Properties

Conductor 173
Order 43
Real No
Primitive Yes
Parity Even
Orbit Label 173.d

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[36]
pari: [g,chi] = znchar(Mod(36,173))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 173
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 43
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.d
Orbit index = 4

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 ]

\(\chi_{173}(6,\cdot)\) \(\chi_{173}(10,\cdot)\) \(\chi_{173}(14,\cdot)\) \(\chi_{173}(16,\cdot)\) \(\chi_{173}(22,\cdot)\) \(\chi_{173}(23,\cdot)\) \(\chi_{173}(29,\cdot)\) \(\chi_{173}(36,\cdot)\) \(\chi_{173}(43,\cdot)\) \(\chi_{173}(47,\cdot)\) \(\chi_{173}(51,\cdot)\) \(\chi_{173}(52,\cdot)\) \(\chi_{173}(57,\cdot)\) \(\chi_{173}(60,\cdot)\) \(\chi_{173}(81,\cdot)\) \(\chi_{173}(83,\cdot)\) \(\chi_{173}(84,\cdot)\) \(\chi_{173}(85,\cdot)\) \(\chi_{173}(95,\cdot)\) \(\chi_{173}(96,\cdot)\) \(\chi_{173}(100,\cdot)\) \(\chi_{173}(106,\cdot)\) \(\chi_{173}(109,\cdot)\) \(\chi_{173}(117,\cdot)\) \(\chi_{173}(118,\cdot)\) \(\chi_{173}(119,\cdot)\) \(\chi_{173}(124,\cdot)\) \(\chi_{173}(132,\cdot)\) \(\chi_{173}(133,\cdot)\) \(\chi_{173}(135,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{14}{43}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{14}{43}\right)\)\(e\left(\frac{34}{43}\right)\)\(e\left(\frac{28}{43}\right)\)\(e\left(\frac{30}{43}\right)\)\(e\left(\frac{5}{43}\right)\)\(e\left(\frac{40}{43}\right)\)\(e\left(\frac{42}{43}\right)\)\(e\left(\frac{25}{43}\right)\)\(e\left(\frac{1}{43}\right)\)\(e\left(\frac{21}{43}\right)\)
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)
\( \tau_{ a }( \chi_{ 173 }(36,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{173}(36,\cdot)) = \sum_{r\in \Z/173\Z} \chi_{173}(36,r) e\left(\frac{2r}{173}\right) = -10.9137470455+7.3409894038i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 173 }(36,·),\chi_{ 173 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{173}(36,\cdot),\chi_{173}(1,\cdot)) = \sum_{r\in \Z/173\Z} \chi_{173}(36,r) \chi_{173}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 173 }(36,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{173}(36,·)) = \sum_{r \in \Z/173\Z} \chi_{173}(36,r) e\left(\frac{1 r + 2 r^{-1}}{173}\right) = 4.8467962155+7.941776499i \)