
// Make newform 324.4.e.b in Magma, downloaded from the LMFDB on 20 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_b();".
// 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_b();".
// 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, -9], [1, -1], [-63, 63], [0, 28], [72, 0], [98, 0], [0, -126], [126, -126], [0, 259], [386, 0], [0, 450], [34, -34], [54, -54], [-693, 0], [0, -180], [280, -280], [0, 586], [504, 0], [161, 0], [-440, 440], [-999, 999], [882, 0], [721, -721], [441, -441], [0, 532], [819, 0], [-1294, 0], [0, -1134], [-1807, 0], [0, 2205], [-1386, 1386], [0, -476], [0, 1575], [-449, 449], [0, -1820], [-1828, 0], [0, 810], [1323, -1323], [-315, 0], [-2800, 0], [3276, -3276], [0, -3221], [3339, 0], [3689, 0], [0, 6022], [952, -952], [-5292, 5292], [0, -2198], [5166, 0], [0, -3402], [-1862, 1862], [-5472, 0], [0, -5292], [-1638, 1638], [-1206, 0], [4319, 0], [2248, -2248], [2016, -2016], [0, 2338], [0, 8514], [6104, 0], [0, 4338], [8155, -8155], [4977, -4977], [-5678, 5678], [0, -2906], [0, -6993], [-7910, 7910], [-2466, 2466], [7182, 0], [11431, -11431], [0, 6616], [-9820, 0], [0, -1440], [-5985, 5985], [-11284, 0], [0, 7308], [0, -6335], [0, -6372], [-3320, 3320], [-2898, 0], [-4291, 0], [8323, -8323], [3780, -3780], [12474, 0], [-16679, 16679], [17271, -17271], [0, -17387], [3087, 0], [5238, -5238], [-4384, 0], [0, 3843], [0, -12566], [-5238, 0], [0, 6309], [-5868, 0], [6776, 0], [-9388, 0], [-15056, 15056], [16569, 0], [0, -14553], [-4284, 4284], [0, -692], [4886, 0], [11025, -11025], [-14562, 0], [0, -11466], [-10955, 10955], [0, -1232], [25622, 0], [0, -21042], [-6524, 6524], [20663, 0], [882, -882], [0, 7252], [21168, 0], [0, -14301], [-15057, 15057], [0, 4690], [5203, -5203], [-6174, 6174], [756, 0], [-17948, 17948], [34083, 0], [5284, -5284], [-24516, 0], [12481, -12481], [0, 3094], [-376, 0], [0, -29484], [0, -11681], [890, 0], [0, 32832], [0, -3185], [-21222, 0], [0, -12320], [0, 11907], [-14994, 0], [-38878, 0], [-3906, 3906], [0, 10207], [8064, 0], [-10486, 0], [33696, -33696], [15778, -15778], [26136, -26136], [0, -23912], [17892, 0], [0, -44162], [-8820, 0], [-4654, 0], [0, -33084], [-47258, 47258], [-36666, 36666], [-2995, 0], [13284, -13284], [-2989, 0], [0, -46485], [-11529, 11529], [37044, 0], [0, -12455], [-20169, 0], [0, 39942], [-54234, 54234], [-26137, 0], [33460, -33460]];
    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_b();".
// 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_b(: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_b();".
// 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_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_324_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![81, 9, 1]>,<7,R![1, -1, 1]>],Snew);
    return Vf;
end function;