
// Make newform 882.2.f.j in Magma, downloaded from the LMFDB on 01 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_882_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_882_f_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_2_f_j();".
// 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_2_f_j();".
// 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 := [4, 0, -2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 2, 2];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_f();"
function MakeCharacter_882_f()
    N := 882;
    order := 3;
    char_gens := [785, 199];
    v := [2, 3];
    // 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_f_Hecke();"
function MakeCharacter_882_f_Hecke( : Kf := false)
    N := 882;
    order := 3;
    char_gens := [785, 199];
    char_values := [[0, 0, -1, 0], [1, 0, 0, 0]];
    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 := 2;
    raw_aps := [[-1, 0, 1, 0], [-1, 0, 0, -1], [0, -1, 1, -1], [0, 0, 0, 0], [-2, 0, 2, 0], [0, 2, 0, 2], [-2, 0, 0, 0], [-5, 2, 0, -1], [0, 0, 1, 0], [2, 2, -2, -4], [0, 0, 6, 0], [2, 8, 0, -4], [0, 4, 0, 4], [-2, -2, 2, 4], [0, 4, 0, -8], [-6, -4, 0, 2], [0, 0, -2, 0], [9, -1, -9, 2], [0, 2, 8, 2], [5, -4, 0, 2], [2, 4, 0, -2], [-3, 2, 3, -4], [2, 0, -2, 0], [12, 4, 0, -2], [-2, 2, 2, -4], [-5, -5, 5, 10], [0, 0, 14, 0], [-12, 0, 0, 0], [-2, 12, 0, -6], [0, -2, 11, -2], [-3, 0, 0, 0], [0, 1, -11, 1], [2, -4, -2, 8], [0, -1, 7, -1], [0, 0, 6, 0], [5, 0, -5, 0], [0, -3, -1, -3], [-10, -8, 0, 4], [0, -6, 4, -6], [-8, 2, 8, -4], [6, 12, 0, -6], [3, 6, 0, -3], [-9, 2, 9, -4], [0, 2, 13, 2], [2, 12, 0, -6], [-2, 4, 0, -2], [0, -2, -8, -2], [-16, 2, 16, -4], [-3, -1, 3, 2], [0, 5, -11, 5], [-7, 0, 0, 0], [0, -4, 3, -4], [-4, 2, 4, -4], [-15, -2, 0, 1], [0, -4, 18, -4], [-21, -2, 21, 4], [11, 6, 0, -3], [-12, 4, 0, -2], [4, 6, -4, -12], [19, 0, -19, 0], [0, -1, -23, -1], [0, -5, 15, -5], [-13, -10, 0, 5], [0, 6, 16, 6], [10, -6, -10, 12], [-6, -6, 6, 12], [10, -6, -10, 12], [0, 12, -6, 12], [0, -8, 0, -8], [16, -2, -16, 4], [-6, 0, 6, 0], [1, -8, 0, 4], [-4, -4, 4, 8], [0, 2, 2, 2], [-2, 20, 0, -10], [0, -2, 2, -2], [20, 2, -20, -4], [-24, -12, 0, 6], [0, 2, 15, 2], [0, -4, -4, -4], [0, 1, 27, 1], [-18, -2, 18, 4], [12, 16, 0, -8], [-2, 8, 0, -4], [12, -4, -12, 8], [10, 2, -10, -4], [1, 16, 0, -8], [-7, 10, 7, -20], [13, -3, -13, 6], [0, -6, 11, -6], [-10, 0, 0, 0], [-10, 8, 10, -16], [-7, -24, 0, 12], [0, -4, -6, -4], [0, 12, -4, 12], [0, 20, 0, -10], [0, 2, -12, 2], [24, -12, 0, 6], [7, 6, 0, -3], [6, -20, 0, 10], [-12, 8, 12, -16], [-8, -4, 0, 2], [0, -11, -13, -11], [30, 0, -30, 0], [0, -4, -24, -4], [-4, 16, 0, -8], [-1, 7, 1, -14], [0, -12, 0, 6], [0, 4, 24, 4], [2, 6, -2, -12], [0, 6, 6, 6], [0, -12, 0, 6], [0, 12, -14, 12], [13, 7, -13, -14], [23, 4, 0, -2], [-17, 10, 17, -20], [0, -12, -10, -12], [36, 12, 0, -6], [0, -4, 0, -4], [10, 6, -10, -12], [0, -1, 7, -1], [-11, 8, 11, -16], [0, 6, 0, -12], [42, 8, 0, -4], [29, 9, -29, -18], [22, -24, 0, 12], [-8, -8, 8, 16], [0, -8, 0, 4], [16, -10, -16, 20], [0, -1, -15, -1], [-6, 16, 0, -8], [0, 0, -36, 0], [0, 8, -21, 8], [-6, -24, 0, 12], [0, 0, 2, 0], [0, -18, 10, -18], [-7, -22, 0, 11], [0, -12, 18, -12], [0, -11, 9, -11], [-26, 8, 0, -4], [2, 0, 0, 0], [-20, 8, 20, -16], [0, 12, -16, 12], [-12, -20, 0, 10], [-16, 12, 0, -6], [-40, 2, 40, -4], [-7, 13, 7, -26], [-30, -2, 30, 4], [0, 0, -10, 0], [7, 4, 0, -2], [0, -10, 2, -10], [0, -16, 0, 8], [-10, -8, 0, 4], [0, 4, -24, 4], [12, -6, -12, 12], [-3, -20, 3, 40], [-11, -12, 0, 6], [0, 14, 0, -28], [-26, -16, 0, 8], [0, -1, 1, -1], [-28, -10, 28, 20], [26, -24, 0, 12], [0, -8, -5, -8], [27, -22, 0, 11], [0, 8, 18, 8], [6, -16, -6, 32], [8, -8, 0, 4], [35, 7, -35, -14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_882_f_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_882_2_f_j();".
// 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_2_f_j(:prec:=4)
    chi := MakeCharacter_882_f();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_2_f_j();".
// 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_2_f_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_882_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![25, 10, 9, -2, 1]>,<11,R![4, 2, 1]>,<13,R![576, 0, 24, 0, 1]>],Snew);
    return Vf;
end function;