Properties

Label 4020.2519
Modulus $4020$
Conductor $4020$
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(4020, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,11,11,6]))
 
pari: [g,chi] = znchar(Mod(2519,4020))
 

Basic properties

Modulus: \(4020\)
Conductor: \(4020\)
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 4020.by

\(\chi_{4020}(59,\cdot)\) \(\chi_{4020}(359,\cdot)\) \(\chi_{4020}(1019,\cdot)\) \(\chi_{4020}(1499,\cdot)\) \(\chi_{4020}(1739,\cdot)\) \(\chi_{4020}(2099,\cdot)\) \(\chi_{4020}(2159,\cdot)\) \(\chi_{4020}(2519,\cdot)\) \(\chi_{4020}(3359,\cdot)\) \(\chi_{4020}(3479,\cdot)\)

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

\((2011,2681,3217,1141)\) → \((-1,-1,-1,e\left(\frac{3}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 4020 }(2519, a) \) \(1\)\(1\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(-1\)\(e\left(\frac{7}{22}\right)\)\(-1\)\(e\left(\frac{21}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4020 }(2519,a) \;\) at \(\;a = \) e.g. 2