Properties

Label 804.259
Modulus $804$
Conductor $268$
Order $22$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(804, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,0,15]))
 
pari: [g,chi] = znchar(Mod(259,804))
 

Basic properties

Modulus: \(804\)
Conductor: \(268\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{268}(259,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 804.u

\(\chi_{804}(43,\cdot)\) \(\chi_{804}(139,\cdot)\) \(\chi_{804}(187,\cdot)\) \(\chi_{804}(259,\cdot)\) \(\chi_{804}(271,\cdot)\) \(\chi_{804}(295,\cdot)\) \(\chi_{804}(343,\cdot)\) \(\chi_{804}(511,\cdot)\) \(\chi_{804}(655,\cdot)\) \(\chi_{804}(715,\cdot)\)

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: 22.22.933750314983937236210361829997141120063111168.1

Values on generators

\((403,269,337)\) → \((-1,1,e\left(\frac{15}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 804 }(259, a) \) \(1\)\(1\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(1\)\(e\left(\frac{6}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 804 }(259,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 804 }(259,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 804 }(259,·),\chi_{ 804 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 804 }(259,·)) \;\) at \(\; a,b = \) e.g. 1,2