Properties

Conductor 373
Order 186
Real No
Primitive Yes
Parity Even
Orbit Label 373.k

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(373)
 
sage: chi = H[10]
 
pari: [g,chi] = znchar(Mod(10,373))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 373
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 186
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 = 373.k
Orbit index = 11

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 ]
 

\(\chi_{373}(3,\cdot)\) \(\chi_{373}(4,\cdot)\) \(\chi_{373}(10,\cdot)\) \(\chi_{373}(25,\cdot)\) \(\chi_{373}(36,\cdot)\) \(\chi_{373}(37,\cdot)\) \(\chi_{373}(48,\cdot)\) \(\chi_{373}(59,\cdot)\) \(\chi_{373}(63,\cdot)\) \(\chi_{373}(71,\cdot)\) \(\chi_{373}(90,\cdot)\) \(\chi_{373}(103,\cdot)\) \(\chi_{373}(106,\cdot)\) \(\chi_{373}(112,\cdot)\) \(\chi_{373}(114,\cdot)\) \(\chi_{373}(116,\cdot)\) \(\chi_{373}(117,\cdot)\) \(\chi_{373}(120,\cdot)\) \(\chi_{373}(121,\cdot)\) \(\chi_{373}(122,\cdot)\) \(\chi_{373}(123,\cdot)\) \(\chi_{373}(134,\cdot)\) \(\chi_{373}(138,\cdot)\) \(\chi_{373}(147,\cdot)\) \(\chi_{373}(153,\cdot)\) \(\chi_{373}(164,\cdot)\) \(\chi_{373}(181,\cdot)\) \(\chi_{373}(194,\cdot)\) \(\chi_{373}(196,\cdot)\) \(\chi_{373}(198,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{85}{186}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{85}{186}\right)\)\(e\left(\frac{71}{93}\right)\)\(e\left(\frac{85}{93}\right)\)\(e\left(\frac{43}{186}\right)\)\(e\left(\frac{41}{186}\right)\)\(e\left(\frac{15}{31}\right)\)\(e\left(\frac{23}{62}\right)\)\(e\left(\frac{49}{93}\right)\)\(e\left(\frac{64}{93}\right)\)\(e\left(\frac{179}{186}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{93})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 373 }(10,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{373}(10,\cdot)) = \sum_{r\in \Z/373\Z} \chi_{373}(10,r) e\left(\frac{2r}{373}\right) = 6.7052604163+-18.1118602785i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 373 }(10,·),\chi_{ 373 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{373}(10,\cdot),\chi_{373}(1,\cdot)) = \sum_{r\in \Z/373\Z} \chi_{373}(10,r) \chi_{373}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 373 }(10,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{373}(10,·)) = \sum_{r \in \Z/373\Z} \chi_{373}(10,r) e\left(\frac{1 r + 2 r^{-1}}{373}\right) = 2.8291671442+20.8102478171i \)