Properties

Conductor 179
Order 178
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 179.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(179)
 
sage: chi = H[11]
 
pari: [g,chi] = znchar(Mod(11,179))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 179
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 178
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 = odd
Orbit label = 179.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 ]
 

\(\chi_{179}(2,\cdot)\) \(\chi_{179}(6,\cdot)\) \(\chi_{179}(7,\cdot)\) \(\chi_{179}(8,\cdot)\) \(\chi_{179}(10,\cdot)\) \(\chi_{179}(11,\cdot)\) \(\chi_{179}(18,\cdot)\) \(\chi_{179}(21,\cdot)\) \(\chi_{179}(23,\cdot)\) \(\chi_{179}(24,\cdot)\) \(\chi_{179}(26,\cdot)\) \(\chi_{179}(28,\cdot)\) \(\chi_{179}(30,\cdot)\) \(\chi_{179}(32,\cdot)\) \(\chi_{179}(33,\cdot)\) \(\chi_{179}(34,\cdot)\) \(\chi_{179}(35,\cdot)\) \(\chi_{179}(37,\cdot)\) \(\chi_{179}(38,\cdot)\) \(\chi_{179}(40,\cdot)\) \(\chi_{179}(41,\cdot)\) \(\chi_{179}(44,\cdot)\) \(\chi_{179}(50,\cdot)\) \(\chi_{179}(53,\cdot)\) \(\chi_{179}(54,\cdot)\) \(\chi_{179}(55,\cdot)\) \(\chi_{179}(58,\cdot)\) \(\chi_{179}(62,\cdot)\) \(\chi_{179}(63,\cdot)\) \(\chi_{179}(69,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{15}{178}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{15}{178}\right)\)\(e\left(\frac{9}{89}\right)\)\(e\left(\frac{15}{89}\right)\)\(e\left(\frac{56}{89}\right)\)\(e\left(\frac{33}{178}\right)\)\(e\left(\frac{73}{178}\right)\)\(e\left(\frac{45}{178}\right)\)\(e\left(\frac{18}{89}\right)\)\(e\left(\frac{127}{178}\right)\)\(e\left(\frac{47}{178}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{89})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 179 }(11,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{179}(11,\cdot)) = \sum_{r\in \Z/179\Z} \chi_{179}(11,r) e\left(\frac{2r}{179}\right) = -9.5274377882+-9.39297233i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 179 }(11,·),\chi_{ 179 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{179}(11,\cdot),\chi_{179}(1,\cdot)) = \sum_{r\in \Z/179\Z} \chi_{179}(11,r) \chi_{179}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 179 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{179}(11,·)) = \sum_{r \in \Z/179\Z} \chi_{179}(11,r) e\left(\frac{1 r + 2 r^{-1}}{179}\right) = -0.2099423823+0.7743966675i \)