
// Make newform 196.4.e.c in Magma, downloaded from the LMFDB on 25 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_4_e_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_196_4_e_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 := [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 := 4;
    raw_aps := [[0, 0], [-4, 4], [0, -6], [0, 0], [12, -12], [-82, 0], [30, -30], [0, -68], [0, -216], [246, 0], [112, -112], [0, -110], [-246, 0], [-172, 0], [0, -192], [-558, 558], [-540, 540], [0, -110], [-140, 140], [-840, 0], [550, -550], [0, 208], [516, 0], [0, 1398], [1586, 0], [1242, -1242], [0, -680], [0, -996], [-1382, 1382], [-750, 0], [176, 0], [0, 1548], [-378, 378], [-2500, 0], [0, -846], [2536, -2536], [1186, -1186], [0, -2108], [-1944, 0], [0, 1362], [-1596, 1596], [-1690, 0], [0, -3552], [2686, -2686], [-1410, 0], [2968, -2968], [-1348, 0], [3872, 0], [-5364, 5364], [0, 874], [0, -378], [1920, 0], [-4322, 4322], [5292, 0], [0, 5118], [-3768, 3768], [-3918, 3918], [0, -4880], [3538, -3538], [-5430, 0], [6436, -6436], [1350, 0], [3332, 0], [4728, -4728], [0, -5114], [0, -7206], [0, -6260], [-5326, 0], [-36, 36], [3134, 0], [-1218, 1218], [0, 10008], [1072, -1072], [0, 274], [7652, 0], [0, -2160], [1074, -1074], [0, -6926], [0, -1938], [9574, -9574], [-5052, 0], [3422, 0], [-2208, 2208], [-6814, 0], [0, -12584], [0, -6996], [9474, 0], [0, -5786], [3438, 0], [9392, 0], [0, 4956], [20592, -20592], [13432, -13432], [-14172, 0], [0, 5956], [16968, 0], [0, -5214], [1398, -1398], [0, 18580], [0, 18970], [-16036, 0], [-8310, 8310], [-7092, 7092], [0, 7158], [-6500, 6500], [-21794, 21794], [9756, 0], [0, -5586], [24, -24], [4298, 0], [0, -8480], [1906, -1906], [7482, 0], [7348, -7348], [4520, 0], [19806, -19806], [-5020, 0], [-28392, 28392], [0, 17562], [4716, 0], [22762, -22762], [4802, 0], [0, -21558], [-3780, 3780], [0, 5500], [10230, 0], [0, -10190], [0, -9408], [-33064, 0], [0, 6322], [20740, -20740], [32040, 0], [0, 12832], [-19906, 0], [0, -10842], [28274, 0], [32346, -32346], [-30116, 30116], [-6594, 0], [43014, -43014], [-14164, 0], [0, -34830], [-31016, 31016], [9876, 0], [3154, -3154], [36936, 0], [9638, 0], [-10266, 10266], [0, 4084], [0, 192], [0, -19910], [14802, 0], [-32548, 0], [0, -1464], [49564, -49564], [8448, 0], [0, -14600], [0, 21102], [-20806, 0], [-24510, 24510], [0, 44148], [27114, 0], [-10264, 0], [0, -51468], [23790, -23790], [-26424, 26424], [-39488, 39488], [-30854, 30854]];
    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_4_e_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_196_4_e_c(:prec:=2)
    chi := MakeCharacter_196_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_4_e_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_196_4_e_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_196_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![16, 4, 1]>,<5,R![36, 6, 1]>],Snew);
    return Vf;
end function;