Properties

Label 3724.2725
Modulus $3724$
Conductor $133$
Order $6$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3724, base_ring=CyclotomicField(6))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,2,1]))
 
pari: [g,chi] = znchar(Mod(2725,3724))
 

Basic properties

Modulus: \(3724\)
Conductor: \(133\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{133}(65,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3724.m

\(\chi_{3724}(753,\cdot)\) \(\chi_{3724}(2725,\cdot)\)

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: \(\Q(\sqrt{-3}) \)
Fixed field: 6.0.5945113699.1

Values on generators

\((1863,3041,3137)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\( \chi_{ 3724 }(2725, a) \) \(-1\)\(1\)\(-1\)\(e\left(\frac{1}{3}\right)\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3724 }(2725,a) \;\) at \(\;a = \) e.g. 2