Properties

Conductor 373
Order 372
Real No
Primitive Yes
Parity Odd
Orbit Label 373.l

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[15]
 
pari: [g,chi] = znchar(Mod(15,373))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 373
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 372
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.l
Orbit index = 12

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 ]
 

\(\chi_{373}(2,\cdot)\) \(\chi_{373}(5,\cdot)\) \(\chi_{373}(6,\cdot)\) \(\chi_{373}(11,\cdot)\) \(\chi_{373}(14,\cdot)\) \(\chi_{373}(15,\cdot)\) \(\chi_{373}(24,\cdot)\) \(\chi_{373}(26,\cdot)\) \(\chi_{373}(32,\cdot)\) \(\chi_{373}(34,\cdot)\) \(\chi_{373}(35,\cdot)\) \(\chi_{373}(42,\cdot)\) \(\chi_{373}(43,\cdot)\) \(\chi_{373}(44,\cdot)\) \(\chi_{373}(47,\cdot)\) \(\chi_{373}(53,\cdot)\) \(\chi_{373}(54,\cdot)\) \(\chi_{373}(57,\cdot)\) \(\chi_{373}(60,\cdot)\) \(\chi_{373}(61,\cdot)\) \(\chi_{373}(62,\cdot)\) \(\chi_{373}(65,\cdot)\) \(\chi_{373}(72,\cdot)\) \(\chi_{373}(76,\cdot)\) \(\chi_{373}(77,\cdot)\) \(\chi_{373}(78,\cdot)\) \(\chi_{373}(79,\cdot)\) \(\chi_{373}(80,\cdot)\) \(\chi_{373}(82,\cdot)\) \(\chi_{373}(85,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{35}{372}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{35}{372}\right)\)\(e\left(\frac{73}{186}\right)\)\(e\left(\frac{35}{186}\right)\)\(e\left(\frac{335}{372}\right)\)\(e\left(\frac{181}{372}\right)\)\(e\left(\frac{39}{62}\right)\)\(e\left(\frac{35}{124}\right)\)\(e\left(\frac{73}{93}\right)\)\(e\left(\frac{185}{186}\right)\)\(e\left(\frac{19}{372}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{372})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 373 }(15,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{373}(15,\cdot)) = \sum_{r\in \Z/373\Z} \chi_{373}(15,r) e\left(\frac{2r}{373}\right) = 4.6258820534+18.7510323777i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 373 }(15,·),\chi_{ 373 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{373}(15,\cdot),\chi_{373}(1,\cdot)) = \sum_{r\in \Z/373\Z} \chi_{373}(15,r) \chi_{373}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 373 }(15,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{373}(15,·)) = \sum_{r \in \Z/373\Z} \chi_{373}(15,r) e\left(\frac{1 r + 2 r^{-1}}{373}\right) = -2.3020454362+7.5600876209i \)