Properties

Conductor 191
Order 19
Real No
Primitive Yes
Parity Even
Orbit Label 191.e

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[121]
pari: [g,chi] = znchar(Mod(121,191))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 191
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 19
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.e
Orbit index = 5

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 ]

\(\chi_{191}(5,\cdot)\) \(\chi_{191}(6,\cdot)\) \(\chi_{191}(25,\cdot)\) \(\chi_{191}(30,\cdot)\) \(\chi_{191}(32,\cdot)\) \(\chi_{191}(36,\cdot)\) \(\chi_{191}(52,\cdot)\) \(\chi_{191}(69,\cdot)\) \(\chi_{191}(107,\cdot)\) \(\chi_{191}(121,\cdot)\) \(\chi_{191}(125,\cdot)\) \(\chi_{191}(136,\cdot)\) \(\chi_{191}(150,\cdot)\) \(\chi_{191}(153,\cdot)\) \(\chi_{191}(154,\cdot)\) \(\chi_{191}(160,\cdot)\) \(\chi_{191}(177,\cdot)\) \(\chi_{191}(180,\cdot)\)

Values on generators

\(19\) → \(e\left(\frac{17}{19}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{7}{19}\right)\)\(e\left(\frac{15}{19}\right)\)\(e\left(\frac{14}{19}\right)\)\(e\left(\frac{14}{19}\right)\)\(e\left(\frac{3}{19}\right)\)\(1\)\(e\left(\frac{2}{19}\right)\)\(e\left(\frac{11}{19}\right)\)\(e\left(\frac{2}{19}\right)\)\(e\left(\frac{1}{19}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{19})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 191 }(121,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{191}(121,\cdot)) = \sum_{r\in \Z/191\Z} \chi_{191}(121,r) e\left(\frac{2r}{191}\right) = -13.4737191499+3.0755312173i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 191 }(121,·),\chi_{ 191 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{191}(121,\cdot),\chi_{191}(1,\cdot)) = \sum_{r\in \Z/191\Z} \chi_{191}(121,r) \chi_{191}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 191 }(121,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{191}(121,·)) = \sum_{r \in \Z/191\Z} \chi_{191}(121,r) e\left(\frac{1 r + 2 r^{-1}}{191}\right) = 4.968950827+11.3280668625i \)