Properties

Conductor 67
Order 11
Real No
Primitive No
Parity Even
Orbit Label 201.i

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(201)
 
sage: chi = H[22]
 
pari: [g,chi] = znchar(Mod(22,201))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 67
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 11
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 = 201.i
Orbit index = 9

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 ]
 

\(\chi_{201}(22,\cdot)\) \(\chi_{201}(25,\cdot)\) \(\chi_{201}(40,\cdot)\) \(\chi_{201}(64,\cdot)\) \(\chi_{201}(76,\cdot)\) \(\chi_{201}(82,\cdot)\) \(\chi_{201}(91,\cdot)\) \(\chi_{201}(148,\cdot)\) \(\chi_{201}(193,\cdot)\) \(\chi_{201}(196,\cdot)\)

Inducing primitive character

\(\chi_{67}(22,\cdot)\)

Values on generators

\((68,136)\) → \((1,e\left(\frac{10}{11}\right))\)

Values

-11245781011131416
\(1\)\(1\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{7}{11}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 201 }(22,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{201}(22,\cdot)) = \sum_{r\in \Z/201\Z} \chi_{201}(22,r) e\left(\frac{2r}{201}\right) = 2.9064282978+-7.6519719386i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 201 }(22,·),\chi_{ 201 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{201}(22,\cdot),\chi_{201}(1,\cdot)) = \sum_{r\in \Z/201\Z} \chi_{201}(22,r) \chi_{201}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 201 }(22,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{201}(22,·)) = \sum_{r \in \Z/201\Z} \chi_{201}(22,r) e\left(\frac{1 r + 2 r^{-1}}{201}\right) = 14.4368421657+-4.2390393342i \)