
// Make newform 169.4.c.a in Magma, downloaded from the LMFDB on 17 May 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_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_169_4_c_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_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 := [[-5, 5], [7, -7], [7, 0], [0, -13], [-26, 26], [0, 0], [0, -77], [0, -126], [96, -96], [82, -82], [-196, 0], [-131, 131], [336, -336], [0, 201], [105, 0], [-432, 0], [0, -294], [0, 56], [478, -478], [0, 9], [-98, 0], [1304, 0], [308, 0], [-1190, 1190], [0, 70], [-420, 420], [588, 0], [684, -684], [-373, 0], [0, 1734], [-1892, 1892], [1435, 0], [0, -1776], [0, 1869], [0, 2466], [3323, 0], [-2730, 0], [0, -544], [1624, -1624], [0, 336], [3029, -3029], [-28, 0], [0, -422], [492, -492], [2991, -2991], [0, 70], [-2851, 2851], [217, -217], [0, -2576], [-455, 0], [3061, 0], [3477, 0], [0, -1610], [0, -1008], [-6041, 6041], [3708, -3708], [0, -8344], [-1617, 1617], [0, 3820], [6214, 0], [5292, -5292], [0, -903], [-2114, 0], [3402, 0], [-10689, 0], [7054, 0], [0, 9704], [-10449, 0], [0, 621], [12481, -12481], [-1400, 1400], [4968, 0], [-8722, 8722], [0, -10012], [-3372, 3372], [0, -847], [11314, 0], [0, 1862], [6820, -6820], [0, -12992], [7343, -7343], [5059, 0], [3243, -3243], [0, -11599], [17374, -17374], [989, 0], [0, -14474], [-1594, 1594], [0, -5915], [11072, 0], [1260, 0], [-12033, 12033], [0, -2280], [-16767, 16767], [-12840, 0], [0, 2198], [-17066, 17066], [2583, 0], [-18620, 18620], [16833, 0], [-8615, 0], [8535, -8535], [0, 4641], [4793, -4793], [-5563, 0], [-24038, 0], [-21224, 21224], [-4354, 0], [7310, 0], [7595, -7595], [0, 826], [14590, -14590], [0, 4888], [11004, 0], [0, -4975], [0, -3950], [0, -3682], [-10402, 10402], [31680, -31680], [0, -21940], [-31374, 31374], [-18013, 18013], [-10640, 0], [0, -9336], [4200, -4200], [9872, 0], [0, 28450], [0, -32718], [-22834, 0], [-7875, 0], [-2140, 2140], [31971, -31971], [7432, -7432], [-20176, 20176], [0, -9478], [-12096, 12096], [0, 17941], [0, 6664], [0, 1442], [-30207, 30207], [-21140, 0], [569, -569], [0, 8538], [32702, 0], [21154, -21154], [0, -2184], [-36687, 0], [36806, 0], [4900, 0], [13697, 0], [0, 6239], [-133, 133], [-26003, 0], [31248, -31248], [38253, -38253], [36374, 0], [0, 27648], [0, 756], [20846, 0], [41321, 0], [54966, -54966], [0, 44553], [27907, 0], [0, 16443], [-45414, 45414], [8981, 0], [17414, -17414], [0, 23702]];
    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_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_169_4_c_a(: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_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_169_4_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_169_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![25, 5, 1]>],Snew);
    return Vf;
end function;