Properties

Label 8470.3019
Modulus $8470$
Conductor $4235$
Order $330$
Real no
Primitive no
Minimal yes
Parity even

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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([165,110,102]))
 
pari: [g,chi] = znchar(Mod(3019,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4235}(3019,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.dh

\(\chi_{8470}(179,\cdot)\) \(\chi_{8470}(289,\cdot)\) \(\chi_{8470}(389,\cdot)\) \(\chi_{8470}(499,\cdot)\) \(\chi_{8470}(599,\cdot)\) \(\chi_{8470}(669,\cdot)\) \(\chi_{8470}(709,\cdot)\) \(\chi_{8470}(779,\cdot)\) \(\chi_{8470}(949,\cdot)\) \(\chi_{8470}(1059,\cdot)\) \(\chi_{8470}(1159,\cdot)\) \(\chi_{8470}(1269,\cdot)\) \(\chi_{8470}(1369,\cdot)\) \(\chi_{8470}(1439,\cdot)\) \(\chi_{8470}(1549,\cdot)\) \(\chi_{8470}(1719,\cdot)\) \(\chi_{8470}(1829,\cdot)\) \(\chi_{8470}(1929,\cdot)\) \(\chi_{8470}(2039,\cdot)\) \(\chi_{8470}(2139,\cdot)\) \(\chi_{8470}(2209,\cdot)\) \(\chi_{8470}(2249,\cdot)\) \(\chi_{8470}(2319,\cdot)\) \(\chi_{8470}(2489,\cdot)\) \(\chi_{8470}(2599,\cdot)\) \(\chi_{8470}(2699,\cdot)\) \(\chi_{8470}(2809,\cdot)\) \(\chi_{8470}(2909,\cdot)\) \(\chi_{8470}(2979,\cdot)\) \(\chi_{8470}(3019,\cdot)\) ...

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

\((6777,6051,7141)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{17}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\( \chi_{ 8470 }(3019, a) \) \(1\)\(1\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{79}{110}\right)\)\(e\left(\frac{323}{330}\right)\)\(e\left(\frac{53}{165}\right)\)\(e\left(\frac{53}{66}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{14}{55}\right)\)\(e\left(\frac{151}{165}\right)\)\(e\left(\frac{49}{330}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(3019,a) \;\) at \(\;a = \) e.g. 2