Properties

Label 3332.61
Modulus $3332$
Conductor $833$
Order $336$
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(336))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,88,63]))
 
pari: [g,chi] = znchar(Mod(61,3332))
 

Basic properties

Modulus: \(3332\)
Conductor: \(833\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(336\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{833}(61,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3332.db

\(\chi_{3332}(5,\cdot)\) \(\chi_{3332}(45,\cdot)\) \(\chi_{3332}(61,\cdot)\) \(\chi_{3332}(73,\cdot)\) \(\chi_{3332}(173,\cdot)\) \(\chi_{3332}(201,\cdot)\) \(\chi_{3332}(241,\cdot)\) \(\chi_{3332}(269,\cdot)\) \(\chi_{3332}(369,\cdot)\) \(\chi_{3332}(381,\cdot)\) \(\chi_{3332}(397,\cdot)\) \(\chi_{3332}(437,\cdot)\) \(\chi_{3332}(453,\cdot)\) \(\chi_{3332}(465,\cdot)\) \(\chi_{3332}(481,\cdot)\) \(\chi_{3332}(537,\cdot)\) \(\chi_{3332}(549,\cdot)\) \(\chi_{3332}(605,\cdot)\) \(\chi_{3332}(649,\cdot)\) \(\chi_{3332}(677,\cdot)\) \(\chi_{3332}(745,\cdot)\) \(\chi_{3332}(789,\cdot)\) \(\chi_{3332}(845,\cdot)\) \(\chi_{3332}(857,\cdot)\) \(\chi_{3332}(873,\cdot)\) \(\chi_{3332}(929,\cdot)\) \(\chi_{3332}(941,\cdot)\) \(\chi_{3332}(957,\cdot)\) \(\chi_{3332}(997,\cdot)\) \(\chi_{3332}(1013,\cdot)\) ...

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: $\Q(\zeta_{336})$
Fixed field: Number field defined by a degree 336 polynomial (not computed)

Values on generators

\((1667,885,785)\) → \((1,e\left(\frac{11}{42}\right),e\left(\frac{3}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 3332 }(61, a) \) \(1\)\(1\)\(e\left(\frac{151}{336}\right)\)\(e\left(\frac{179}{336}\right)\)\(e\left(\frac{151}{168}\right)\)\(e\left(\frac{265}{336}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{55}{56}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{257}{336}\right)\)\(e\left(\frac{11}{168}\right)\)\(e\left(\frac{39}{112}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3332 }(61,a) \;\) at \(\;a = \) e.g. 2