Properties

Label 3332.81
Modulus $3332$
Conductor $833$
Order $84$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,8,21]))
 
pari: [g,chi] = znchar(Mod(81,3332))
 

Basic properties

Modulus: \(3332\)
Conductor: \(833\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{833}(81,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3332.cn

\(\chi_{3332}(81,\cdot)\) \(\chi_{3332}(149,\cdot)\) \(\chi_{3332}(429,\cdot)\) \(\chi_{3332}(625,\cdot)\) \(\chi_{3332}(837,\cdot)\) \(\chi_{3332}(905,\cdot)\) \(\chi_{3332}(1033,\cdot)\) \(\chi_{3332}(1101,\cdot)\) \(\chi_{3332}(1313,\cdot)\) \(\chi_{3332}(1381,\cdot)\) \(\chi_{3332}(1509,\cdot)\) \(\chi_{3332}(1577,\cdot)\) \(\chi_{3332}(1789,\cdot)\) \(\chi_{3332}(1857,\cdot)\) \(\chi_{3332}(1985,\cdot)\) \(\chi_{3332}(2053,\cdot)\) \(\chi_{3332}(2265,\cdot)\) \(\chi_{3332}(2461,\cdot)\) \(\chi_{3332}(2741,\cdot)\) \(\chi_{3332}(2809,\cdot)\) \(\chi_{3332}(2937,\cdot)\) \(\chi_{3332}(3005,\cdot)\) \(\chi_{3332}(3217,\cdot)\) \(\chi_{3332}(3285,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1667,885,785)\) → \((1,e\left(\frac{2}{21}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 3332 }(81, a) \) \(1\)\(1\)\(e\left(\frac{29}{84}\right)\)\(e\left(\frac{1}{84}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{47}{84}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{31}{84}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{1}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3332 }(81,a) \;\) at \(\;a = \) e.g. 2