Properties

Label 8470.1761
Modulus $8470$
Conductor $847$
Order $33$
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(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,44,60]))
 
pari: [g,chi] = znchar(Mod(1761,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{847}(67,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.ca

\(\chi_{8470}(221,\cdot)\) \(\chi_{8470}(331,\cdot)\) \(\chi_{8470}(991,\cdot)\) \(\chi_{8470}(1101,\cdot)\) \(\chi_{8470}(1761,\cdot)\) \(\chi_{8470}(1871,\cdot)\) \(\chi_{8470}(2531,\cdot)\) \(\chi_{8470}(2641,\cdot)\) \(\chi_{8470}(3301,\cdot)\) \(\chi_{8470}(3411,\cdot)\) \(\chi_{8470}(4071,\cdot)\) \(\chi_{8470}(4181,\cdot)\) \(\chi_{8470}(4951,\cdot)\) \(\chi_{8470}(5611,\cdot)\) \(\chi_{8470}(5721,\cdot)\) \(\chi_{8470}(6381,\cdot)\) \(\chi_{8470}(6491,\cdot)\) \(\chi_{8470}(7151,\cdot)\) \(\chi_{8470}(7921,\cdot)\) \(\chi_{8470}(8031,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: 33.33.1190468723186094710476914290572582621182874672631260195591246568700644178487220449.1

Values on generators

\((6777,6051,7141)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{10}{11}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{32}{33}\right)\)\(1\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{17}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(1761,a) \;\) at \(\;a = \) e.g. 2