Properties

Conductor 199
Order 99
Real No
Primitive Yes
Parity Even
Orbit Label 199.k

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(199)
sage: chi = H[50]
pari: [g,chi] = znchar(Mod(50,199))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 199
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 99
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = Yes
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Even
Orbit label = 199.k
Orbit index = 11

Galois orbit

sage: chi.sage_character().galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

\(\chi_{199}(2,\cdot)\) \(\chi_{199}(4,\cdot)\) \(\chi_{199}(7,\cdot)\) \(\chi_{199}(9,\cdot)\) \(\chi_{199}(10,\cdot)\) \(\chi_{199}(13,\cdot)\) \(\chi_{199}(14,\cdot)\) \(\chi_{199}(16,\cdot)\) \(\chi_{199}(20,\cdot)\) \(\chi_{199}(23,\cdot)\) \(\chi_{199}(26,\cdot)\) \(\chi_{199}(29,\cdot)\) \(\chi_{199}(31,\cdot)\) \(\chi_{199}(32,\cdot)\) \(\chi_{199}(33,\cdot)\) \(\chi_{199}(35,\cdot)\) \(\chi_{199}(36,\cdot)\) \(\chi_{199}(45,\cdot)\) \(\chi_{199}(46,\cdot)\) \(\chi_{199}(47,\cdot)\) \(\chi_{199}(49,\cdot)\) \(\chi_{199}(50,\cdot)\) \(\chi_{199}(51,\cdot)\) \(\chi_{199}(53,\cdot)\) \(\chi_{199}(56,\cdot)\) \(\chi_{199}(57,\cdot)\) \(\chi_{199}(65,\cdot)\) \(\chi_{199}(66,\cdot)\) \(\chi_{199}(70,\cdot)\) \(\chi_{199}(72,\cdot)\) ...

Values on generators

\(3\) → \(e\left(\frac{92}{99}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{50}{99}\right)\)\(e\left(\frac{92}{99}\right)\)\(e\left(\frac{1}{99}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{43}{99}\right)\)\(e\left(\frac{95}{99}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{85}{99}\right)\)\(e\left(\frac{74}{99}\right)\)\(e\left(\frac{7}{11}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{99})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 199 }(50,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{199}(50,\cdot)) = \sum_{r\in \Z/199\Z} \chi_{199}(50,r) e\left(\frac{2r}{199}\right) = -13.5244702072+4.0110728756i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 199 }(50,·),\chi_{ 199 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{199}(50,\cdot),\chi_{199}(1,\cdot)) = \sum_{r\in \Z/199\Z} \chi_{199}(50,r) \chi_{199}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 199 }(50,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{199}(50,·)) = \sum_{r \in \Z/199\Z} \chi_{199}(50,r) e\left(\frac{1 r + 2 r^{-1}}{199}\right) = 0.1840820423+-11.6008629155i \)