Properties

Label 179.41
Modulus $179$
Conductor $179$
Order $178$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(179, base_ring=CyclotomicField(178))
 
M = H._module
 
chi = DirichletCharacter(H, M([155]))
 
pari: [g,chi] = znchar(Mod(41,179))
 

Basic properties

Modulus: \(179\)
Conductor: \(179\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(178\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 179.d

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

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{89})$
Fixed field: Number field defined by a degree 178 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{155}{178}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 179 }(41, a) \) \(-1\)\(1\)\(e\left(\frac{155}{178}\right)\)\(e\left(\frac{4}{89}\right)\)\(e\left(\frac{66}{89}\right)\)\(e\left(\frac{15}{89}\right)\)\(e\left(\frac{163}{178}\right)\)\(e\left(\frac{161}{178}\right)\)\(e\left(\frac{109}{178}\right)\)\(e\left(\frac{8}{89}\right)\)\(e\left(\frac{7}{178}\right)\)\(e\left(\frac{11}{178}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 179 }(41,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 179 }(41,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 179 }(41,·),\chi_{ 179 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 179 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2