Properties

Label 8470.6161
Modulus $8470$
Conductor $121$
Order $11$
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([0,0,4]))
 
pari: [g,chi] = znchar(Mod(6161,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(121\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(11\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{121}(111,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.bc

\(\chi_{8470}(771,\cdot)\) \(\chi_{8470}(1541,\cdot)\) \(\chi_{8470}(2311,\cdot)\) \(\chi_{8470}(3081,\cdot)\) \(\chi_{8470}(3851,\cdot)\) \(\chi_{8470}(4621,\cdot)\) \(\chi_{8470}(5391,\cdot)\) \(\chi_{8470}(6161,\cdot)\) \(\chi_{8470}(6931,\cdot)\) \(\chi_{8470}(7701,\cdot)\)

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: 11.11.672749994932560009201.1

Values on generators

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

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(1\)\(1\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(1\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{7}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(6161,a) \;\) at \(\;a = \) e.g. 2