Properties

Label 6422.59
Modulus $6422$
Conductor $3211$
Order $468$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6422, base_ring=CyclotomicField(468))
 
M = H._module
 
chi = DirichletCharacter(H, M([105,26]))
 
pari: [g,chi] = znchar(Mod(59,6422))
 

Basic properties

Modulus: \(6422\)
Conductor: \(3211\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(468\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3211}(59,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6422.dn

\(\chi_{6422}(15,\cdot)\) \(\chi_{6422}(33,\cdot)\) \(\chi_{6422}(59,\cdot)\) \(\chi_{6422}(67,\cdot)\) \(\chi_{6422}(71,\cdot)\) \(\chi_{6422}(97,\cdot)\) \(\chi_{6422}(167,\cdot)\) \(\chi_{6422}(219,\cdot)\) \(\chi_{6422}(345,\cdot)\) \(\chi_{6422}(383,\cdot)\) \(\chi_{6422}(431,\cdot)\) \(\chi_{6422}(509,\cdot)\) \(\chi_{6422}(527,\cdot)\) \(\chi_{6422}(553,\cdot)\) \(\chi_{6422}(561,\cdot)\) \(\chi_{6422}(565,\cdot)\) \(\chi_{6422}(583,\cdot)\) \(\chi_{6422}(591,\cdot)\) \(\chi_{6422}(661,\cdot)\) \(\chi_{6422}(713,\cdot)\) \(\chi_{6422}(839,\cdot)\) \(\chi_{6422}(877,\cdot)\) \(\chi_{6422}(1003,\cdot)\) \(\chi_{6422}(1021,\cdot)\) \(\chi_{6422}(1047,\cdot)\) \(\chi_{6422}(1055,\cdot)\) \(\chi_{6422}(1059,\cdot)\) \(\chi_{6422}(1077,\cdot)\) \(\chi_{6422}(1085,\cdot)\) \(\chi_{6422}(1155,\cdot)\) ...

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: $\Q(\zeta_{468})$
Fixed field: Number field defined by a degree 468 polynomial (not computed)

Values on generators

\((4903,4733)\) → \((e\left(\frac{35}{156}\right),e\left(\frac{1}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(21\)\(23\)\(25\)
\( \chi_{ 6422 }(59, a) \) \(1\)\(1\)\(e\left(\frac{127}{234}\right)\)\(e\left(\frac{425}{468}\right)\)\(e\left(\frac{53}{156}\right)\)\(e\left(\frac{10}{117}\right)\)\(e\left(\frac{121}{156}\right)\)\(e\left(\frac{211}{468}\right)\)\(e\left(\frac{73}{234}\right)\)\(e\left(\frac{413}{468}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{191}{234}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6422 }(59,a) \;\) at \(\;a = \) e.g. 2