Properties

Conductor 191
Order 95
Real No
Primitive Yes
Parity Even
Orbit Label 191.g

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(191)
sage: chi = H[13]
pari: [g,chi] = znchar(Mod(13,191))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 191
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 95
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 = 191.g
Orbit index = 7

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 ]

\(\chi_{191}(2,\cdot)\) \(\chi_{191}(3,\cdot)\) \(\chi_{191}(4,\cdot)\) \(\chi_{191}(8,\cdot)\) \(\chi_{191}(9,\cdot)\) \(\chi_{191}(10,\cdot)\) \(\chi_{191}(12,\cdot)\) \(\chi_{191}(13,\cdot)\) \(\chi_{191}(15,\cdot)\) \(\chi_{191}(16,\cdot)\) \(\chi_{191}(17,\cdot)\) \(\chi_{191}(18,\cdot)\) \(\chi_{191}(20,\cdot)\) \(\chi_{191}(23,\cdot)\) \(\chi_{191}(24,\cdot)\) \(\chi_{191}(26,\cdot)\) \(\chi_{191}(27,\cdot)\) \(\chi_{191}(34,\cdot)\) \(\chi_{191}(40,\cdot)\) \(\chi_{191}(43,\cdot)\) \(\chi_{191}(45,\cdot)\) \(\chi_{191}(46,\cdot)\) \(\chi_{191}(48,\cdot)\) \(\chi_{191}(50,\cdot)\) \(\chi_{191}(51,\cdot)\) \(\chi_{191}(54,\cdot)\) \(\chi_{191}(59,\cdot)\) \(\chi_{191}(60,\cdot)\) \(\chi_{191}(64,\cdot)\) \(\chi_{191}(65,\cdot)\) ...

Values on generators

\(19\) → \(e\left(\frac{56}{95}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{89}{95}\right)\)\(e\left(\frac{36}{95}\right)\)\(e\left(\frac{83}{95}\right)\)\(e\left(\frac{9}{19}\right)\)\(e\left(\frac{6}{19}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{77}{95}\right)\)\(e\left(\frac{72}{95}\right)\)\(e\left(\frac{39}{95}\right)\)\(e\left(\frac{2}{19}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{95})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 191 }(13,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{191}(13,\cdot)) = \sum_{r\in \Z/191\Z} \chi_{191}(13,r) e\left(\frac{2r}{191}\right) = 12.618627056+-5.6365105538i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 191 }(13,·),\chi_{ 191 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{191}(13,\cdot),\chi_{191}(1,\cdot)) = \sum_{r\in \Z/191\Z} \chi_{191}(13,r) \chi_{191}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 191 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{191}(13,·)) = \sum_{r \in \Z/191\Z} \chi_{191}(13,r) e\left(\frac{1 r + 2 r^{-1}}{191}\right) = -11.2087013342+2.25364243i \)