
// Make newform 882.4.g.bh in Magma, downloaded from the LMFDB on 22 September 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_bh();".
// 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_bh();".
// 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 := [3364, 0, 58, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 58, 29];
    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_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, 0], [-1, 0, -1, 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 := 4;
    raw_aps := [[2, 0, 2, 0], [0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0], [0, 0, -2, 0], [0, 0, 0, 2], [0, 3, 0, 3], [0, -10, 0, 0], [30, 0, 30, 0], [-212, 0, 0, 0], [0, 14, 0, 14], [-246, 0, -246, 0], [0, 0, 0, -21], [-284, 0, 0, 0], [0, 4, 0, 0], [0, 0, -548, 0], [0, 44, 0, 44], [0, 34, 0, 0], [0, 0, 652, 0], [-770, 0, 0, 0], [0, -64, 0, -64], [-472, 0, -472, 0], [0, 0, 0, 12], [0, 47, 0, 0], [0, 0, 0, -20], [0, -67, 0, -67], [0, 38, 0, 0], [-1646, 0, -1646, 0], [0, 0, 982, 0], [-1288, 0, 0, 0], [-1072, 0, 0, 0], [0, -180, 0, 0], [0, 0, -2936, 0], [0, 0, 0, -12], [-964, 0, -964, 0], [0, 0, -1688, 0], [0, 226, 0, 226], [3324, 0, 3324, 0], [0, 0, 0, 200], [0, 215, 0, 0], [0, 0, -1254, 0], [0, 0, 0, 202], [906, 0, 906, 0], [0, 0, 182, 0], [-3468, 0, 0, 0], [0, -256, 0, -256], [-2620, 0, 0, 0], [0, 0, 0, -92], [0, 256, 0, 256], [0, 362, 0, 0], [-712, 0, -712, 0], [2586, 0, 0, 0], [0, -148, 0, -148], [0, 0, 0, -296], [0, 299, 0, 0], [0, 0, 2578, 0], [0, -61, 0, -61], [0, 74, 0, 0], [0, 0, 1510, 0], [4008, 0, 0, 0], [0, 158, 0, 158], [0, 0, 0, -345], [0, 0, 0, 418], [0, 484, 0, 484], [0, -36, 0, 0], [3780, 0, 3780, 0], [-6260, 0, -6260, 0], [-3166, 0, 0, 0], [0, 0, 3618, 0], [0, 0, 0, 294], [0, 243, 0, 243], [130, 0, 130, 0], [0, 512, 0, 512], [50, 0, 50, 0], [-4956, 0, 0, 0], [0, 444, 0, 0], [0, 0, 13140, 0], [0, -250, 0, 0], [4824, 0, 4824, 0], [0, 348, 0, 348], [0, 0, 0, -692], [-4458, 0, 0, 0], [0, 0, 16214, 0], [0, 0, 0, 236], [0, 564, 0, 0], [15006, 0, 15006, 0], [-1824, 0, 0, 0], [586, 0, 586, 0], [0, 0, 0, 1011], [-88, 0, 0, 0], [0, -640, 0, 0], [0, -324, 0, -324], [0, 0, -824, 0], [15426, 0, 0, 0], [-5844, 0, -5844, 0], [0, 0, 0, 668], [0, -775, 0, 0], [0, 103, 0, 103], [0, 1256, 0, 0], [6626, 0, 6626, 0], [-19964, 0, 0, 0], [0, 0, -6780, 0], [0, -1244, 0, -1244], [-192, 0, -192, 0], [0, 0, -23028, 0], [0, 704, 0, 704], [0, 0, 0, -1280], [0, -879, 0, 0], [0, 0, -18066, 0], [0, 0, 0, 1304], [0, -348, 0, 0], [0, 0, 15938, 0], [11696, 0, 0, 0], [0, 884, 0, 884], [11856, 0, 0, 0], [0, 0, -5608, 0], [0, 0, 0, 1662], [0, -384, 0, -384], [-28636, 0, -28636, 0], [-31786, 0, 0, 0], [0, 622, 0, 622], [24986, 0, 0, 0], [0, 3, 0, 0], [0, 0, -30094, 0], [0, -1112, 0, 0], [-7660, 0, 0, 0], [-10654, 0, -10654, 0], [0, 2436, 0, 0], [0, 0, 0, 1166], [0, 166, 0, 0], [0, 0, -8156, 0], [2910, 0, 0, 0], [-13584, 0, -13584, 0], [19054, 0, 0, 0], [0, -329, 0, 0], [0, 0, 0, 1412], [0, 1557, 0, 1557], [0, -2428, 0, -2428], [0, 0, 0, 351], [0, 0, -19776, 0], [0, 0, 0, -1048], [29972, 0, 29972, 0], [0, 0, -38816, 0], [-34386, 0, 0, 0], [0, -862, 0, -862], [0, 0, 0, 1776], [0, 0, 0, -970], [0, -1287, 0, -1287], [0, 822, 0, 0], [30774, 0, 30774, 0], [16758, 0, 16758, 0], [0, 0, 0, -1655], [-5468, 0, 0, 0], [0, 1776, 0, 0], [0, 0, -5876, 0], [-17962, 0, 0, 0], [-53424, 0, -53424, 0], [0, -393, 0, 0], [0, 0, 0, -240], [0, -3489, 0, -3489], [10202, 0, 10202, 0], [8856, 0, 0, 0], [-40760, 0, 0, 0], [0, -3032, 0, 0], [0, 0, -23360, 0], [0, 824, 0, 824], [0, 0, -19480, 0], [0, 2398, 0, 2398]];
    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_bh();".
// 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_bh(:prec:=4)
    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_bh();".
// 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_bh( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_882_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![53824, 0, 232, 0, 1]>,<11,R![4, -2, 1]>,<13,R![-928, 0, 1]>],Snew);
    return Vf;
end function;