Properties

Label 8470.79
Modulus $8470$
Conductor $4235$
Order $330$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(8470, base_ring=CyclotomicField(330))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([165,110,123]))
 
pari: [g,chi] = znchar(Mod(79,8470))
 

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 \(\chi_{4235}(79,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.dk

\(\chi_{8470}(39,\cdot)\) \(\chi_{8470}(79,\cdot)\) \(\chi_{8470}(149,\cdot)\) \(\chi_{8470}(249,\cdot)\) \(\chi_{8470}(359,\cdot)\) \(\chi_{8470}(459,\cdot)\) \(\chi_{8470}(569,\cdot)\) \(\chi_{8470}(739,\cdot)\) \(\chi_{8470}(809,\cdot)\) \(\chi_{8470}(849,\cdot)\) \(\chi_{8470}(919,\cdot)\) \(\chi_{8470}(1019,\cdot)\) \(\chi_{8470}(1229,\cdot)\) \(\chi_{8470}(1339,\cdot)\) \(\chi_{8470}(1509,\cdot)\) \(\chi_{8470}(1579,\cdot)\) \(\chi_{8470}(1619,\cdot)\) \(\chi_{8470}(1689,\cdot)\) \(\chi_{8470}(1789,\cdot)\) \(\chi_{8470}(1899,\cdot)\) \(\chi_{8470}(1999,\cdot)\) \(\chi_{8470}(2109,\cdot)\) \(\chi_{8470}(2279,\cdot)\) \(\chi_{8470}(2349,\cdot)\) \(\chi_{8470}(2389,\cdot)\) \(\chi_{8470}(2459,\cdot)\) \(\chi_{8470}(2559,\cdot)\) \(\chi_{8470}(2669,\cdot)\) \(\chi_{8470}(2769,\cdot)\) \(\chi_{8470}(2879,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ 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

\((6777,6051,7141)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{41}{110}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{8}{55}\right)\)\(e\left(\frac{16}{165}\right)\)\(e\left(\frac{199}{330}\right)\)\(e\left(\frac{17}{66}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{37}{110}\right)\)\(e\left(\frac{64}{165}\right)\)\(e\left(\frac{271}{330}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(79,a) \;\) at \(\;a = \) e.g. 2