Properties

Label 8470.1299
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([33,44,36]))
 
pari: [g,chi] = znchar(Mod(1299,8470))
 

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 \(\chi_{4235}(1299,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.cq

\(\chi_{8470}(529,\cdot)\) \(\chi_{8470}(639,\cdot)\) \(\chi_{8470}(1299,\cdot)\) \(\chi_{8470}(1409,\cdot)\) \(\chi_{8470}(2069,\cdot)\) \(\chi_{8470}(2839,\cdot)\) \(\chi_{8470}(2949,\cdot)\) \(\chi_{8470}(3609,\cdot)\) \(\chi_{8470}(3719,\cdot)\) \(\chi_{8470}(4379,\cdot)\) \(\chi_{8470}(4489,\cdot)\) \(\chi_{8470}(5149,\cdot)\) \(\chi_{8470}(5259,\cdot)\) \(\chi_{8470}(5919,\cdot)\) \(\chi_{8470}(6029,\cdot)\) \(\chi_{8470}(6689,\cdot)\) \(\chi_{8470}(6799,\cdot)\) \(\chi_{8470}(7459,\cdot)\) \(\chi_{8470}(7569,\cdot)\) \(\chi_{8470}(8339,\cdot)\)

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

\((6777,6051,7141)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{6}{11}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{1}{66}\right)\)\(-1\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{49}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(1299,a) \;\) at \(\;a = \) e.g. 2