Properties

Conductor 373
Order 62
Real No
Primitive Yes
Parity Even
Orbit Label 373.h

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[22]
 
pari: [g,chi] = znchar(Mod(22,373))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 373
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 62
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.h
Orbit index = 8

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 ]
 

\(\chi_{373}(7,\cdot)\) \(\chi_{373}(13,\cdot)\) \(\chi_{373}(17,\cdot)\) \(\chi_{373}(22,\cdot)\) \(\chi_{373}(27,\cdot)\) \(\chi_{373}(31,\cdot)\) \(\chi_{373}(55,\cdot)\) \(\chi_{373}(64,\cdot)\) \(\chi_{373}(84,\cdot)\) \(\chi_{373}(87,\cdot)\) \(\chi_{373}(137,\cdot)\) \(\chi_{373}(152,\cdot)\) \(\chi_{373}(156,\cdot)\) \(\chi_{373}(158,\cdot)\) \(\chi_{373}(160,\cdot)\) \(\chi_{373}(184,\cdot)\) \(\chi_{373}(204,\cdot)\) \(\chi_{373}(210,\cdot)\) \(\chi_{373}(219,\cdot)\) \(\chi_{373}(229,\cdot)\) \(\chi_{373}(254,\cdot)\) \(\chi_{373}(262,\cdot)\) \(\chi_{373}(264,\cdot)\) \(\chi_{373}(282,\cdot)\) \(\chi_{373}(287,\cdot)\) \(\chi_{373}(298,\cdot)\) \(\chi_{373}(324,\cdot)\) \(\chi_{373}(332,\cdot)\) \(\chi_{373}(343,\cdot)\) \(\chi_{373}(361,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{41}{62}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{41}{62}\right)\)\(e\left(\frac{12}{31}\right)\)\(e\left(\frac{10}{31}\right)\)\(e\left(\frac{47}{62}\right)\)\(e\left(\frac{3}{62}\right)\)\(e\left(\frac{29}{31}\right)\)\(e\left(\frac{61}{62}\right)\)\(e\left(\frac{24}{31}\right)\)\(e\left(\frac{13}{31}\right)\)\(e\left(\frac{1}{62}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{31})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 373 }(22,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{373}(22,\cdot)) = \sum_{r\in \Z/373\Z} \chi_{373}(22,r) e\left(\frac{2r}{373}\right) = 16.4181570435+10.1707482171i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 373 }(22,·),\chi_{ 373 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{373}(22,\cdot),\chi_{373}(1,\cdot)) = \sum_{r\in \Z/373\Z} \chi_{373}(22,r) \chi_{373}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 373 }(22,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{373}(22,·)) = \sum_{r \in \Z/373\Z} \chi_{373}(22,r) e\left(\frac{1 r + 2 r^{-1}}{373}\right) = -9.3052829801+16.7649078596i \)