Properties

Conductor 163
Order 81
Real No
Primitive Yes
Parity Even
Orbit Label 163.i

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(163)
 
sage: chi = H[26]
 
pari: [g,chi] = znchar(Mod(26,163))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 163
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 81
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 = 163.i
Orbit index = 9

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 ]
 

\(\chi_{163}(4,\cdot)\) \(\chi_{163}(9,\cdot)\) \(\chi_{163}(10,\cdot)\) \(\chi_{163}(14,\cdot)\) \(\chi_{163}(15,\cdot)\) \(\chi_{163}(16,\cdot)\) \(\chi_{163}(24,\cdot)\) \(\chi_{163}(26,\cdot)\) \(\chi_{163}(33,\cdot)\) \(\chi_{163}(34,\cdot)\) \(\chi_{163}(35,\cdot)\) \(\chi_{163}(39,\cdot)\) \(\chi_{163}(41,\cdot)\) \(\chi_{163}(43,\cdot)\) \(\chi_{163}(46,\cdot)\) \(\chi_{163}(47,\cdot)\) \(\chi_{163}(49,\cdot)\) \(\chi_{163}(51,\cdot)\) \(\chi_{163}(54,\cdot)\) \(\chi_{163}(55,\cdot)\) \(\chi_{163}(56,\cdot)\) \(\chi_{163}(57,\cdot)\) \(\chi_{163}(60,\cdot)\) \(\chi_{163}(62,\cdot)\) \(\chi_{163}(69,\cdot)\) \(\chi_{163}(71,\cdot)\) \(\chi_{163}(74,\cdot)\) \(\chi_{163}(81,\cdot)\) \(\chi_{163}(83,\cdot)\) \(\chi_{163}(84,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{26}{81}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{26}{81}\right)\)\(e\left(\frac{34}{81}\right)\)\(e\left(\frac{52}{81}\right)\)\(e\left(\frac{22}{27}\right)\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{35}{81}\right)\)\(e\left(\frac{26}{27}\right)\)\(e\left(\frac{68}{81}\right)\)\(e\left(\frac{11}{81}\right)\)\(e\left(\frac{7}{81}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{81})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 163 }(26,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{163}(26,\cdot)) = \sum_{r\in \Z/163\Z} \chi_{163}(26,r) e\left(\frac{2r}{163}\right) = 12.6957654854+1.3481612441i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 163 }(26,·),\chi_{ 163 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{163}(26,\cdot),\chi_{163}(1,\cdot)) = \sum_{r\in \Z/163\Z} \chi_{163}(26,r) \chi_{163}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 163 }(26,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{163}(26,·)) = \sum_{r \in \Z/163\Z} \chi_{163}(26,r) e\left(\frac{1 r + 2 r^{-1}}{163}\right) = 5.1476781498+8.1673474034i \)