Properties

Label 3312.1885
Modulus $3312$
Conductor $3312$
Order $12$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3312, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,4,6]))
 
pari: [g,chi] = znchar(Mod(1885,3312))
 

Basic properties

Modulus: \(3312\)
Conductor: \(3312\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3312.bp

\(\chi_{3312}(229,\cdot)\) \(\chi_{3312}(781,\cdot)\) \(\chi_{3312}(1885,\cdot)\) \(\chi_{3312}(2437,\cdot)\)

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: 12.0.54739011255265747059867648.1

Values on generators

\((415,2485,2945,2305)\) → \((1,-i,e\left(\frac{1}{3}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 3312 }(1885, a) \) \(-1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(-1\)\(-i\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3312 }(1885,a) \;\) at \(\;a = \) e.g. 2