Properties

Label 8670.37
Modulus $8670$
Conductor $1445$
Order $272$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8670, base_ring=CyclotomicField(272))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,68,129]))
 
pari: [g,chi] = znchar(Mod(37,8670))
 

Basic properties

Modulus: \(8670\)
Conductor: \(1445\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(272\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1445}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8670.cn

\(\chi_{8670}(37,\cdot)\) \(\chi_{8670}(97,\cdot)\) \(\chi_{8670}(163,\cdot)\) \(\chi_{8670}(193,\cdot)\) \(\chi_{8670}(277,\cdot)\) \(\chi_{8670}(283,\cdot)\) \(\chi_{8670}(313,\cdot)\) \(\chi_{8670}(337,\cdot)\) \(\chi_{8670}(547,\cdot)\) \(\chi_{8670}(607,\cdot)\) \(\chi_{8670}(673,\cdot)\) \(\chi_{8670}(703,\cdot)\) \(\chi_{8670}(787,\cdot)\) \(\chi_{8670}(793,\cdot)\) \(\chi_{8670}(823,\cdot)\) \(\chi_{8670}(847,\cdot)\) \(\chi_{8670}(1057,\cdot)\) \(\chi_{8670}(1117,\cdot)\) \(\chi_{8670}(1183,\cdot)\) \(\chi_{8670}(1213,\cdot)\) \(\chi_{8670}(1297,\cdot)\) \(\chi_{8670}(1303,\cdot)\) \(\chi_{8670}(1333,\cdot)\) \(\chi_{8670}(1357,\cdot)\) \(\chi_{8670}(1567,\cdot)\) \(\chi_{8670}(1627,\cdot)\) \(\chi_{8670}(1693,\cdot)\) \(\chi_{8670}(1723,\cdot)\) \(\chi_{8670}(1807,\cdot)\) \(\chi_{8670}(1813,\cdot)\) ...

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: $\Q(\zeta_{272})$
Fixed field: Number field defined by a degree 272 polynomial (not computed)

Values on generators

\((2891,6937,6361)\) → \((1,i,e\left(\frac{129}{272}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 8670 }(37, a) \) \(1\)\(1\)\(e\left(\frac{207}{272}\right)\)\(e\left(\frac{247}{272}\right)\)\(e\left(\frac{12}{17}\right)\)\(e\left(\frac{19}{136}\right)\)\(e\left(\frac{139}{272}\right)\)\(e\left(\frac{213}{272}\right)\)\(e\left(\frac{73}{272}\right)\)\(e\left(\frac{117}{272}\right)\)\(e\left(\frac{27}{272}\right)\)\(e\left(\frac{127}{136}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8670 }(37,a) \;\) at \(\;a = \) e.g. 2