Properties

Label 199.11
Modulus $199$
Conductor $199$
Order $22$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(199, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([21]))
 
pari: [g,chi] = znchar(Mod(11,199))
 

Basic properties

Modulus: \(199\)
Conductor: \(199\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 199.h

\(\chi_{199}(11,\cdot)\) \(\chi_{199}(60,\cdot)\) \(\chi_{199}(74,\cdot)\) \(\chi_{199}(78,\cdot)\) \(\chi_{199}(85,\cdot)\) \(\chi_{199}(96,\cdot)\) \(\chi_{199}(136,\cdot)\) \(\chi_{199}(137,\cdot)\) \(\chi_{199}(138,\cdot)\) \(\chi_{199}(181,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: Number field defined by a degree 22 polynomial

Values on generators

\(3\) → \(e\left(\frac{21}{22}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 199 }(11, a) \) \(-1\)\(1\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{9}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 199 }(11,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 199 }(11,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 199 }(11,·),\chi_{ 199 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 199 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2