Properties

Label 8470.7141
Modulus $8470$
Conductor $121$
Order $110$
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(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,1]))
 
pari: [g,chi] = znchar(Mod(7141,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(121\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{121}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.cr

\(\chi_{8470}(211,\cdot)\) \(\chi_{8470}(281,\cdot)\) \(\chi_{8470}(491,\cdot)\) \(\chi_{8470}(701,\cdot)\) \(\chi_{8470}(981,\cdot)\) \(\chi_{8470}(1051,\cdot)\) \(\chi_{8470}(1261,\cdot)\) \(\chi_{8470}(1471,\cdot)\) \(\chi_{8470}(1751,\cdot)\) \(\chi_{8470}(1821,\cdot)\) \(\chi_{8470}(2031,\cdot)\) \(\chi_{8470}(2241,\cdot)\) \(\chi_{8470}(2521,\cdot)\) \(\chi_{8470}(2591,\cdot)\) \(\chi_{8470}(2801,\cdot)\) \(\chi_{8470}(3011,\cdot)\) \(\chi_{8470}(3291,\cdot)\) \(\chi_{8470}(3571,\cdot)\) \(\chi_{8470}(3781,\cdot)\) \(\chi_{8470}(4061,\cdot)\) \(\chi_{8470}(4131,\cdot)\) \(\chi_{8470}(4341,\cdot)\) \(\chi_{8470}(4551,\cdot)\) \(\chi_{8470}(4901,\cdot)\) \(\chi_{8470}(5111,\cdot)\) \(\chi_{8470}(5601,\cdot)\) \(\chi_{8470}(5671,\cdot)\) \(\chi_{8470}(5881,\cdot)\) \(\chi_{8470}(6091,\cdot)\) \(\chi_{8470}(6371,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\( \chi_{ 8470 }(7141, a) \) \(-1\)\(1\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{101}{110}\right)\)\(e\left(\frac{49}{110}\right)\)\(e\left(\frac{83}{110}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{17}{110}\right)\)\(e\left(\frac{43}{55}\right)\)\(e\left(\frac{21}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(7141,a) \;\) at \(\;a = \) e.g. 2