], [0, -15032304], [7412296, 0], [0, -2779282], [12104620, -12104620], [4469256, 0], [0, -28846256], [9608678, 0], [0, 4545882], [21592318, 0], [-14840010, 14840010], [-24878524, 24878524], [-31607982, 0], [3100086, -3100086], [-1871804, 0], [0, 20018394], [-15311768, 15311768], [9593100, 0], [25220942, -25220942], [17762328, 0], [486970, 0], [-1926342, 1926342], [0, 22353836], [0, -18583776], [0, 29104834], [3146958, 0], [15999524, 0], [0, -34587432], [-17439644, 17439644], [-2595888, 0], [0, 17641096], [0, 39678498], [-39341258, 0], [46250622, -46250622], [0, 37902492], [-26646246, 0], [40979048, 0], [0, -27203364], [-25355538, 25355538], [11913912, -11913912], [-29193056, 29193056], [-17300074, 17300074]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_196_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_196_6_e_d();".
// 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_196_6_e_d(:prec:=2)
    chi := MakeCharacter_196_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_196_6_e_d();".
// 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_196_6_e_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_196_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![144, 12, 1]>],Snew);
    return Vf;
end function;