Properties

Label 5077.28
Modulus $5077$
Conductor $5077$
Order $846$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5077, base_ring=CyclotomicField(846))
 
M = H._module
 
chi = DirichletCharacter(H, M([229]))
 
pari: [g,chi] = znchar(Mod(28,5077))
 

Basic properties

Modulus: \(5077\)
Conductor: \(5077\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(846\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5077.t

\(\chi_{5077}(3,\cdot)\) \(\chi_{5077}(28,\cdot)\) \(\chi_{5077}(53,\cdot)\) \(\chi_{5077}(64,\cdot)\) \(\chi_{5077}(104,\cdot)\) \(\chi_{5077}(107,\cdot)\) \(\chi_{5077}(127,\cdot)\) \(\chi_{5077}(145,\cdot)\) \(\chi_{5077}(151,\cdot)\) \(\chi_{5077}(169,\cdot)\) \(\chi_{5077}(187,\cdot)\) \(\chi_{5077}(198,\cdot)\) \(\chi_{5077}(225,\cdot)\) \(\chi_{5077}(236,\cdot)\) \(\chi_{5077}(238,\cdot)\) \(\chi_{5077}(239,\cdot)\) \(\chi_{5077}(243,\cdot)\) \(\chi_{5077}(267,\cdot)\) \(\chi_{5077}(291,\cdot)\) \(\chi_{5077}(345,\cdot)\) \(\chi_{5077}(385,\cdot)\) \(\chi_{5077}(402,\cdot)\) \(\chi_{5077}(503,\cdot)\) \(\chi_{5077}(514,\cdot)\) \(\chi_{5077}(529,\cdot)\) \(\chi_{5077}(537,\cdot)\) \(\chi_{5077}(543,\cdot)\) \(\chi_{5077}(544,\cdot)\) \(\chi_{5077}(564,\cdot)\) \(\chi_{5077}(570,\cdot)\) ...

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: $\Q(\zeta_{423})$
Fixed field: Number field defined by a degree 846 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{229}{846}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 5077 }(28, a) \) \(1\)\(1\)\(e\left(\frac{229}{846}\right)\)\(e\left(\frac{103}{141}\right)\)\(e\left(\frac{229}{423}\right)\)\(e\left(\frac{37}{94}\right)\)\(e\left(\frac{1}{846}\right)\)\(e\left(\frac{161}{423}\right)\)\(e\left(\frac{229}{282}\right)\)\(e\left(\frac{65}{141}\right)\)\(e\left(\frac{281}{423}\right)\)\(e\left(\frac{5}{846}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5077 }(28,a) \;\) at \(\;a = \) e.g. 2