Properties

Conductor 373
Order 31
Real No
Primitive Yes
Parity Even
Orbit Label 373.g

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[30]
 
pari: [g,chi] = znchar(Mod(30,373))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 373
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 31
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.g
Orbit index = 7

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 ]
 

\(\chi_{373}(12,\cdot)\) \(\chi_{373}(30,\cdot)\) \(\chi_{373}(41,\cdot)\) \(\chi_{373}(49,\cdot)\) \(\chi_{373}(75,\cdot)\) \(\chi_{373}(86,\cdot)\) \(\chi_{373}(91,\cdot)\) \(\chi_{373}(109,\cdot)\) \(\chi_{373}(111,\cdot)\) \(\chi_{373}(119,\cdot)\) \(\chi_{373}(144,\cdot)\) \(\chi_{373}(154,\cdot)\) \(\chi_{373}(163,\cdot)\) \(\chi_{373}(169,\cdot)\) \(\chi_{373}(189,\cdot)\) \(\chi_{373}(213,\cdot)\) \(\chi_{373}(215,\cdot)\) \(\chi_{373}(217,\cdot)\) \(\chi_{373}(221,\cdot)\) \(\chi_{373}(236,\cdot)\) \(\chi_{373}(286,\cdot)\) \(\chi_{373}(289,\cdot)\) \(\chi_{373}(309,\cdot)\) \(\chi_{373}(318,\cdot)\) \(\chi_{373}(342,\cdot)\) \(\chi_{373}(346,\cdot)\) \(\chi_{373}(351,\cdot)\) \(\chi_{373}(356,\cdot)\) \(\chi_{373}(360,\cdot)\) \(\chi_{373}(366,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{3}{31}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{3}{31}\right)\)\(e\left(\frac{1}{31}\right)\)\(e\left(\frac{6}{31}\right)\)\(e\left(\frac{11}{31}\right)\)\(e\left(\frac{4}{31}\right)\)\(e\left(\frac{5}{31}\right)\)\(e\left(\frac{9}{31}\right)\)\(e\left(\frac{2}{31}\right)\)\(e\left(\frac{14}{31}\right)\)\(e\left(\frac{22}{31}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 373 }(30,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{373}(30,\cdot)) = \sum_{r\in \Z/373\Z} \chi_{373}(30,r) e\left(\frac{2r}{373}\right) = 4.4182691508+18.8010344851i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 373 }(30,·),\chi_{ 373 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{373}(30,\cdot),\chi_{373}(1,\cdot)) = \sum_{r\in \Z/373\Z} \chi_{373}(30,r) \chi_{373}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 373 }(30,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{373}(30,·)) = \sum_{r \in \Z/373\Z} \chi_{373}(30,r) e\left(\frac{1 r + 2 r^{-1}}{373}\right) = -7.0479808399+-2.2113182544i \)