Properties

Label 5077.19
Modulus $5077$
Conductor $5077$
Order $2538$
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(2538))
 
M = H._module
 
chi = DirichletCharacter(H, M([2455]))
 
pari: [g,chi] = znchar(Mod(19,5077))
 

Basic properties

Modulus: \(5077\)
Conductor: \(5077\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2538\)
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.w

\(\chi_{5077}(4,\cdot)\) \(\chi_{5077}(10,\cdot)\) \(\chi_{5077}(19,\cdot)\) \(\chi_{5077}(34,\cdot)\) \(\chi_{5077}(36,\cdot)\) \(\chi_{5077}(47,\cdot)\) \(\chi_{5077}(48,\cdot)\) \(\chi_{5077}(55,\cdot)\) \(\chi_{5077}(58,\cdot)\) \(\chi_{5077}(61,\cdot)\) \(\chi_{5077}(70,\cdot)\) \(\chi_{5077}(71,\cdot)\) \(\chi_{5077}(78,\cdot)\) \(\chi_{5077}(79,\cdot)\) \(\chi_{5077}(85,\cdot)\) \(\chi_{5077}(90,\cdot)\) \(\chi_{5077}(103,\cdot)\) \(\chi_{5077}(121,\cdot)\) \(\chi_{5077}(137,\cdot)\) \(\chi_{5077}(138,\cdot)\) \(\chi_{5077}(146,\cdot)\) \(\chi_{5077}(147,\cdot)\) \(\chi_{5077}(154,\cdot)\) \(\chi_{5077}(160,\cdot)\) \(\chi_{5077}(166,\cdot)\) \(\chi_{5077}(171,\cdot)\) \(\chi_{5077}(173,\cdot)\) \(\chi_{5077}(175,\cdot)\) \(\chi_{5077}(177,\cdot)\) \(\chi_{5077}(186,\cdot)\) ...

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: $\Q(\zeta_{1269})$
Fixed field: Number field defined by a degree 2538 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{2455}{2538}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 5077 }(19, a) \) \(1\)\(1\)\(e\left(\frac{2455}{2538}\right)\)\(e\left(\frac{148}{423}\right)\)\(e\left(\frac{1186}{1269}\right)\)\(e\left(\frac{27}{94}\right)\)\(e\left(\frac{805}{2538}\right)\)\(e\left(\frac{167}{1269}\right)\)\(e\left(\frac{763}{846}\right)\)\(e\left(\frac{296}{423}\right)\)\(e\left(\frac{323}{1269}\right)\)\(e\left(\frac{2333}{2538}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5077 }(19,a) \;\) at \(\;a = \) e.g. 2