Properties

Conductor 373
Order 124
Real No
Primitive Yes
Parity Odd
Orbit Label 373.j

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[18]
 
pari: [g,chi] = znchar(Mod(18,373))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 373
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 124
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 = Odd
Orbit label = 373.j
Orbit index = 10

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 ]
 

\(\chi_{373}(8,\cdot)\) \(\chi_{373}(18,\cdot)\) \(\chi_{373}(19,\cdot)\) \(\chi_{373}(20,\cdot)\) \(\chi_{373}(23,\cdot)\) \(\chi_{373}(33,\cdot)\) \(\chi_{373}(45,\cdot)\) \(\chi_{373}(50,\cdot)\) \(\chi_{373}(56,\cdot)\) \(\chi_{373}(58,\cdot)\) \(\chi_{373}(67,\cdot)\) \(\chi_{373}(74,\cdot)\) \(\chi_{373}(96,\cdot)\) \(\chi_{373}(97,\cdot)\) \(\chi_{373}(113,\cdot)\) \(\chi_{373}(125,\cdot)\) \(\chi_{373}(126,\cdot)\) \(\chi_{373}(129,\cdot)\) \(\chi_{373}(133,\cdot)\) \(\chi_{373}(136,\cdot)\) \(\chi_{373}(139,\cdot)\) \(\chi_{373}(140,\cdot)\) \(\chi_{373}(142,\cdot)\) \(\chi_{373}(145,\cdot)\) \(\chi_{373}(146,\cdot)\) \(\chi_{373}(157,\cdot)\) \(\chi_{373}(161,\cdot)\) \(\chi_{373}(167,\cdot)\) \(\chi_{373}(176,\cdot)\) \(\chi_{373}(185,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{35}{124}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{35}{124}\right)\)\(e\left(\frac{11}{62}\right)\)\(e\left(\frac{35}{62}\right)\)\(e\left(\frac{87}{124}\right)\)\(e\left(\frac{57}{124}\right)\)\(e\left(\frac{55}{62}\right)\)\(e\left(\frac{105}{124}\right)\)\(e\left(\frac{11}{31}\right)\)\(e\left(\frac{61}{62}\right)\)\(e\left(\frac{19}{124}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{124})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 373 }(18,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{373}(18,\cdot)) = \sum_{r\in \Z/373\Z} \chi_{373}(18,r) e\left(\frac{2r}{373}\right) = 12.1993575263+-14.9724973183i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 373 }(18,·),\chi_{ 373 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{373}(18,\cdot),\chi_{373}(1,\cdot)) = \sum_{r\in \Z/373\Z} \chi_{373}(18,r) \chi_{373}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 373 }(18,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{373}(18,·)) = \sum_{r \in \Z/373\Z} \chi_{373}(18,r) e\left(\frac{1 r + 2 r^{-1}}{373}\right) = 17.364136031+-14.1585882792i \)