Properties

Label 4830.211
Modulus $4830$
Conductor $23$
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(4830, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,0,4]))
 
pari: [g,chi] = znchar(Mod(211,4830))
 

Basic properties

Modulus: \(4830\)
Conductor: \(23\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(11\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{23}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4830.bo

\(\chi_{4830}(211,\cdot)\) \(\chi_{4830}(841,\cdot)\) \(\chi_{4830}(1051,\cdot)\) \(\chi_{4830}(1681,\cdot)\) \(\chi_{4830}(2101,\cdot)\) \(\chi_{4830}(3361,\cdot)\) \(\chi_{4830}(3571,\cdot)\) \(\chi_{4830}(3781,\cdot)\) \(\chi_{4830}(3991,\cdot)\) \(\chi_{4830}(4411,\cdot)\)

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: \(\Q(\zeta_{23})^+\)

Values on generators

\((3221,967,2761,1891)\) → \((1,1,1,e\left(\frac{2}{11}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\(1\)\(1\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4830 }(211,a) \;\) at \(\;a = \) e.g. 2