Properties

Conductor 268
Order 22
Real No
Primitive No
Parity Even
Orbit Label 804.u

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(804)
 
sage: chi = H[43]
 
pari: [g,chi] = znchar(Mod(43,804))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 268
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 22
Real = No
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = No
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 804.u
Orbit index = 21

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 ]
 

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

Inducing primitive character

\(\chi_{268}(43,\cdot)\)

Values on generators

\((403,269,337)\) → \((-1,1,e\left(\frac{3}{22}\right))\)

Values

-11571113171923252931
\(1\)\(1\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{1}{11}\right)\)\(1\)\(e\left(\frac{10}{11}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{11})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 804 }(43,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{804}(43,\cdot)) = \sum_{r\in \Z/804\Z} \chi_{804}(43,r) e\left(\frac{r}{402}\right) = -0.0 \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 804 }(43,·),\chi_{ 804 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{804}(43,\cdot),\chi_{804}(1,\cdot)) = \sum_{r\in \Z/804\Z} \chi_{804}(43,r) \chi_{804}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 804 }(43,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{804}(43,·)) = \sum_{r \in \Z/804\Z} \chi_{804}(43,r) e\left(\frac{1 r + 2 r^{-1}}{804}\right) = -34.2846576923+-15.6572785409i \)