Properties

Conductor 177
Order 58
Real No
Primitive Yes
Parity Even
Orbit Label 177.f

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(177)
 
sage: chi = H[11]
 
pari: [g,chi] = znchar(Mod(11,177))
 

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 = Yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 177.f
Orbit index = 6

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 ]
 

\(\chi_{177}(2,\cdot)\) \(\chi_{177}(8,\cdot)\) \(\chi_{177}(11,\cdot)\) \(\chi_{177}(14,\cdot)\) \(\chi_{177}(23,\cdot)\) \(\chi_{177}(32,\cdot)\) \(\chi_{177}(38,\cdot)\) \(\chi_{177}(44,\cdot)\) \(\chi_{177}(47,\cdot)\) \(\chi_{177}(50,\cdot)\) \(\chi_{177}(56,\cdot)\) \(\chi_{177}(65,\cdot)\) \(\chi_{177}(77,\cdot)\) \(\chi_{177}(83,\cdot)\) \(\chi_{177}(89,\cdot)\) \(\chi_{177}(92,\cdot)\) \(\chi_{177}(98,\cdot)\) \(\chi_{177}(101,\cdot)\) \(\chi_{177}(113,\cdot)\) \(\chi_{177}(128,\cdot)\) \(\chi_{177}(131,\cdot)\) \(\chi_{177}(149,\cdot)\) \(\chi_{177}(152,\cdot)\) \(\chi_{177}(155,\cdot)\) \(\chi_{177}(158,\cdot)\) \(\chi_{177}(161,\cdot)\) \(\chi_{177}(170,\cdot)\) \(\chi_{177}(173,\cdot)\)

Values on generators

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

Values

-11245781011131416
\(1\)\(1\)\(e\left(\frac{27}{29}\right)\)\(e\left(\frac{25}{29}\right)\)\(e\left(\frac{5}{58}\right)\)\(e\left(\frac{22}{29}\right)\)\(e\left(\frac{23}{29}\right)\)\(e\left(\frac{1}{58}\right)\)\(e\left(\frac{8}{29}\right)\)\(e\left(\frac{23}{58}\right)\)\(e\left(\frac{20}{29}\right)\)\(e\left(\frac{21}{29}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 177 }(11,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{177}(11,\cdot)) = \sum_{r\in \Z/177\Z} \chi_{177}(11,r) e\left(\frac{2r}{177}\right) = 12.5613427257+-4.3832258585i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 177 }(11,·),\chi_{ 177 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{177}(11,\cdot),\chi_{177}(1,\cdot)) = \sum_{r\in \Z/177\Z} \chi_{177}(11,r) \chi_{177}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 177 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{177}(11,·)) = \sum_{r \in \Z/177\Z} \chi_{177}(11,r) e\left(\frac{1 r + 2 r^{-1}}{177}\right) = 0.0 \)