
// Make newform 169.4.b.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_169_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_169_b_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_b_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_b_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, 0, 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_b();"
function MakeCharacter_169_b()
    N := 169;
    order := 2;
    char_gens := [2];
    v := [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_169_b_Hecke();"
function MakeCharacter_169_b_Hecke(Kf)
    N := 169;
    order := 2;
    char_gens := [2];
    char_values := [[-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 := [[0, 4], [2, 0], [0, 17], [0, -20], [0, 32], [0, 0], [13, 0], [0, 30], [-78, 0], [197, 0], [0, -74], [0, 227], [0, -165], [156, 0], [0, 162], [93, 0], [0, 864], [145, 0], [0, 862], [0, 654], [0, -215], [-76, 0], [0, 628], [0, 266], [0, 238], [819, 0], [-1638, 0], [522, 0], [0, -1634], [327, 0], [2158, 0], [730, 0], [0, -1671], [912, 0], [0, -2115], [0, -514], [2901, 0], [0, -2360], [0, -280], [-1326, 0], [-4264, 0], [403, 0], [-1246, 0], [0, -267], [0, 1278], [-4238, 0], [3070, 0], [0, -5378], [0, -3974], [0, 6298], [-4030, 0], [0, -984], [0, -943], [2730, 0], [1885, 0], [4032, 0], [4006, 0], [0, 4296], [5551, 0], [0, 5557], [-3120, 0], [0, -8301], [0, -8678], [-8658, 0], [-5250, 0], [0, 6413], [0, 3488], [1833, 0], [7230, 0], [0, 5258], [0, 3163], [0, 10068], [7438, 0], [-9683, 0], [0, -1062], [0, -3532], [11063, 0], [0, 5986], [0, -5935], [0, -15089], [-10814, 0], [0, -6535], [0, 1980], [6929, 0], [4576, 0], [-8812, 0], [0, -1918], [0, -11761], [0, 901], [0, -1372], [6396, 0], [0, -3270], [0, 19920], [-6552, 0], [0, 1746], [14692, 0], [0, 8077], [11247, 0], [2732, 0], [0, 18375], [-10346, 0], [0, -345], [8580, 0], [19682, 0], [-26624, 0], [0, -14101], [0, 1408], [0, 1241], [11078, 0], [-13817, 0], [8270, 0], [0, 22273], [0, -18989], [0, -72], [0, 23380], [-6383, 0], [0, -17104], [-6994, 0], [-5250, 0], [-4340, 0], [0, 4179], [-22867, 0], [5410, 0], [0, 13578], [0, 12744], [-16406, 0], [0, -709], [7644, 0], [15808, 0], [0, -2583], [0, -4076], [0, -34056], [364, 0], [-6914, 0], [0, -13982], [0, -18066], [0, 14434], [0, 15398], [36842, 0], [41511, 0], [0, 23066], [0, 28838], [-27456, 0], [0, 33572], [45799, 0], [0, -32286], [0, -20937], [7189, 0], [-32498, 0], [0, 8428], [0, -6847], [-29731, 0], [23738, 0], [27588, 0], [37128, 0], [20516, 0], [-21006, 0], [0, 20427], [33191, 0], [0, -36422], [0, 39630], [-57642, 0], [0, 2162], [-19758, 0], [0, -12489], [0, 28658], [-42794, 0], [-52583, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_169_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_169_4_b_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_b_c(:prec:=2)
    chi := MakeCharacter_169_b();
    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_b_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_b_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_169_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![16, 0, 1]>],Snew);
    return Vf;
end function;