Properties

Label 4830.559
Modulus $4830$
Conductor $805$
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([0,11,11,19]))
 
pari: [g,chi] = znchar(Mod(559,4830))
 

Basic properties

Modulus: \(4830\)
Conductor: \(805\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{805}(559,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4830.ce

\(\chi_{4830}(559,\cdot)\) \(\chi_{4830}(769,\cdot)\) \(\chi_{4830}(1399,\cdot)\) \(\chi_{4830}(2029,\cdot)\) \(\chi_{4830}(2449,\cdot)\) \(\chi_{4830}(2659,\cdot)\) \(\chi_{4830}(2869,\cdot)\) \(\chi_{4830}(3079,\cdot)\) \(\chi_{4830}(4339,\cdot)\) \(\chi_{4830}(4759,\cdot)\)

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: 22.22.3810948187032486523644418763541659564892578125.1

Values on generators

\((3221,967,2761,1891)\) → \((1,-1,-1,e\left(\frac{19}{22}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\(1\)\(1\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4830 }(559,a) \;\) at \(\;a = \) e.g. 2