Properties

Conductor 173
Order 86
Real No
Primitive Yes
Parity Even
Orbit Label 173.e

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(173)
 
sage: chi = H[78]
 
pari: [g,chi] = znchar(Mod(78,173))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 173
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 86
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 = 173.e
Orbit index = 5

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 ]
 

\(\chi_{173}(4,\cdot)\) \(\chi_{173}(9,\cdot)\) \(\chi_{173}(13,\cdot)\) \(\chi_{173}(15,\cdot)\) \(\chi_{173}(21,\cdot)\) \(\chi_{173}(24,\cdot)\) \(\chi_{173}(25,\cdot)\) \(\chi_{173}(31,\cdot)\) \(\chi_{173}(33,\cdot)\) \(\chi_{173}(34,\cdot)\) \(\chi_{173}(35,\cdot)\) \(\chi_{173}(37,\cdot)\) \(\chi_{173}(38,\cdot)\) \(\chi_{173}(40,\cdot)\) \(\chi_{173}(41,\cdot)\) \(\chi_{173}(49,\cdot)\) \(\chi_{173}(54,\cdot)\) \(\chi_{173}(55,\cdot)\) \(\chi_{173}(56,\cdot)\) \(\chi_{173}(64,\cdot)\) \(\chi_{173}(67,\cdot)\) \(\chi_{173}(73,\cdot)\) \(\chi_{173}(77,\cdot)\) \(\chi_{173}(78,\cdot)\) \(\chi_{173}(88,\cdot)\) \(\chi_{173}(89,\cdot)\) \(\chi_{173}(90,\cdot)\) \(\chi_{173}(92,\cdot)\) \(\chi_{173}(113,\cdot)\) \(\chi_{173}(116,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{79}{86}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{79}{86}\right)\)\(e\left(\frac{69}{86}\right)\)\(e\left(\frac{36}{43}\right)\)\(e\left(\frac{71}{86}\right)\)\(e\left(\frac{31}{43}\right)\)\(e\left(\frac{23}{86}\right)\)\(e\left(\frac{65}{86}\right)\)\(e\left(\frac{26}{43}\right)\)\(e\left(\frac{32}{43}\right)\)\(e\left(\frac{11}{86}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{43})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 173 }(78,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{173}(78,\cdot)) = \sum_{r\in \Z/173\Z} \chi_{173}(78,r) e\left(\frac{2r}{173}\right) = 3.3462773341+-12.7201583325i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 173 }(78,·),\chi_{ 173 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{173}(78,\cdot),\chi_{173}(1,\cdot)) = \sum_{r\in \Z/173\Z} \chi_{173}(78,r) \chi_{173}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 173 }(78,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{173}(78,·)) = \sum_{r \in \Z/173\Z} \chi_{173}(78,r) e\left(\frac{1 r + 2 r^{-1}}{173}\right) = -1.567459647+0.4097876496i \)