
// Make newform 648.4.i.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_648_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_648_i_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_648_4_i_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_648_4_i_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_648_i();"
function MakeCharacter_648_i()
    N := 648;
    order := 3;
    char_gens := [487, 325, 569];
    v := [3, 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_648_i_Hecke();"
function MakeCharacter_648_i_Hecke(Kf)
    N := 648;
    order := 3;
    char_gens := [487, 325, 569];
    char_values := [[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], [0, 0], [0, -16], [12, -12], [-64, 64], [0, -58], [32, 0], [-136, 0], [0, 128], [144, -144], [0, -20], [-18, 0], [0, 288], [200, -200], [-384, 384], [496, 0], [0, 128], [458, -458], [0, 496], [512, 0], [-602, 0], [-1108, 1108], [-704, 704], [-960, 0], [-206, 206], [-432, 432], [0, 68], [-384, 0], [-518, 0], [0, 960], [796, 0], [0, -512], [-1824, 1824], [0, 2160], [0, 688], [844, -844], [0, -118], [3576, 0], [0, -384], [-2448, 2448], [-4224, 0], [-510, 0], [384, -384], [0, 3454], [-3216, 0], [1708, 0], [0, -2320], [116, -116], [-1344, 1344], [0, -4594], [5056, 0], [0, 3712], [978, -978], [1856, 0], [0, -7808], [-1024, 1024], [1328, 0], [-5812, 0], [-8386, 8386], [640, -640], [0, -4832], [0, -6384], [-3312, 0], [0, -9984], [-2586, 2586], [2832, -2832], [5920, -5920], [0, 4674], [0, 9024], [4362, -4362], [-8768, 8768], [-6144, 0], [-4564, 4564], [0, 8770], [-1096, 0], [0, -10368], [3248, -3248], [-6106, 0], [0, -7008], [0, -1590], [0, 192], [-9074, 9074], [-5248, 0], [-8222, 0], [-16236, 16236], [-14528, 14528], [6304, 0], [-1958, 1958], [-4048, 4048], [0, -16988], [6720, 0], [-9728, 9728], [-8444, 0], [0, 15360], [0, -6624], [-6912, 0], [0, 19920], [3680, 0], [-11720, 0], [11754, 0], [18904, -18904], [3088, 0], [0, 21440], [-22624, 22624], [0, 6000], [19922, 0], [3584, -3584], [1984, 0], [0, -14976], [-25738, 25738], [0, 8548], [8558, 0], [0, 10368], [13088, -13088], [-4412, 0], [30176, -30176], [0, -21288], [17024, 0], [0, -2256], [-23808, 23808], [0, 26242], [24590, -24590], [2864, -2864], [-7616, 0], [-2168, 2168], [-18000, 0], [-3506, 3506], [15616, 0], [15036, -15036], [0, 19126], [-17392, 0], [0, 32384], [0, 27708], [-37246, 0], [0, 4192], [0, -26882], [-17232, 0], [0, -31816], [0, -8272], [-9184, 0], [-19832, 0], [-15216, 15216], [0, 39772], [-18304, 0], [4906, 0], [-15360, 15360], [24802, -24802], [-15072, 15072], [0, -1800], [7552, 0], [0, -20838], [47744, 0], [28280, 0], [0, -7424], [5912, -5912], [26240, -26240], [35620, 0], [3232, -3232], [-11478, 0], [0, 39984], [-24192, 24192], [-39456, 0], [0, 41668], [-51648, 0], [0, -55776], [36096, -36096], [42532, 0], [-29806, 29806]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_648_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_648_4_i_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_648_4_i_a(:prec:=2)
    chi := MakeCharacter_648_i();
    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_648_4_i_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_648_4_i_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_648_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![256, 16, 1]>],Snew);
    return Vf;
end function;