Properties

Label 3332.271
Modulus $3332$
Conductor $3332$
Order $42$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,17,21]))
 
pari: [g,chi] = znchar(Mod(271,3332))
 

Basic properties

Modulus: \(3332\)
Conductor: \(3332\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3332.ca

\(\chi_{3332}(271,\cdot)\) \(\chi_{3332}(339,\cdot)\) \(\chi_{3332}(747,\cdot)\) \(\chi_{3332}(1223,\cdot)\) \(\chi_{3332}(1291,\cdot)\) \(\chi_{3332}(1699,\cdot)\) \(\chi_{3332}(1767,\cdot)\) \(\chi_{3332}(2243,\cdot)\) \(\chi_{3332}(2651,\cdot)\) \(\chi_{3332}(2719,\cdot)\) \(\chi_{3332}(3127,\cdot)\) \(\chi_{3332}(3195,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((1667,885,785)\) → \((-1,e\left(\frac{17}{42}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 3332 }(271, a) \) \(1\)\(1\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{3}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3332 }(271,a) \;\) at \(\;a = \) e.g. 2