Properties

Conductor 103
Order 17
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 103.e

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(103)
 
sage: chi = H[72]
 
pari: [g,chi] = znchar(Mod(72,103))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 103
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 17
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 103.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 ]
 

\(\chi_{103}(8,\cdot)\) \(\chi_{103}(9,\cdot)\) \(\chi_{103}(13,\cdot)\) \(\chi_{103}(14,\cdot)\) \(\chi_{103}(23,\cdot)\) \(\chi_{103}(30,\cdot)\) \(\chi_{103}(34,\cdot)\) \(\chi_{103}(61,\cdot)\) \(\chi_{103}(64,\cdot)\) \(\chi_{103}(66,\cdot)\) \(\chi_{103}(72,\cdot)\) \(\chi_{103}(76,\cdot)\) \(\chi_{103}(79,\cdot)\) \(\chi_{103}(81,\cdot)\) \(\chi_{103}(93,\cdot)\) \(\chi_{103}(100,\cdot)\)

Values on generators

\(5\) → \(e\left(\frac{1}{17}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{3}{17}\right)\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{4}{17}\right)\)\(e\left(\frac{13}{17}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{10}{17}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{17})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 103 }(72,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{103}(72,\cdot)) = \sum_{r\in \Z/103\Z} \chi_{103}(72,r) e\left(\frac{2r}{103}\right) = -6.4956166673+-7.7978820273i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 103 }(72,·),\chi_{ 103 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{103}(72,\cdot),\chi_{103}(1,\cdot)) = \sum_{r\in \Z/103\Z} \chi_{103}(72,r) \chi_{103}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 103 }(72,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{103}(72,·)) = \sum_{r \in \Z/103\Z} \chi_{103}(72,r) e\left(\frac{1 r + 2 r^{-1}}{103}\right) = 2.6977382822+-9.4815665709i \)