Properties

Conductor 69
Order 22
Real No
Primitive No
Parity Odd
Orbit Label 276.n

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(276)
 
sage: chi = H[29]
 
pari: [g,chi] = znchar(Mod(29,276))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 69
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 = Odd
Orbit label = 276.n
Orbit index = 14

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 ]
 

\(\chi_{276}(29,\cdot)\) \(\chi_{276}(41,\cdot)\) \(\chi_{276}(77,\cdot)\) \(\chi_{276}(101,\cdot)\) \(\chi_{276}(173,\cdot)\) \(\chi_{276}(197,\cdot)\) \(\chi_{276}(209,\cdot)\) \(\chi_{276}(233,\cdot)\) \(\chi_{276}(257,\cdot)\) \(\chi_{276}(269,\cdot)\)

Inducing primitive character

\(\chi_{69}(29,\cdot)\)

Values on generators

\((139,185,97)\) → \((1,-1,e\left(\frac{9}{11}\right))\)

Values

-11571113171925293135
\(-1\)\(1\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{19}{22}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 276 }(29,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{276}(29,\cdot)) = \sum_{r\in \Z/276\Z} \chi_{276}(29,r) e\left(\frac{r}{138}\right) = -12.8241144245+10.5613488356i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 276 }(29,·),\chi_{ 276 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{276}(29,\cdot),\chi_{276}(1,\cdot)) = \sum_{r\in \Z/276\Z} \chi_{276}(29,r) \chi_{276}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 276 }(29,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{276}(29,·)) = \sum_{r \in \Z/276\Z} \chi_{276}(29,r) e\left(\frac{1 r + 2 r^{-1}}{276}\right) = 0.0 \)