
// Make newform 147.4.e.g in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_147_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_147_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_147_4_e_g();".
// 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_147_4_e_g();".
// 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_147_e();"
function MakeCharacter_147_e()
    N := 147;
    order := 3;
    char_gens := [50, 52];
    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_147_e_Hecke();"
function MakeCharacter_147_e_Hecke(Kf)
    N := 147;
    order := 3;
    char_gens := [50, 52];
    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, 3], [-3, 3], [0, -18], [0, 0], [36, -36], [34, 0], [42, -42], [0, -124], [0, 0], [102, 0], [-160, 160], [0, -398], [318, 0], [-268, 0], [0, 240], [498, -498], [-132, 132], [0, 398], [-92, 92], [-720, 0], [-502, 502], [0, 1024], [204, 0], [0, 354], [286, 0], [414, -414], [0, 56], [0, -12], [-1478, 1478], [402, 0], [1280, 0], [0, 1764], [2358, -2358], [52, 0], [0, 1746], [232, -232], [1694, -1694], [0, 2932], [-1176, 0], [0, 870], [2316, -2316], [106, 0], [0, 1128], [-4034, 4034], [-1314, 0], [5096, -5096], [-3076, 0], [1888, 0], [-4716, 4716], [0, -1690], [0, -138], [1896, 0], [-3598, 3598], [3060, 0], [0, -6822], [-2592, 2592], [8214, -8214], [0, -5344], [6514, -6514], [6618, 0], [3260, -3260], [-5118, 0], [-452, 0], [5016, -5016], [0, 5402], [0, -10086], [0, 8044], [4178, 0], [-156, 156], [12418, 0], [-7830, 7830], [0, 9312], [-3760, 3760], [0, -5870], [-1852, 0], [0, 2160], [6786, -6786], [0, -6514], [0, -3330], [-5398, 5398], [-13092, 0], [-322, 0], [-2616, 2616], [-4322, 0], [0, -9016], [0, 5268], [-5310, 0], [0, -15770], [5370, 0], [-3328, 0], [0, 4548], [-8064, 8064], [-16616, 16616], [-7140, 0], [0, 9124], [6552, 0], [0, 2790], [-14862, 14862], [0, 17660], [0, 19834], [20972, 0], [-21174, 21174], [-17772, 17772], [0, -8250], [-20756, 20756], [2, -2], [-26364, 0], [0, 2298], [-3072, 3072], [-24554, 0], [0, 16832], [2482, -2482], [-15798, 0], [-15460, 15460], [-7720, 0], [17262, -17262], [12220, 0], [13560, -13560], [0, -23094], [22548, 0], [17462, -17462], [-22462, 0], [0, -25554], [-9276, 9276], [0, 27380], [25830, 0], [0, 6226], [0, -15072], [32920, 0], [0, -6946], [2356, -2356], [-23520, 0], [0, -3008], [-20770, 0], [0, 11538], [-8498, 0], [-32322, 32322], [26228, -26228], [43338, 0], [28902, -28902], [-27164, 0], [0, 17202], [5992, -5992], [25884, 0], [-1474, 1474], [-33528, 0], [-1190, 0], [34578, -34578], [0, -44404], [0, 38328], [0, 38842], [35046, 0], [14204, 0], [0, -26136], [9052, -9052], [5016, 0], [0, -44552], [0, 24234], [13894, 0], [46758, -46758], [0, -13812], [-58518, 0], [19640, 0], [0, -58308], [23550, -23550], [15768, -15768], [-35264, 35264], [-29338, 29338]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_147_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_147_4_e_g();".
// 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_147_4_e_g(:prec:=2)
    chi := MakeCharacter_147_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_147_4_e_g();".
// 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_147_4_e_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_147_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![9, -3, 1]>,<5,R![324, 18, 1]>],Snew);
    return Vf;
end function;