
// Make newform 336.4.bj.c in Magma, downloaded from the LMFDB on 02 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_336_bj();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_336_bj_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_336_4_bj_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_336_4_bj_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_336_bj();"
function MakeCharacter_336_bj()
    N := 336;
    order := 6;
    char_gens := [127, 85, 113, 241];
    v := [3, 6, 3, 4];
    // 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_336_bj_Hecke();"
function MakeCharacter_336_bj_Hecke( : Kf := false)
    N := 336;
    order := 6;
    char_gens := [127, 85, 113, 241];
    char_values := [[-1, 0], [1, 0], [-1, 0], [-1, 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 := 4;
    raw_aps := [[0, 0], [6, -3], [0, 0], [-19, 1], [0, 0], [-89, 0], [0, 0], [-17, -17], [0, 0], [0, 0], [-398, 199], [433, -433], [0, 0], [-197, 394], [0, 0], [0, 0], [0, 0], [-182, 182], [-502, 251], [0, 0], [0, -919], [-757, -757], [0, 0], [0, 0], [1330, 0], [0, 0], [1207, 1207], [0, 0], [0, -1567], [0, 0], [-629, 1258], [0, 0], [0, 0], [703, -1406], [0, 0], [3780, -1890], [0, 3850], [1386, 1386], [0, 0], [0, 0], [0, 0], [-1241, 0], [0, 0], [0, -5111], [0, 0], [2340, -1170], [-630, 1260], [3366, -6732], [0, 0], [-6823, 6823], [0, 0], [0, 0], [0, -7378], [0, 0], [0, 0], [0, 0], [0, 0], [-5130, -5130], [0, -5167], [0, 0], [-10046, 5023], [0, 0], [-4861, 9722], [0, 0], [901, -901], [0, 0], [-3961, -3961], [7363, 0], [0, 0], [-11914, 0], [0, 0], [0, 0], [-3382, 1691], [-251, 251], [827, -1654], [0, 0], [0, 0], [-13067, 13067], [0, 0], [0, 8297], [0, 0], [10459, 0], [0, 0], [-14149, 0], [-6210, -6210], [0, 0], [0, 0], [-19207, 19207], [0, 0], [10583, -21166], [0, 0], [0, 0], [18002, -9001], [0, 0], [-12293, -12293], [0, 0], [0, 0], [0, 0], [11987, 11987], [20789, -20789], [-14742, 29484], [0, 0], [0, 0], [0, 0], [-31502, 15751], [0, -27323], [0, 0], [0, 0], [0, 0], [11951, 0], [11519, 11519], [0, -17390], [0, 0], [-35566, 17783], [18270, -36540], [0, 0], [-2033, 4066], [0, 0], [0, 0], [0, 0], [0, 13249], [-33949, 0], [0, 0], [0, 0], [-1369, -1369], [0, 0], [36146, -36146], [0, 0], [16271, -32542], [24281, -24281], [-16886, 8443], [0, 0], [-1961, -1961], [41470, 0], [0, 0], [39023, 0], [0, 0], [-9396, 4698], [0, 0], [0, 0], [-13950, 27900], [0, 0], [-52668, 26334], [0, 0], [0, -47143], [0, 0], [38411, 0], [0, 0], [22770, 22770], [0, 0], [-50150, 50150], [0, 0], [24263, -48526], [0, 0], [62858, -31429], [0, 0], [14687, 14687], [0, 0], [15227, 0], [0, 0], [0, 0], [0, 0], [15371, -30742], [0, 0], [0, 0], [0, 0], [21058, -10529], [0, -33983]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_336_bj_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_336_4_bj_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_336_4_bj_c(:prec:=2)
    chi := MakeCharacter_336_bj();
    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_336_4_bj_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_336_4_bj_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_336_bj();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![0, 1]>,<13,R![89, 1]>,<19,R![867, 51, 1]>],Snew);
    return Vf;
end function;