Properties

Label 8470.6929
Modulus $8470$
Conductor $4235$
Order $22$
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(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,11,1]))
 
pari: [g,chi] = znchar(Mod(6929,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4235}(2694,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.br

\(\chi_{8470}(769,\cdot)\) \(\chi_{8470}(1539,\cdot)\) \(\chi_{8470}(2309,\cdot)\) \(\chi_{8470}(3079,\cdot)\) \(\chi_{8470}(3849,\cdot)\) \(\chi_{8470}(4619,\cdot)\) \(\chi_{8470}(5389,\cdot)\) \(\chi_{8470}(6159,\cdot)\) \(\chi_{8470}(6929,\cdot)\) \(\chi_{8470}(7699,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: Number field defined by a degree 22 polynomial

Values on generators

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

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(1\)\(1\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{15}{22}\right)\)\(1\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{9}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(6929,a) \;\) at \(\;a = \) e.g. 2