Properties

Label 277.26
Modulus $277$
Conductor $277$
Order $92$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(277, base_ring=CyclotomicField(92))
 
M = H._module
 
chi = DirichletCharacter(H, M([31]))
 
pari: [g,chi] = znchar(Mod(26,277))
 

Basic properties

Modulus: \(277\)
Conductor: \(277\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(92\)
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 277.j

\(\chi_{277}(2,\cdot)\) \(\chi_{277}(8,\cdot)\) \(\chi_{277}(15,\cdot)\) \(\chi_{277}(26,\cdot)\) \(\chi_{277}(32,\cdot)\) \(\chi_{277}(33,\cdot)\) \(\chi_{277}(37,\cdot)\) \(\chi_{277}(38,\cdot)\) \(\chi_{277}(42,\cdot)\) \(\chi_{277}(51,\cdot)\) \(\chi_{277}(54,\cdot)\) \(\chi_{277}(61,\cdot)\) \(\chi_{277}(73,\cdot)\) \(\chi_{277}(82,\cdot)\) \(\chi_{277}(104,\cdot)\) \(\chi_{277}(109,\cdot)\) \(\chi_{277}(118,\cdot)\) \(\chi_{277}(125,\cdot)\) \(\chi_{277}(128,\cdot)\) \(\chi_{277}(129,\cdot)\) \(\chi_{277}(132,\cdot)\) \(\chi_{277}(138,\cdot)\) \(\chi_{277}(139,\cdot)\) \(\chi_{277}(145,\cdot)\) \(\chi_{277}(148,\cdot)\) \(\chi_{277}(149,\cdot)\) \(\chi_{277}(152,\cdot)\) \(\chi_{277}(159,\cdot)\) \(\chi_{277}(168,\cdot)\) \(\chi_{277}(173,\cdot)\) ...

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: $\Q(\zeta_{92})$
Fixed field: Number field defined by a degree 92 polynomial

Values on generators

\(5\) → \(e\left(\frac{31}{92}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 277 }(26, a) \) \(-1\)\(1\)\(e\left(\frac{49}{92}\right)\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{3}{46}\right)\)\(e\left(\frac{31}{92}\right)\)\(e\left(\frac{81}{92}\right)\)\(e\left(\frac{19}{46}\right)\)\(e\left(\frac{55}{92}\right)\)\(e\left(\frac{16}{23}\right)\)\(e\left(\frac{20}{23}\right)\)\(e\left(\frac{33}{92}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 277 }(26,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 277 }(26,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 277 }(26,·),\chi_{ 277 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 277 }(26,·)) \;\) at \(\; a,b = \) e.g. 1,2