
// Make newform 336.4.bc.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_336_bc();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_336_bc_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_bc_a();".
// 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_bc_a();".
// 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_bc();"
function MakeCharacter_336_bc()
    N := 336;
    order := 6;
    char_gens := [127, 85, 113, 241];
    v := [6, 6, 3, 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_336_bc_Hecke();"
function MakeCharacter_336_bc_Hecke(Kf)
    N := 336;
    order := 6;
    char_gens := [127, 85, 113, 241];
    char_values := [[1, 0], [1, 0], [-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], [-3, -3], [0, 0], [-18, -1], [0, 0], [-17, 34], [0, 0], [-34, 17], [0, 0], [0, 0], [199, 199], [0, -433], [0, 0], [449, 0], [0, 0], [0, 0], [0, 0], [1080, -540], [1007, -1007], [0, 0], [487, 487], [0, 503], [0, 0], [0, 0], [-792, 1584], [0, 0], [-2414, 1207], [0, 0], [-1567, 1567], [0, 0], [2647, 0], [0, 0], [0, 0], [703, -1406], [0, 0], [-1748, 1748], [468, 468], [0, -3400], [0, 0], [0, 0], [0, 0], [-2719, 5438], [0, 0], [5111, -5111], [0, 0], [1170, 1170], [6032, 0], [-3366, 6732], [0, 0], [1406, -703], [0, 0], [0, 0], [720, 720], [0, 0], [0, 0], [0, 0], [0, 0], [10260, -5130], [5167, -5167], [0, 0], [-5023, -5023], [0, 0], [4861, -9722], [0, 0], [12746, -6373], [0, 0], [0, -9899], [7363, 0], [0, 0], [-3060, 6120], [0, 0], [0, 0], [-1691, -1691], [0, -251], [14687, 0], [0, 0], [0, 0], [-10298, 5149], [0, 0], [-8263, -8263], [0, 0], [-10459, 0], [0, 0], [-6443, 12886], [-12420, 6210], [0, 0], [0, 0], [0, -19207], [0, 0], [-7811, 0], [0, 0], [0, 0], [14797, -14797], [0, 0], [0, 6607], [0, 0], [0, 0], [0, 0], [-23974, 11987], [0, -20789], [-1640, 0], [0, 0], [0, 0], [0, 0], [629, -629], [2699, 2699], [0, 0], [0, 0], [0, 0], [15551, -31102], [23038, -11519], [-17390, 17390], [0, 0], [17783, 17783], [-1892, 0], [0, 0], [2033, -4066], [0, 0], [0, 0], [0, 0], [18071, 18071], [33949, 0], [0, 0], [0, 0], [2738, -1369], [0, 0], [0, 36146], [0, 0], [16271, -32542], [-36254, 18127], [-37423, 37423], [0, 0], [0, -41021], [41470, 0], [0, 0], [9937, -19874], [0, 0], [4698, 4698], [0, 0], [0, 0], [-13950, 27900], [0, 0], [-12220, 12220], [0, 0], [4337, 4337], [0, 0], [18323, -36646], [0, 0], [45540, -22770], [0, 0], [0, -50150], [0, 0], [-31429, 0], [0, 0], [-4607, 4607], [0, 0], [0, 49573], [0, 0], [31931, -63862], [0, 0], [0, 0], [0, 0], [-53927, 0], [0, 0], [0, 0], [0, 0], [-59669, 59669], [30601, 30601]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_336_bc_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_336_4_bc_a();".
// 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_bc_a(:prec:=2)
    chi := MakeCharacter_336_bc();
    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_bc_a();".
// 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_bc_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_336_bc();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![0, 1]>,<13,R![867, 0, 1]>],Snew);
    return Vf;
end function;