Properties

Label 3630.2771
Modulus $3630$
Conductor $363$
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(3630, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,0,7]))
 
pari: [g,chi] = znchar(Mod(2771,3630))
 

Basic properties

Modulus: \(3630\)
Conductor: \(363\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{363}(230,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3630.bd

\(\chi_{3630}(131,\cdot)\) \(\chi_{3630}(461,\cdot)\) \(\chi_{3630}(791,\cdot)\) \(\chi_{3630}(1121,\cdot)\) \(\chi_{3630}(1781,\cdot)\) \(\chi_{3630}(2111,\cdot)\) \(\chi_{3630}(2441,\cdot)\) \(\chi_{3630}(2771,\cdot)\) \(\chi_{3630}(3101,\cdot)\) \(\chi_{3630}(3431,\cdot)\)

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

\((1211,727,3511)\) → \((-1,1,e\left(\frac{7}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 3630 }(2771, a) \) \(1\)\(1\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{21}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3630 }(2771,a) \;\) at \(\;a = \) e.g. 2