Properties

Label 8470.3111
Modulus $8470$
Conductor $847$
Order $330$
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(330))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,55,78]))
 
pari: [g,chi] = znchar(Mod(3111,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{847}(570,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.di

\(\chi_{8470}(31,\cdot)\) \(\chi_{8470}(201,\cdot)\) \(\chi_{8470}(311,\cdot)\) \(\chi_{8470}(411,\cdot)\) \(\chi_{8470}(521,\cdot)\) \(\chi_{8470}(621,\cdot)\) \(\chi_{8470}(691,\cdot)\) \(\chi_{8470}(731,\cdot)\) \(\chi_{8470}(801,\cdot)\) \(\chi_{8470}(1081,\cdot)\) \(\chi_{8470}(1181,\cdot)\) \(\chi_{8470}(1391,\cdot)\) \(\chi_{8470}(1501,\cdot)\) \(\chi_{8470}(1571,\cdot)\) \(\chi_{8470}(1741,\cdot)\) \(\chi_{8470}(1851,\cdot)\) \(\chi_{8470}(1951,\cdot)\) \(\chi_{8470}(2061,\cdot)\) \(\chi_{8470}(2161,\cdot)\) \(\chi_{8470}(2231,\cdot)\) \(\chi_{8470}(2271,\cdot)\) \(\chi_{8470}(2341,\cdot)\) \(\chi_{8470}(2511,\cdot)\) \(\chi_{8470}(2621,\cdot)\) \(\chi_{8470}(2721,\cdot)\) \(\chi_{8470}(2831,\cdot)\) \(\chi_{8470}(3001,\cdot)\) \(\chi_{8470}(3041,\cdot)\) \(\chi_{8470}(3111,\cdot)\) \(\chi_{8470}(3281,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((6777,6051,7141)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{13}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\( \chi_{ 8470 }(3111, a) \) \(-1\)\(1\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{41}{110}\right)\)\(e\left(\frac{247}{330}\right)\)\(e\left(\frac{149}{330}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{55}\right)\)\(e\left(\frac{163}{330}\right)\)\(e\left(\frac{43}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(3111,a) \;\) at \(\;a = \) e.g. 2