Properties

Label 9360.931
Modulus $9360$
Conductor $1872$
Order $12$
Real no
Primitive no
Minimal yes
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([6,9,4,0,3]))
 
pari: [g,chi] = znchar(Mod(931,9360))
 

Basic properties

Modulus: \(9360\)
Conductor: \(1872\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1872}(931,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9360.ox

\(\chi_{9360}(931,\cdot)\) \(\chi_{9360}(3931,\cdot)\) \(\chi_{9360}(7051,\cdot)\) \(\chi_{9360}(7171,\cdot)\)

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: 12.12.3921210015059966692568334336.1

Values on generators

\((8191,2341,2081,5617,5761)\) → \((-1,-i,e\left(\frac{1}{3}\right),1,i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9360 }(931, a) \) \(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(-1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9360 }(931,a) \;\) at \(\;a = \) e.g. 2