Properties

Label 4830.209
Modulus $4830$
Conductor $2415$
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(4830, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,11,11,2]))
 
pari: [g,chi] = znchar(Mod(209,4830))
 

Basic properties

Modulus: \(4830\)
Conductor: \(2415\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2415}(209,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4830.ch

\(\chi_{4830}(209,\cdot)\) \(\chi_{4830}(629,\cdot)\) \(\chi_{4830}(1889,\cdot)\) \(\chi_{4830}(2099,\cdot)\) \(\chi_{4830}(2309,\cdot)\) \(\chi_{4830}(2519,\cdot)\) \(\chi_{4830}(2939,\cdot)\) \(\chi_{4830}(3569,\cdot)\) \(\chi_{4830}(4199,\cdot)\) \(\chi_{4830}(4409,\cdot)\)

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

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

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\(1\)\(1\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{21}{22}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4830 }(209,a) \;\) at \(\;a = \) e.g. 2