Properties

Conductor 373
Order 93
Real No
Primitive Yes
Parity Even
Orbit Label 373.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(373)
 
sage: chi = H[170]
 
pari: [g,chi] = znchar(Mod(170,373))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 373
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 93
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.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 ]
 

\(\chi_{373}(9,\cdot)\) \(\chi_{373}(16,\cdot)\) \(\chi_{373}(21,\cdot)\) \(\chi_{373}(28,\cdot)\) \(\chi_{373}(29,\cdot)\) \(\chi_{373}(38,\cdot)\) \(\chi_{373}(39,\cdot)\) \(\chi_{373}(40,\cdot)\) \(\chi_{373}(46,\cdot)\) \(\chi_{373}(51,\cdot)\) \(\chi_{373}(52,\cdot)\) \(\chi_{373}(66,\cdot)\) \(\chi_{373}(68,\cdot)\) \(\chi_{373}(70,\cdot)\) \(\chi_{373}(73,\cdot)\) \(\chi_{373}(81,\cdot)\) \(\chi_{373}(83,\cdot)\) \(\chi_{373}(93,\cdot)\) \(\chi_{373}(94,\cdot)\) \(\chi_{373}(95,\cdot)\) \(\chi_{373}(100,\cdot)\) \(\chi_{373}(101,\cdot)\) \(\chi_{373}(107,\cdot)\) \(\chi_{373}(108,\cdot)\) \(\chi_{373}(115,\cdot)\) \(\chi_{373}(124,\cdot)\) \(\chi_{373}(130,\cdot)\) \(\chi_{373}(148,\cdot)\) \(\chi_{373}(165,\cdot)\) \(\chi_{373}(170,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{74}{93}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{74}{93}\right)\)\(e\left(\frac{35}{93}\right)\)\(e\left(\frac{55}{93}\right)\)\(e\left(\frac{44}{93}\right)\)\(e\left(\frac{16}{93}\right)\)\(e\left(\frac{17}{31}\right)\)\(e\left(\frac{12}{31}\right)\)\(e\left(\frac{70}{93}\right)\)\(e\left(\frac{25}{93}\right)\)\(e\left(\frac{88}{93}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 373 }(170,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{373}(170,\cdot)) = \sum_{r\in \Z/373\Z} \chi_{373}(170,r) e\left(\frac{2r}{373}\right) = 5.779029146+18.4283157703i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 373 }(170,·),\chi_{ 373 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{373}(170,\cdot),\chi_{373}(1,\cdot)) = \sum_{r\in \Z/373\Z} \chi_{373}(170,r) \chi_{373}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 373 }(170,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{373}(170,·)) = \sum_{r \in \Z/373\Z} \chi_{373}(170,r) e\left(\frac{1 r + 2 r^{-1}}{373}\right) = -29.0952489837+21.7455952787i \)