Properties

Label 9360.1463
Modulus $9360$
Conductor $4680$
Order $12$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9360, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,6,10,9,11]))
 
pari: [g,chi] = znchar(Mod(1463,9360))
 

Basic properties

Modulus: \(9360\)
Conductor: \(4680\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4680}(3803,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9360.mj

\(\chi_{9360}(1463,\cdot)\) \(\chi_{9360}(7607,\cdot)\) \(\chi_{9360}(8807,\cdot)\) \(\chi_{9360}(8903,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.12.355491663986814578735616000000000.3

Values on generators

\((8191,2341,2081,5617,5761)\) → \((-1,-1,e\left(\frac{5}{6}\right),-i,e\left(\frac{11}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9360 }(1463, a) \) \(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(i\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(-1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9360 }(1463,a) \;\) at \(\;a = \) e.g. 2