Properties

Label 8470.2323
Modulus $8470$
Conductor $4235$
Order $220$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8470, base_ring=CyclotomicField(220))
 
M = H._module
 
chi = DirichletCharacter(H, M([165,110,182]))
 
pari: [g,chi] = znchar(Mod(2323,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(220\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4235}(2323,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.de

\(\chi_{8470}(13,\cdot)\) \(\chi_{8470}(83,\cdot)\) \(\chi_{8470}(167,\cdot)\) \(\chi_{8470}(237,\cdot)\) \(\chi_{8470}(293,\cdot)\) \(\chi_{8470}(447,\cdot)\) \(\chi_{8470}(503,\cdot)\) \(\chi_{8470}(657,\cdot)\) \(\chi_{8470}(783,\cdot)\) \(\chi_{8470}(853,\cdot)\) \(\chi_{8470}(937,\cdot)\) \(\chi_{8470}(1007,\cdot)\) \(\chi_{8470}(1063,\cdot)\) \(\chi_{8470}(1217,\cdot)\) \(\chi_{8470}(1273,\cdot)\) \(\chi_{8470}(1427,\cdot)\) \(\chi_{8470}(1553,\cdot)\) \(\chi_{8470}(1623,\cdot)\) \(\chi_{8470}(1707,\cdot)\) \(\chi_{8470}(1777,\cdot)\) \(\chi_{8470}(1833,\cdot)\) \(\chi_{8470}(1987,\cdot)\) \(\chi_{8470}(2043,\cdot)\) \(\chi_{8470}(2197,\cdot)\) \(\chi_{8470}(2323,\cdot)\) \(\chi_{8470}(2477,\cdot)\) \(\chi_{8470}(2547,\cdot)\) \(\chi_{8470}(2603,\cdot)\) \(\chi_{8470}(2757,\cdot)\) \(\chi_{8470}(2813,\cdot)\) ...

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: $\Q(\zeta_{220})$
Fixed field: Number field defined by a degree 220 polynomial (not computed)

Values on generators

\((6777,6051,7141)\) → \((-i,-1,e\left(\frac{91}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\( \chi_{ 8470 }(2323, a) \) \(-1\)\(1\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{67}{220}\right)\)\(e\left(\frac{173}{220}\right)\)\(e\left(\frac{73}{110}\right)\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{31}{55}\right)\)\(e\left(\frac{71}{110}\right)\)\(e\left(\frac{109}{220}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(2323,a) \;\) at \(\;a = \) e.g. 2