
// Make newform 169.4.c.c in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_169_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_169_c_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_169_4_c_c();".
// 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_169_4_c_c();".
// 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_169_c();"
function MakeCharacter_169_c()
    N := 169;
    order := 3;
    char_gens := [2];
    v := [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_169_c_Hecke();"
function MakeCharacter_169_c_Hecke(Kf)
    N := 169;
    order := 3;
    char_gens := [2];
    char_values := [[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 := [[3, -3], [1, -1], [9, 0], [0, 15], [-48, 48], [0, 0], [0, -45], [0, 6], [162, -162], [144, -144], [-264, 0], [303, -303], [-192, 192], [0, -97], [-111, 0], [-414, 0], [0, 522], [0, -376], [-36, 36], [0, 357], [1098, 0], [-830, 0], [438, 0], [-438, 438], [0, -852], [396, -396], [-182, 0], [612, -612], [-1083, 0], [0, -90], [2086, -2086], [-1467, 0], [0, -414], [0, 2419], [0, -930], [1683, 0], [1874, 0], [0, -1194], [-2388, 2388], [0, -1566], [-657, 657], [1222, 0], [0, -1260], [342, -342], [81, -81], [0, 1996], [-2833, 2833], [-3507, 3507], [0, -228], [-5493, 0], [-3627, 0], [-6075, 0], [0, 210], [0, 7092], [-5805, 5805], [-792, 792], [0, -5472], [2331, -2331], [0, -1384], [4062, 0], [-3764, 3764], [0, 4227], [-306, 0], [2106, 0], [10051, 0], [2154, 0], [0, 10770], [2171, 0], [0, 7047], [-6873, 6873], [-9318, 9318], [-4128, 0], [2536, -2536], [0, 92], [10182, -10182], [0, -579], [-2106, 0], [0, -1974], [11886, -11886], [0, 1254], [-5823, 5823], [7341, 0], [-7485, 7485], [0, -15203], [-1762, 1762], [-7317, 0], [0, -5016], [9870, -9870], [0, -14541], [2112, 0], [-3276, 0], [-15453, 15453], [0, -3660], [747, -747], [15804, 0], [0, 12078], [16110, -16110], [3915, 0], [-16184, 16184], [7923, 0], [-14389, 0], [-10383, 10383], [0, -16425], [12213, -12213], [6383, 0], [-6426, 0], [-21330, 21330], [-12084, 0], [2394, 0], [21971, -21971], [0, 15406], [-9630, 9630], [0, 14748], [3672, 0], [0, -19875], [0, 1710], [0, -16452], [25902, -25902], [-18108, 18108], [0, 32904], [15318, -15318], [-7729, 7729], [19242, 0], [0, 22518], [9168, -9168], [-1170, 0], [0, -1662], [0, 30960], [8372, 0], [2739, 0], [-6756, 6756], [29643, -29643], [18128, -18128], [6410, -6410], [0, 28290], [-27960, 27960], [0, -5649], [0, 756], [0, 31194], [-17055, 17055], [-35520, 0], [1095, -1095], [0, -2554], [-21522, 0], [-13124, 13124], [0, -23424], [-31077, 0], [19422, 0], [1744, 0], [-19179, 0], [0, 29217], [-15633, 15633], [-30589, 0], [25884, -25884], [-12305, 12305], [29772, 0], [0, -47644], [0, 21924], [32398, 0], [-2097, 0], [-20016, 20016], [0, 24993], [40959, 0], [0, 48933], [47388, -47388], [-16803, 0], [57526, -57526], [0, 25000]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_169_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_169_4_c_c();".
// 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_169_4_c_c(:prec:=2)
    chi := MakeCharacter_169_c();
    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_169_4_c_c();".
// 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_169_4_c_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_169_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![9, -3, 1]>],Snew);
    return Vf;
end function;