
// Make newform 117.4.g.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_117_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_117_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_117_4_g_a();".
// 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_117_4_g_a();".
// 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_117_g();"
function MakeCharacter_117_g()
    N := 117;
    order := 3;
    char_gens := [92, 28];
    v := [3, 1];
    // 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_117_g_Hecke();"
function MakeCharacter_117_g_Hecke(Kf)
    N := 117;
    order := 3;
    char_gens := [92, 28];
    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 := [[-1, 1], [0, 0], [-7, 0], [0, 10], [-22, 22], [-39, -13], [0, 37], [0, -30], [-162, 162], [-113, 113], [196, 0], [-13, 13], [285, -285], [0, 246], [462, 0], [537, 0], [0, 576], [0, 635], [-202, 202], [0, -1086], [-805, 0], [884, 0], [-518, 0], [194, -194], [0, 1202], [-429, 429], [-1302, 0], [-1338, 1338], [-1034, 0], [0, 1077], [988, -988], [-560, 0], [0, -519], [0, 348], [0, -645], [2914, 0], [-2079, 0], [0, -1700], [3680, -3680], [0, 4146], [3674, -3674], [-3283, 0], [0, -596], [393, -393], [-3522, 3522], [0, -2018], [-160, 160], [-4072, 4072], [0, -5794], [6482, 0], [-6890, 0], [-2466, 0], [0, 3617], [0, 4860], [565, -565], [-498, 498], [0, 5546], [2256, -2256], [0, -2309], [-5833, 0], [-1650, 1650], [0, 2991], [-2422, 0], [3402, 0], [2310, 0], [257, 0], [0, -1028], [2487, 0], [0, -2850], [2018, -2018], [-5287, 5287], [7278, 0], [4202, -4202], [0, 1583], [2052, -2052], [0, -6872], [11653, 0], [0, -6134], [-10795, 10795], [0, 8489], [1496, -1496], [-11695, 0], [10590, -10590], [0, 13949], [10726, -10726], [-16228, 0], [0, 7538], [-15539, 15539], [0, 4811], [562, 0], [-4914, 0], [-3600, 3600], [0, 17130], [-11838, 11838], [8976, 0], [0, 1682], [15167, -15167], [6783, 0], [13918, -13918], [-1335, 0], [-3806, 0], [-1905, 1905], [0, -4800], [14678, -14678], [-586, 0], [8939, 0], [-13792, 13792], [-9569, 0], [5192, 0], [3677, -3677], [0, 10960], [26027, -26027], [0, 17681], [3192, 0], [0, -7580], [0, -27707], [0, -11216], [-2536, 2536], [17730, -17730], [0, 18920], [-5241, 5241], [-20467, 20467], [70, 0], [0, 6432], [6666, -6666], [14054, 0], [0, 71], [0, 3936], [34202, 0], [-27363, 0], [-21776, 21776], [2484, -2484], [-32906, 32906], [3914, -3914], [0, -33038], [17586, -17586], [0, 18314], [0, 42068], [0, -4282], [40221, -40221], [-7084, 0], [17338, -17338], [0, -35496], [14992, 0], [20659, -20659], [0, 28716], [13377, 0], [27419, 0], [2422, 0], [34522, 0], [0, -13733], [-22759, 22759], [-2168, 0], [15888, -15888], [11628, -11628], [12584, 0], [0, -17184], [0, 12777], [9191, 0], [-50498, 0], [1560, -1560], [0, -21498], [-418, 0], [0, 18132], [12501, -12501], [43708, 0], [39614, -39614], [0, 36503]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_117_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_117_4_g_a();".
// 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_117_4_g_a(:prec:=2)
    chi := MakeCharacter_117_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_117_4_g_a();".
// 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_117_4_g_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_117_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![1, 1, 1]>],Snew);
    return Vf;
end function;