
// Make newform 324.4.e.f in Magma, downloaded from the LMFDB on 13 May 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_f();".
// 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_f();".
// 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, 3], [4, -4], [-24, 24], [0, 25], [21, 0], [-52, 0], [0, 168], [-177, 177], [0, 124], [-265, 0], [0, 426], [160, -160], [-540, 540], [258, 0], [0, 528], [505, -505], [0, 244], [-204, 0], [-397, 0], [-200, 200], [-540, 540], [453, 0], [-290, 290], [1374, -1374], [0, -1268], [-1164, 0], [587, 0], [0, -1233], [1496, 0], [0, -2004], [2775, -2775], [0, -2012], [0, -2145], [-3176, 3176], [0, -515], [3272, 0], [0, -684], [3123, -3123], [-2892, 0], [-1426, 0], [-1596, 1596], [0, -1115], [1245, 0], [-700, 0], [0, -704], [880, -880], [-3924, 3924], [0, 6145], [3885, 0], [0, 4860], [4549, -4549], [-4800, 0], [0, 2511], [-4548, 4548], [2517, 0], [4040, 0], [7090, -7090], [-3165, 3165], [0, 6748], [0, 6495], [-8980, 0], [0, 3864], [-5495, 5495], [-885, 885], [10924, -10924], [0, 7870], [0, -4716], [1642, -1642], [3090, -3090], [5688, 0], [-6776, 6776], [0, -6158], [7484, 0], [0, 4620], [-930, 930], [1919, 0], [0, -10005], [0, -7655], [0, -14040], [-11219, 11219], [1884, 0], [17507, 0], [3508, -3508], [12420, -12420], [-8658, 0], [-335, 335], [246, -246], [0, -14312], [-8184, 0], [18108, -18108], [13700, 0], [0, -20136], [0, 13408], [21600, 0], [0, -5946], [-8970, 0], [-12520, 0], [3911, 0], [17020, -17020], [-13875, 0], [0, 540], [-12513, 12513], [0, 20416], [-1801, 0], [-5520, 5520], [11937, 0], [0, 16188], [11941, -11941], [0, 12460], [4670, 0], [0, 9435], [13960, -13960], [-18904, 0], [-21381, 21381], [0, 1420], [21204, 0], [0, 23190], [-9804, 9804], [0, 30241], [-803, 803], [-14466, 14466], [7020, 0], [24460, -24460], [-27651, 0], [4921, -4921], [20232, 0], [-6500, 6500], [0, -13718], [-13588, 0], [0, 34092], [0, 940], [-5410, 0], [0, -6849], [0, 9133], [37845, 0], [0, 33976], [0, -23805], [-6567, 0], [-8656, 0], [5991, -5991], [0, -45812], [-35160, 0], [-9898, 0], [4032, -4032], [-18638, 18638], [37575, -37575], [0, 6268], [44832, 0], [0, -14495], [-25986, 0], [34868, 0], [0, -6720], [17296, -17296], [-14256, 14256], [15752, 0], [31563, -31563], [-3901, 0], [0, -53445], [9036, -9036], [25065, 0], [0, -14444], [-50004, 0], [0, -45390], [35952, -35952], [14156, 0], [51541, -51541]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_324_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![9, -3, 1]>,<7,R![16, -4, 1]>],Snew);
    return Vf;
end function;