
// Make newform 882.4.g.q in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_882_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_882_g_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_882_4_g_q();".
// 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_882_4_g_q();".
// 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_882_g();"
function MakeCharacter_882_g()
    N := 882;
    order := 3;
    char_gens := [785, 199];
    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_882_g_Hecke();"
function MakeCharacter_882_g_Hecke(Kf)
    N := 882;
    order := 3;
    char_gens := [785, 199];
    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, 2], [0, 0], [0, -6], [0, 0], [30, -30], [-2, 0], [-66, 66], [0, -52], [0, 114], [-72, 0], [-196, 196], [0, 286], [-378, 0], [164, 0], [0, 228], [-348, 348], [348, -348], [0, -106], [-596, 596], [-630, 0], [-1042, 1042], [0, 88], [-1440, 0], [0, -1374], [34, 0], [438, -438], [0, 1676], [0, 2022], [502, -502], [-2016, 0], [1784, 0], [0, -1608], [-2580, 2580], [-2144, 0], [0, 1500], [1240, -1240], [614, -614], [0, -92], [-3924, 0], [0, 1902], [-6, 6], [2878, 0], [0, -354], [4858, -4858], [396, 0], [1712, -1712], [-772, 0], [-776, 0], [1788, -1788], [0, 5402], [0, 3012], [-3546, 0], [-3562, 3562], [3348, 0], [0, -366], [4170, -4170], [-6078, 6078], [0, 2468], [394, -394], [396, 0], [-1348, 1348], [7506, 0], [-1748, 0], [1140, -1140], [0, 146], [0, -8148], [0, 9700], [8174, 0], [-4038, 4038], [-10766, 0], [-3666, 3666], [0, -5106], [-5776, 5776], [0, -8462], [6860, 0], [0, 696], [11136, -11136], [0, 10838], [0, -8364], [-1762, 1762], [14580, 0], [8534, 0], [5934, -5934], [14758, 0], [0, -11392], [0, 7026], [3384, 0], [0, 4282], [16650, 0], [-9664, 0], [0, 12324], [-18660, 18660], [3400, -3400], [-2970, 0], [0, 988], [5184, 0], [0, -16854], [4398, -4398], [0, -10672], [0, -20702], [-22876, 0], [-12876, 12876], [6900, -6900], [0, 14676], [-380, 380], [-11806, 11806], [-19188, 0], [0, -690], [-20490, 20490], [11590, 0], [0, -6424], [9682, -9682], [-5076, 0], [22664, -22664], [-8584, 0], [372, -372], [-3188, 0], [12732, -12732], [0, -3576], [-11430, 0], [22646, -22646], [-13570, 0], [0, 2838], [-6558, 6558], [0, -21832], [-16200, 0], [0, -36722], [0, -13776], [-34220, 0], [0, -13750], [-39836, 39836], [-34470, 0], [0, -5240], [18578, 0], [0, -30534], [39958, 0], [3966, -3966], [-3760, 3760], [-24102, 0], [11712, -11712], [-37424, 0], [0, 13452], [-20432, 20432], [-24390, 0], [-28510, 28510], [-36972, 0], [14074, 0], [8826, -8826], [0, -20500], [0, 7674], [0, 8890], [738, 0], [20, 0], [0, -39804], [-29180, 29180], [48258, 0], [0, -25760], [0, 44778], [44494, 0], [7458, -7458], [0, 17790], [5832, 0], [-13264, 0], [0, -3984], [-38940, 38940], [8808, -8808], [-18488, 18488], [-5002, 5002]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_882_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_882_4_g_q();".
// 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_882_4_g_q(:prec:=2)
    chi := MakeCharacter_882_g();
    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_882_4_g_q();".
// 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_882_4_g_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_882_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![36, 6, 1]>,<11,R![900, -30, 1]>,<13,R![2, 1]>],Snew);
    return Vf;
end function;