Properties

Conductor 167
Order 83
Real No
Primitive Yes
Parity Even
Orbit Label 167.c

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(167)
 
sage: chi = H[18]
 
pari: [g,chi] = znchar(Mod(18,167))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 167
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 83
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 = 167.c
Orbit index = 3

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 ]
 

\(\chi_{167}(2,\cdot)\) \(\chi_{167}(3,\cdot)\) \(\chi_{167}(4,\cdot)\) \(\chi_{167}(6,\cdot)\) \(\chi_{167}(7,\cdot)\) \(\chi_{167}(8,\cdot)\) \(\chi_{167}(9,\cdot)\) \(\chi_{167}(11,\cdot)\) \(\chi_{167}(12,\cdot)\) \(\chi_{167}(14,\cdot)\) \(\chi_{167}(16,\cdot)\) \(\chi_{167}(18,\cdot)\) \(\chi_{167}(19,\cdot)\) \(\chi_{167}(21,\cdot)\) \(\chi_{167}(22,\cdot)\) \(\chi_{167}(24,\cdot)\) \(\chi_{167}(25,\cdot)\) \(\chi_{167}(27,\cdot)\) \(\chi_{167}(28,\cdot)\) \(\chi_{167}(29,\cdot)\) \(\chi_{167}(31,\cdot)\) \(\chi_{167}(32,\cdot)\) \(\chi_{167}(33,\cdot)\) \(\chi_{167}(36,\cdot)\) \(\chi_{167}(38,\cdot)\) \(\chi_{167}(42,\cdot)\) \(\chi_{167}(44,\cdot)\) \(\chi_{167}(47,\cdot)\) \(\chi_{167}(48,\cdot)\) \(\chi_{167}(49,\cdot)\) ...

Values on generators

\(5\) → \(e\left(\frac{31}{83}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{78}{83}\right)\)\(e\left(\frac{9}{83}\right)\)\(e\left(\frac{73}{83}\right)\)\(e\left(\frac{31}{83}\right)\)\(e\left(\frac{4}{83}\right)\)\(e\left(\frac{6}{83}\right)\)\(e\left(\frac{68}{83}\right)\)\(e\left(\frac{18}{83}\right)\)\(e\left(\frac{26}{83}\right)\)\(e\left(\frac{38}{83}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{83})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 167 }(18,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{167}(18,\cdot)) = \sum_{r\in \Z/167\Z} \chi_{167}(18,r) e\left(\frac{2r}{167}\right) = 9.9274998334+8.273134053i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 167 }(18,·),\chi_{ 167 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{167}(18,\cdot),\chi_{167}(1,\cdot)) = \sum_{r\in \Z/167\Z} \chi_{167}(18,r) \chi_{167}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 167 }(18,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{167}(18,·)) = \sum_{r \in \Z/167\Z} \chi_{167}(18,r) e\left(\frac{1 r + 2 r^{-1}}{167}\right) = 17.2797031471+-3.3098385474i \)