
// Make newform 882.4.g.bi in Magma, downloaded from the LMFDB on 25 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_bi();".
// 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_bi();".
// 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 := [484, 0, 22, 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, 22, 11];
    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)
    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 := 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 := [[2, 0, 2, 0], [0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0], [0, 0, -20, 0], [0, 0, 0, -7], [0, -6, 0, -6], [0, -1, 0, 0], [48, 0, 48, 0], [166, 0, 0, 0], [0, -22, 0, -22], [78, 0, 78, 0], [0, 0, 0, 42], [436, 0, 0, 0], [0, 22, 0, 0], [0, 0, -62, 0], [0, 71, 0, 71], [0, -29, 0, 0], [0, 0, 580, 0], [544, 0, 0, 0], [0, -64, 0, -64], [680, 0, 680, 0], [0, 0, 0, 21], [0, -160, 0, 0], [0, 0, 0, 70], [0, -13, 0, -13], [0, 146, 0, 0], [-260, 0, -260, 0], [0, 0, 1882, 0], [1286, 0, 0, 0], [2312, 0, 0, 0], [0, -27, 0, 0], [0, 0, 1114, 0], [0, 0, 0, -147], [-946, 0, -946, 0], [0, 0, 832, 0], [0, 307, 0, 307], [-636, 0, -636, 0], [0, 0, 0, -70], [0, 71, 0, 0], [0, 0, 3228, 0], [0, 0, 0, 301], [-2136, 0, -2136, 0], [0, 0, 1658, 0], [978, 0, 0, 0], [0, -526, 0, -526], [1556, 0, 0, 0], [0, 0, 0, -308], [0, 211, 0, 211], [0, -295, 0, 0], [-6490, 0, -6490, 0], [4296, 0, 0, 0], [0, 482, 0, 482], [0, 0, 0, 595], [0, -160, 0, 0], [0, 0, 400, 0], [0, 29, 0, 29], [0, -736, 0, 0], [0, 0, -6770, 0], [-1878, 0, 0, 0], [0, 41, 0, 41], [0, 0, 0, -399], [0, 0, 0, -77], [0, -776, 0, -776], [0, -162, 0, 0], [2358, 0, 2358, 0], [-2372, 0, -2372, 0], [-250, 0, 0, 0], [0, 0, -9540, 0], [0, 0, 0, 609], [0, -468, 0, -468], [1840, 0, 1840, 0], [0, -316, 0, -316], [-3982, 0, -3982, 0], [2676, 0, 0, 0], [0, 750, 0, 0], [0, 0, -8658, 0], [0, 965, 0, 0], [-5706, 0, -5706, 0], [0, 258, 0, 258], [0, 0, 0, -161], [-16770, 0, 0, 0], [0, 0, -1336, 0], [0, 0, 0, 1190], [0, 384, 0, 0], [6348, 0, 6348, 0], [-7170, 0, 0, 0], [-6866, 0, -6866, 0], [0, 0, 0, 147], [2648, 0, 0, 0], [0, -1315, 0, 0], [0, 1422, 0, 1422], [0, 0, 13936, 0], [12276, 0, 0, 0], [2220, 0, 2220, 0], [0, 0, 0, -1204], [0, -1693, 0, 0], [0, -1238, 0, -1238], [0, -1345, 0, 0], [-1798, 0, -1798, 0], [1276, 0, 0, 0], [0, 0, -2694, 0], [0, 1681, 0, 1681], [12606, 0, 12606, 0], [0, 0, 6852, 0], [0, 1532, 0, 1532], [0, 0, 0, 2023], [0, 876, 0, 0], [0, 0, 19104, 0], [0, 0, 0, -2296], [0, -1464, 0, 0], [0, 0, 28034, 0], [8258, 0, 0, 0], [0, -547, 0, -547], [912, 0, 0, 0], [0, 0, 890, 0], [0, 0, 0, 3129], [0, -1266, 0, -1266], [-21526, 0, -21526, 0], [-23452, 0, 0, 0], [0, -2843, 0, -2843], [-13858, 0, 0, 0], [0, 3459, 0, 0], [0, 0, 27812, 0], [0, 139, 0, 0], [-22906, 0, 0, 0], [15086, 0, 15086, 0], [0, -2190, 0, 0], [0, 0, 0, 770], [0, -3137, 0, 0], [0, 0, 32668, 0], [37056, 0, 0, 0], [19608, 0, 19608, 0], [19378, 0, 0, 0], [0, -1490, 0, 0], [0, 0, 0, -910], [0, 3123, 0, 3123], [0, 1469, 0, 1469], [0, 0, 0, -3717], [0, 0, -14034, 0], [0, 0, 0, 707], [28622, 0, 28622, 0], [0, 0, 24688, 0], [30756, 0, 0, 0], [0, 2477, 0, 2477], [0, 0, 0, -2562], [0, 0, 0, 2513], [0, -2610, 0, -2610], [0, 3513, 0, 0], [40872, 0, 40872, 0], [12006, 0, 12006, 0], [0, 0, 0, 3808], [19588, 0, 0, 0], [0, -4290, 0, 0], [0, 0, 15868, 0], [-39832, 0, 0, 0], [30528, 0, 30528, 0], [0, -1770, 0, 0], [0, 0, 0, -3192], [0, -573, 0, -573], [45212, 0, 45212, 0], [-34218, 0, 0, 0], [14464, 0, 0, 0], [0, 4033, 0, 0], [0, 0, -42062, 0], [0, 4586, 0, 4586], [0, 0, 21272, 0], [0, 13, 0, 13]];
    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_bi();".
// 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_bi(: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_bi();".
// 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_bi( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_882_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![7744, 0, 88, 0, 1]>,<11,R![400, -20, 1]>,<13,R![-4312, 0, 1]>],Snew);
    return Vf;
end function;