Properties

Label 6030.829
Modulus $6030$
Conductor $335$
Order $22$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6030, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,11,10]))
 
pari: [g,chi] = znchar(Mod(829,6030))
 

Basic properties

Modulus: \(6030\)
Conductor: \(335\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{335}(159,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6030.cn

\(\chi_{6030}(829,\cdot)\) \(\chi_{6030}(2359,\cdot)\) \(\chi_{6030}(2809,\cdot)\) \(\chi_{6030}(3079,\cdot)\) \(\chi_{6030}(3439,\cdot)\) \(\chi_{6030}(3709,\cdot)\) \(\chi_{6030}(4699,\cdot)\) \(\chi_{6030}(5419,\cdot)\) \(\chi_{6030}(5509,\cdot)\) \(\chi_{6030}(5869,\cdot)\)

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

\((4691,1207,3151)\) → \((1,-1,e\left(\frac{5}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 6030 }(829, a) \) \(1\)\(1\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{5}{22}\right)\)\(1\)\(e\left(\frac{4}{11}\right)\)\(-1\)\(e\left(\frac{1}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6030 }(829,a) \;\) at \(\;a = \) e.g. 2