Properties

Conductor 177
Order 58
Real No
Primitive No
Parity Even
Orbit Label 354.g

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(354)
 
sage: chi = H[11]
 
pari: [g,chi] = znchar(Mod(11,354))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 177
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 58
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 = 354.g
Orbit index = 7

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 ]
 

\(\chi_{354}(11,\cdot)\) \(\chi_{354}(23,\cdot)\) \(\chi_{354}(47,\cdot)\) \(\chi_{354}(65,\cdot)\) \(\chi_{354}(77,\cdot)\) \(\chi_{354}(83,\cdot)\) \(\chi_{354}(89,\cdot)\) \(\chi_{354}(101,\cdot)\) \(\chi_{354}(113,\cdot)\) \(\chi_{354}(131,\cdot)\) \(\chi_{354}(149,\cdot)\) \(\chi_{354}(155,\cdot)\) \(\chi_{354}(161,\cdot)\) \(\chi_{354}(173,\cdot)\) \(\chi_{354}(179,\cdot)\) \(\chi_{354}(185,\cdot)\) \(\chi_{354}(191,\cdot)\) \(\chi_{354}(209,\cdot)\) \(\chi_{354}(215,\cdot)\) \(\chi_{354}(221,\cdot)\) \(\chi_{354}(227,\cdot)\) \(\chi_{354}(233,\cdot)\) \(\chi_{354}(269,\cdot)\) \(\chi_{354}(275,\cdot)\) \(\chi_{354}(305,\cdot)\) \(\chi_{354}(329,\cdot)\) \(\chi_{354}(335,\cdot)\) \(\chi_{354}(347,\cdot)\)

Inducing primitive character

\(\chi_{177}(11,\cdot)\)

Values on generators

\((119,61)\) → \((-1,e\left(\frac{25}{58}\right))\)

Values

-11571113171923252931
\(1\)\(1\)\(e\left(\frac{5}{58}\right)\)\(e\left(\frac{22}{29}\right)\)\(e\left(\frac{8}{29}\right)\)\(e\left(\frac{23}{58}\right)\)\(e\left(\frac{43}{58}\right)\)\(e\left(\frac{11}{29}\right)\)\(e\left(\frac{28}{29}\right)\)\(e\left(\frac{5}{29}\right)\)\(e\left(\frac{33}{58}\right)\)\(e\left(\frac{7}{58}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{29})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 354 }(11,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{354}(11,\cdot)) = \sum_{r\in \Z/354\Z} \chi_{354}(11,r) e\left(\frac{r}{177}\right) = 9.5598971282+-9.2524789596i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 354 }(11,·),\chi_{ 354 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{354}(11,\cdot),\chi_{354}(1,\cdot)) = \sum_{r\in \Z/354\Z} \chi_{354}(11,r) \chi_{354}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 354 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{354}(11,·)) = \sum_{r \in \Z/354\Z} \chi_{354}(11,r) e\left(\frac{1 r + 2 r^{-1}}{354}\right) = 0.0 \)