
// Make newform 882.4.g.d in Magma, downloaded from the LMFDB on 25 November 2025.
// 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_d();".
// 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_d();".
// 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 := false)
    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 := ISA(Type(Kf), Fld), 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]];
    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, -7], [0, 0], [35, -35], [-66, 0], [-59, 59], [0, 137], [0, -7], [-106, 0], [75, -75], [0, -11], [-498, 0], [260, 0], [0, 171], [-417, 417], [17, -17], [0, 51], [-439, 439], [784, 0], [295, -295], [0, 495], [932, 0], [0, 873], [290, 0], [1085, -1085], [0, 1553], [0, 129], [965, -965], [50, 0], [936, 0], [0, 755], [-2357, 2357], [-28, 0], [0, 2295], [1109, -1109], [1559, -1559], [0, 2251], [2788, 0], [0, -1579], [2451, -2451], [1170, 0], [0, -1275], [-35, 35], [2734, 0], [2243, -2243], [1172, 0], [-2024, 0], [-2571, 2571], [0, 895], [0, 1787], [5100, 0], [-4177, 4177], [-4680, 0], [0, 1749], [-4473, 4473], [-1975, 1975], [0, -8439], [-527, 527], [202, 0], [-7949, 7949], [318, 0], [8132, 0], [929, -929], [0, -209], [0, 7131], [0, 6571], [-11466, 0], [-9777, 9777], [-11914, 0], [-9123, 9123], [0, 8149], [9671, -9671], [0, 4109], [-3488, 0], [0, -8717], [163, -163], [0, 999], [0, -14757], [-133, 133], [-6420, 0], [10266, 0], [-15213, 15213], [1378, 0], [0, -2763], [0, 5849], [-4582, 0], [0, -11551], [-9494, 0], [-10160, 0], [0, 1307], [-18287, 18287], [14953, -14953], [-14352, 0], [0, 5531], [8400, 0], [0, 2385], [9153, -9153], [0, -13807], [0, -8175], [4656, 0], [7003, -7003], [19753, -19753], [0, -6897], [-24915, 24915], [127, -127], [9044, 0], [0, 10701], [20799, -20799], [1402, 0], [0, 6525], [-15051, 15051], [-11150, 0], [3415, -3415], [-21184, 0], [-10705, 10705], [-6860, 0], [-14463, 14463], [0, 5979], [6940, 0], [13399, -13399], [29510, 0], [0, 26001], [-8805, 8805], [0, 28685], [3146, 0], [0, -1259], [0, -16425], [6032, 0], [0, 15243], [10053, -10053], [-24384, 0], [0, -11589], [14562, 0], [0, 22765], [-3766, 0], [26861, -26861], [-2097, 2097], [-35334, 0], [42535, -42535], [-30676, 0], [0, 37343], [-2815, 2815], [9276, 0], [18571, -18571], [29048, 0], [-32090, 0], [24537, -24537], [0, 20825], [0, -22847], [0, 42737], [6162, 0], [7748, 0], [0, 25923], [-31935, 31935], [-3408, 0], [0, -13909], [0, 24537], [32758, 0], [38561, -38561], [0, 39661], [46618, 0], [14816, 0], [0, 16875], [-15837, 15837], [-9915, 9915], [43681, -43681], [-47113, 47113]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_882_g_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_882_4_g_d();".
// 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_d(: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_d();".
// 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_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_882_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![49, 7, 1]>,<11,R![1225, -35, 1]>,<13,R![66, 1]>],Snew);
    return Vf;
end function;