Properties

Label 4840.309
Modulus $4840$
Conductor $4840$
Order $22$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,11,11,20]))
 
pari: [g,chi] = znchar(Mod(309,4840))
 

Basic properties

Modulus: \(4840\)
Conductor: \(4840\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4840.cj

\(\chi_{4840}(309,\cdot)\) \(\chi_{4840}(749,\cdot)\) \(\chi_{4840}(1189,\cdot)\) \(\chi_{4840}(1629,\cdot)\) \(\chi_{4840}(2069,\cdot)\) \(\chi_{4840}(2509,\cdot)\) \(\chi_{4840}(2949,\cdot)\) \(\chi_{4840}(3829,\cdot)\) \(\chi_{4840}(4269,\cdot)\) \(\chi_{4840}(4709,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ 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

\((3631,2421,1937,4721)\) → \((1,-1,-1,e\left(\frac{10}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4840 }(309, a) \) \(1\)\(1\)\(1\)\(e\left(\frac{19}{22}\right)\)\(1\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(1\)\(e\left(\frac{21}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4840 }(309,a) \;\) at \(\;a = \) e.g. 2