Properties

Label 8470.5669
Modulus $8470$
Conductor $4235$
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([55,55,12]))
 
pari: [g,chi] = znchar(Mod(5669,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4235}(1434,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.ct

\(\chi_{8470}(69,\cdot)\) \(\chi_{8470}(279,\cdot)\) \(\chi_{8470}(489,\cdot)\) \(\chi_{8470}(559,\cdot)\) \(\chi_{8470}(839,\cdot)\) \(\chi_{8470}(1259,\cdot)\) \(\chi_{8470}(1329,\cdot)\) \(\chi_{8470}(1609,\cdot)\) \(\chi_{8470}(1819,\cdot)\) \(\chi_{8470}(2029,\cdot)\) \(\chi_{8470}(2099,\cdot)\) \(\chi_{8470}(2379,\cdot)\) \(\chi_{8470}(2589,\cdot)\) \(\chi_{8470}(2799,\cdot)\) \(\chi_{8470}(2869,\cdot)\) \(\chi_{8470}(3359,\cdot)\) \(\chi_{8470}(3569,\cdot)\) \(\chi_{8470}(3919,\cdot)\) \(\chi_{8470}(4129,\cdot)\) \(\chi_{8470}(4339,\cdot)\) \(\chi_{8470}(4409,\cdot)\) \(\chi_{8470}(4689,\cdot)\) \(\chi_{8470}(4899,\cdot)\) \(\chi_{8470}(5179,\cdot)\) \(\chi_{8470}(5459,\cdot)\) \(\chi_{8470}(5669,\cdot)\) \(\chi_{8470}(5879,\cdot)\) \(\chi_{8470}(5949,\cdot)\) \(\chi_{8470}(6229,\cdot)\) \(\chi_{8470}(6439,\cdot)\) ...

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

\((6777,6051,7141)\) → \((-1,-1,e\left(\frac{6}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\( \chi_{ 8470 }(5669, a) \) \(-1\)\(1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{55}\right)\)\(e\left(\frac{19}{55}\right)\)\(e\left(\frac{61}{110}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{47}{55}\right)\)\(e\left(\frac{97}{110}\right)\)\(e\left(\frac{9}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(5669,a) \;\) at \(\;a = \) e.g. 2