Properties

Label 5077.7
Modulus $5077$
Conductor $5077$
Order $1269$
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([686]))
 
pari: [g,chi] = znchar(Mod(7,5077))
 

Basic properties

Modulus: \(5077\)
Conductor: \(5077\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1269\)
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.u

\(\chi_{5077}(7,\cdot)\) \(\chi_{5077}(12,\cdot)\) \(\chi_{5077}(16,\cdot)\) \(\chi_{5077}(26,\cdot)\) \(\chi_{5077}(30,\cdot)\) \(\chi_{5077}(46,\cdot)\) \(\chi_{5077}(49,\cdot)\) \(\chi_{5077}(57,\cdot)\) \(\chi_{5077}(59,\cdot)\) \(\chi_{5077}(62,\cdot)\) \(\chi_{5077}(63,\cdot)\) \(\chi_{5077}(76,\cdot)\) \(\chi_{5077}(82,\cdot)\) \(\chi_{5077}(86,\cdot)\) \(\chi_{5077}(88,\cdot)\) \(\chi_{5077}(100,\cdot)\) \(\chi_{5077}(102,\cdot)\) \(\chi_{5077}(108,\cdot)\) \(\chi_{5077}(112,\cdot)\) \(\chi_{5077}(131,\cdot)\) \(\chi_{5077}(136,\cdot)\) \(\chi_{5077}(141,\cdot)\) \(\chi_{5077}(143,\cdot)\) \(\chi_{5077}(144,\cdot)\) \(\chi_{5077}(155,\cdot)\) \(\chi_{5077}(163,\cdot)\) \(\chi_{5077}(165,\cdot)\) \(\chi_{5077}(174,\cdot)\) \(\chi_{5077}(182,\cdot)\) \(\chi_{5077}(183,\cdot)\) ...

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 1269 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{343}{1269}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 5077 }(7, a) \) \(1\)\(1\)\(e\left(\frac{343}{1269}\right)\)\(e\left(\frac{5}{423}\right)\)\(e\left(\frac{686}{1269}\right)\)\(e\left(\frac{13}{47}\right)\)\(e\left(\frac{358}{1269}\right)\)\(e\left(\frac{1066}{1269}\right)\)\(e\left(\frac{343}{423}\right)\)\(e\left(\frac{10}{423}\right)\)\(e\left(\frac{694}{1269}\right)\)\(e\left(\frac{98}{1269}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5077 }(7,a) \;\) at \(\;a = \) e.g. 2