Properties

Label 8470.4399
Modulus $8470$
Conductor $4235$
Order $66$
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([33,11,15]))
 
pari: [g,chi] = znchar(Mod(4399,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4235}(164,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.cm

\(\chi_{8470}(439,\cdot)\) \(\chi_{8470}(549,\cdot)\) \(\chi_{8470}(1319,\cdot)\) \(\chi_{8470}(1979,\cdot)\) \(\chi_{8470}(2089,\cdot)\) \(\chi_{8470}(2749,\cdot)\) \(\chi_{8470}(2859,\cdot)\) \(\chi_{8470}(3519,\cdot)\) \(\chi_{8470}(4289,\cdot)\) \(\chi_{8470}(4399,\cdot)\) \(\chi_{8470}(5059,\cdot)\) \(\chi_{8470}(5169,\cdot)\) \(\chi_{8470}(5829,\cdot)\) \(\chi_{8470}(5939,\cdot)\) \(\chi_{8470}(6599,\cdot)\) \(\chi_{8470}(6709,\cdot)\) \(\chi_{8470}(7369,\cdot)\) \(\chi_{8470}(7479,\cdot)\) \(\chi_{8470}(8139,\cdot)\) \(\chi_{8470}(8249,\cdot)\)

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: Number field defined by a degree 66 polynomial

Values on generators

\((6777,6051,7141)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{5}{22}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{53}{66}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{49}{66}\right)\)\(1\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{47}{66}\right)\)\(e\left(\frac{25}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(4399,a) \;\) at \(\;a = \) e.g. 2