
// Make newform 324.4.e.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_324_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_324_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_324_4_e_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_324_4_e_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_324_e();"
function MakeCharacter_324_e()
    N := 324;
    order := 3;
    char_gens := [163, 245];
    v := [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_324_e_Hecke();"
function MakeCharacter_324_e_Hecke(Kf)
    N := 324;
    order := 3;
    char_gens := [163, 245];
    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], [0, 0], [0, -18], [-8, 8], [36, -36], [0, 10], [-18, 0], [-100, 0], [0, 72], [-234, 234], [0, 16], [-226, 0], [0, 90], [-452, 452], [432, -432], [-414, 0], [0, -684], [-422, 422], [0, -332], [360, 0], [26, 0], [-512, 512], [-1188, 1188], [630, 0], [1054, -1054], [558, -558], [0, -8], [-1764, 0], [1622, 0], [0, -1134], [-592, 0], [0, -1908], [954, -954], [0, -2564], [0, -738], [2440, -2440], [0, 2554], [-820, 0], [0, 1944], [-1242, 1242], [-1116, 0], [1070, 0], [-576, 576], [0, 1342], [-1422, 0], [872, 0], [0, -1340], [-4880, 4880], [2700, -2700], [0, -254], [-4410, 0], [0, -3888], [-5138, 5138], [-4788, 0], [0, -5886], [2232, -2232], [666, 0], [-5536, 0], [-2126, 2126], [-2934, 2934], [0, -2036], [0, 2286], [1244, 0], [0, 1224], [-1898, 1898], [-9162, 9162], [4348, -4348], [0, -7154], [0, -1836], [-5894, 5894], [11106, -11106], [-13176, 0], [6112, -6112], [0, 13618], [692, 0], [0, -8064], [12654, -12654], [-106, 0], [0, -4014], [0, -3914], [0, 4428], [15490, -15490], [-6768, 0], [1298, 0], [2248, -2248], [-9612, 9612], [-162, 0], [-1370, 1370], [-15354, 15354], [0, 13024], [14436, 0], [12096, -12096], [6056, 0], [0, 7524], [0, -5276], [-4968, 0], [0, 10998], [8838, 0], [22436, 0], [-4762, 0], [6004, -6004], [5274, 0], [0, -12420], [-21366, 21366], [0, -21140], [3266, 0], [17028, -17028], [-9522, 0], [0, -10296], [3382, -3382], [0, 20656], [-22114, 0], [0, 19962], [604, -604], [152, 0], [4194, -4194], [0, 7252], [6696, 0], [0, 28422], [-19908, 19908], [0, -14318], [-30050, 30050], [22158, -22158], [3132, 0], [20932, -20932], [21834, 0], [-12446, 12446], [12528, 0], [-11576, 11576], [0, 29338], [2540, 0], [0, -18792], [0, -4832], [-20818, 0], [0, 12042], [0, -13058], [11826, 0], [0, -11996], [0, 6966], [40806, 0], [-17980, 0], [-12834, 12834], [0, 37864], [-42516, 0], [45638, 0], [17496, -17496], [-32174, 32174], [-38934, 38934], [0, -29780], [48096, 0], [0, -21302], [7470, 0], [764, 0], [0, 32328], [36316, -36316], [-13392, 13392], [38072, 0], [-12798, 12798], [34874, 0], [0, 17190], [40284, -40284], [-15498, 0], [0, -37160], [-18468, 0], [0, 10386], [44136, -44136], [-28432, 0], [39778, -39778]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_324_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_324_4_e_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_324_4_e_a(:prec:=2)
    chi := MakeCharacter_324_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_324_4_e_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_324_4_e_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_324_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![324, 18, 1]>,<7,R![64, 8, 1]>],Snew);
    return Vf;
end function;