Properties

Label 8470.19
Modulus $8470$
Conductor $4235$
Order $330$
Real no
Primitive no
Minimal yes
Parity even

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([165,275,249]))
 
pari: [g,chi] = znchar(Mod(19,8470))
 

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 \(\chi_{4235}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.dl

\(\chi_{8470}(19,\cdot)\) \(\chi_{8470}(129,\cdot)\) \(\chi_{8470}(299,\cdot)\) \(\chi_{8470}(369,\cdot)\) \(\chi_{8470}(409,\cdot)\) \(\chi_{8470}(479,\cdot)\) \(\chi_{8470}(579,\cdot)\) \(\chi_{8470}(689,\cdot)\) \(\chi_{8470}(789,\cdot)\) \(\chi_{8470}(899,\cdot)\) \(\chi_{8470}(1069,\cdot)\) \(\chi_{8470}(1139,\cdot)\) \(\chi_{8470}(1179,\cdot)\) \(\chi_{8470}(1249,\cdot)\) \(\chi_{8470}(1349,\cdot)\) \(\chi_{8470}(1459,\cdot)\) \(\chi_{8470}(1559,\cdot)\) \(\chi_{8470}(1669,\cdot)\) \(\chi_{8470}(1839,\cdot)\) \(\chi_{8470}(1949,\cdot)\) \(\chi_{8470}(2019,\cdot)\) \(\chi_{8470}(2119,\cdot)\) \(\chi_{8470}(2229,\cdot)\) \(\chi_{8470}(2329,\cdot)\) \(\chi_{8470}(2439,\cdot)\) \(\chi_{8470}(2609,\cdot)\) \(\chi_{8470}(2679,\cdot)\) \(\chi_{8470}(2719,\cdot)\) \(\chi_{8470}(2789,\cdot)\) \(\chi_{8470}(2889,\cdot)\) ...

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

\((6777,6051,7141)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{83}{110}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{23}{110}\right)\)\(e\left(\frac{101}{330}\right)\)\(e\left(\frac{131}{165}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{91}{110}\right)\)\(e\left(\frac{239}{330}\right)\)\(e\left(\frac{283}{330}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(19,a) \;\) at \(\;a = \) e.g. 2