
// Make newform 169.4.c.i in Magma, downloaded from the LMFDB on 19 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_i();".
// 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_i();".
// 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, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1], [0, 2, 0, -1]];
    Rf_basisdens := [1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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 := [[-1, 1, 0, 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, 0, -2, 0], [0, 7, 0, 0], [0, 0, 0, 8], [0, 0, -13, 13], [0, 0, 13, 0], [0, 0, 0, 0], [27, -27, 0, 0], [0, 0, -51, 51], [0, 57, 0, 0], [0, 69, 0, 0], [0, 0, 0, -42], [0, 0, 23, 0], [0, 0, -227, 0], [-85, 85, 0, 0], [0, 0, 0, 198], [426, 0, 0, 0], [0, 0, 11, -11], [17, -17, 0, 0], [0, 0, 95, 0], [0, 0, 337, -337], [0, 0, 0, -580], [-1244, 0, 0, 0], [0, 0, 0, -246], [0, 0, -177, 0], [0, 0, 713, -713], [0, 1959, 0, 0], [-1856, 0, 0, 0], [0, 255, 0, 0], [0, 0, 0, -352], [-411, 411, 0, 0], [0, -2243, 0, 0], [-372, 0, 0, 0], [0, 0, 687, -687], [2545, -2545, 0, 0], [0, 0, 753, -753], [0, 0, 0, 50], [-1534, 0, 0, 0], [0, 0, 943, -943], [0, 0, -939, 0], [-873, 873, 0, 0], [0, -1287, 0, 0], [-2, 0, 0, 0], [2841, -2841, 0, 0], [0, 0, 2451, 0], [0, 0, 1589, 0], [-1685, 1685, 0, 0], [0, -1681, 0, 0], [0, 0, -2365, 0], [0, 0, 253, -253], [0, 0, 0, -104], [5778, 0, 0, 0], [0, 0, 0, -1074], [0, 0, -1189, 1189], [4491, -4491, 0, 0], [0, 5451, 0, 0], [0, 783, 0, 0], [5085, -5085, 0, 0], [0, 0, 765, 0], [-3421, 3421, 0, 0], [0, 0, 0, -468], [0, 7177, 0, 0], [0, 0, -5377, 5377], [0, 0, 0, 2758], [-6192, 0, 0, 0], [-770, 0, 0, 0], [0, 0, 0, 4652], [0, 0, -3047, 3047], [-8278, 0, 0, 0], [-6867, 6867, 0, 0], [0, 0, -7017, 0], [0, 0, 3353, 0], [0, 0, 0, -774], [0, -3665, 0, 0], [-5371, 5371, 0, 0], [0, 0, -6645, 0], [0, 0, 1397, -1397], [9858, 0, 0, 0], [0, 0, 5035, -5035], [0, 0, 4379, 0], [0, 0, 2485, -2485], [0, -5397, 0, 0], [0, 0, 0, 4192], [0, 0, -281, 0], [12139, -12139, 0, 0], [0, 461, 0, 0], [12156, 0, 0, 0], [0, 0, 171, -171], [0, 0, 353, 0], [0, 0, -7579, 7579], [0, 0, 0, 482], [-14496, 0, 0, 0], [0, 0, 5137, 0], [0, 0, 2745, -2745], [0, -1635, 0, 0], [0, 0, 0, -8334], [-12687, 12687, 0, 0], [0, 0, -3319, 0], [6054, 0, 0, 0], [0, 14803, 0, 0], [0, 0, 0, 12396], [-13516, 0, 0, 0], [0, 0, -1669, 0], [11583, -11583, 0, 0], [0, 12879, 0, 0], [11636, 0, 0, 0], [0, 0, 0, 7108], [0, 0, -9033, 0], [0, 0, 0, -14908], [1668, 0, 0, 0], [0, -13699, 0, 0], [23173, -23173, 0, 0], [0, 0, 9593, 0], [0, 0, 16393, -16393], [0, 0, 0, 3606], [0, 0, -12921, 12921], [19827, -19827, 0, 0], [0, 0, -4879, 4879], [0, 2949, 0, 0], [0, -12039, 0, 0], [-3363, 3363, 0, 0], [0, 0, -5865, 0], [0, -18169, 0, 0], [9042, 0, 0, 0], [0, 0, 7195, -7195], [0, 0, -2493, 0], [18270, 0, 0, 0], [0, 0, -941, 941], [9831, -9831, 0, 0], [15464, 0, 0, 0], [0, 0, 0, -7284], [0, 0, 9401, 0], [0, 0, 6239, 0], [0, -13615, 0, 0], [0, -5551, 0, 0], [0, 0, -5821, 5821], [0, 0, 17181, 0], [0, 0, 16013, -16013], [0, 0, 18215, -18215], [1455, -1455, 0, 0], [0, -1659, 0, 0], [0, 0, 0, 2542], [0, 0, -16571, 0], [15779, -15779, 0, 0], [0, 0, 0, 4246], [0, 17371, 0, 0], [0, 0, -17017, 17017], [0, 0, 0, -1680], [5346, 0, 0, 0], [24244, 0, 0, 0], [0, 0, 0, 18926], [0, 0, -2623, 2623], [0, -20517, 0, 0], [23852, 0, 0, 0], [0, -38757, 0, 0], [0, -39071, 0, 0], [-53040, 0, 0, 0], [-367, 367, 0, 0], [0, 0, -17283, 17283], [42166, 0, 0, 0], [0, 0, 0, -20220], [0, 0, 1501, 0], [10623, -10623, 0, 0], [0, 0, 0, 11662], [2325, -2325, 0, 0], [0, 0, 19017, 0], [0, 0, 0, -24654], [0, 4843, 0, 0], [-10943, 10943, 0, 0]];
    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_i();".
// 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_i(:prec:=4)
    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_i();".
// 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_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_169_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![144, 0, 12, 0, 1]>],Snew);
    return Vf;
end function;