
// Make newform 882.4.g.k in Magma, downloaded from the LMFDB on 17 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_k();".
// 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_k();".
// 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, 14], [0, 0], [-28, 28], [-18, 0], [-74, 74], [0, 80], [0, -112], [-190, 0], [72, -72], [0, 346], [162, 0], [-412, 0], [0, -24], [318, -318], [200, -200], [0, -198], [716, -716], [-392, 0], [538, -538], [0, -240], [-1072, 0], [0, -810], [-1354, 0], [1358, -1358], [0, -832], [0, 444], [-1870, 1870], [-1378, 0], [1944, 0], [0, 848], [-2966, 2966], [-2800, 0], [0, 510], [-592, 592], [-2686, 2686], [0, 1012], [544, 0], [0, -1858], [-300, 300], [2358, 0], [0, 1392], [-1778, 1778], [-1214, 0], [1040, -1040], [-3868, 0], [-3968, 0], [3936, -3936], [0, 4810], [0, -2182], [3000, 0], [2042, -2042], [-528, 0], [0, -5634], [168, -168], [1310, -1310], [0, -2208], [-5294, 5294], [-3242, 0], [-1592, 1592], [-5022, 0], [9536, 0], [968, -968], [0, 3058], [0, -4986], [0, -8612], [-10206, 0], [2004, -2004], [-1330, 0], [-978, 978], [0, -9680], [-8656, 8656], [0, -5278], [6340, 0], [0, 6232], [-14810, 14810], [0, 5154], [0, 3282], [5810, -5810], [13560, 0], [-738, 0], [1272, -1272], [5062, 0], [0, 5640], [0, 13388], [3230, 0], [0, 10646], [7282, 0], [12688, 0], [0, 2816], [3160, -3160], [14176, -14176], [11268, 0], [0, 4460], [-1512, 0], [0, 11790], [-1362, 1362], [0, 6968], [0, -7062], [-8196, 0], [-7466, 7466], [-24968, 24968], [0, 14250], [-6372, 6372], [-8366, 8366], [20384, 0], [0, -9378], [-9000, 9000], [-7562, 0], [0, -2976], [-4278, 4278], [-18794, 0], [18040, -18040], [-21688, 0], [-10558, 10558], [26152, 0], [-25584, 25584], [0, 15198], [6100, 0], [-2318, 2318], [-10222, 0], [0, -25434], [-8532, 8532], [0, 20672], [21458, 0], [0, 9850], [0, 18840], [-37504, 0], [0, 13338], [-17100, 17100], [19632, 0], [0, -33912], [-31386, 0], [0, 34558], [-39130, 0], [25982, -25982], [35424, -35424], [-30606, 0], [16810, -16810], [9368, 0], [0, 34382], [4472, -4472], [1716, 0], [-7910, 7910], [-19360, 0], [-698, 0], [23406, -23406], [0, 7280], [0, 9808], [0, 8066], [25842, 0], [-5692, 0], [0, 13536], [-17004, 17004], [14568, 0], [0, 1400], [0, 13830], [24166, 0], [10838, -10838], [0, -40916], [-56618, 0], [17504, 0], [0, -23112], [23874, -23874], [15312, -15312], [16528, -16528], [-28606, 28606]];
    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_k();".
// 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_k(: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_k();".
// 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_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_882_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![196, -14, 1]>,<11,R![784, 28, 1]>,<13,R![18, 1]>],Snew);
    return Vf;
end function;