Properties

Label 3332.11
Modulus $3332$
Conductor $3332$
Order $336$
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(336))
 
M = H._module
 
chi = DirichletCharacter(H, M([168,320,147]))
 
pari: [g,chi] = znchar(Mod(11,3332))
 

Basic properties

Modulus: \(3332\)
Conductor: \(3332\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(336\)
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.cy

\(\chi_{3332}(11,\cdot)\) \(\chi_{3332}(23,\cdot)\) \(\chi_{3332}(39,\cdot)\) \(\chi_{3332}(95,\cdot)\) \(\chi_{3332}(107,\cdot)\) \(\chi_{3332}(163,\cdot)\) \(\chi_{3332}(207,\cdot)\) \(\chi_{3332}(235,\cdot)\) \(\chi_{3332}(303,\cdot)\) \(\chi_{3332}(347,\cdot)\) \(\chi_{3332}(403,\cdot)\) \(\chi_{3332}(415,\cdot)\) \(\chi_{3332}(431,\cdot)\) \(\chi_{3332}(487,\cdot)\) \(\chi_{3332}(499,\cdot)\) \(\chi_{3332}(515,\cdot)\) \(\chi_{3332}(555,\cdot)\) \(\chi_{3332}(571,\cdot)\) \(\chi_{3332}(583,\cdot)\) \(\chi_{3332}(639,\cdot)\) \(\chi_{3332}(683,\cdot)\) \(\chi_{3332}(711,\cdot)\) \(\chi_{3332}(751,\cdot)\) \(\chi_{3332}(779,\cdot)\) \(\chi_{3332}(823,\cdot)\) \(\chi_{3332}(879,\cdot)\) \(\chi_{3332}(891,\cdot)\) \(\chi_{3332}(907,\cdot)\) \(\chi_{3332}(947,\cdot)\) \(\chi_{3332}(963,\cdot)\) ...

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

\((1667,885,785)\) → \((-1,e\left(\frac{20}{21}\right),e\left(\frac{7}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 3332 }(11, a) \) \(1\)\(1\)\(e\left(\frac{299}{336}\right)\)\(e\left(\frac{271}{336}\right)\)\(e\left(\frac{131}{168}\right)\)\(e\left(\frac{221}{336}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{39}{56}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{85}{336}\right)\)\(e\left(\frac{103}{168}\right)\)\(e\left(\frac{75}{112}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3332 }(11,a) \;\) at \(\;a = \) e.g. 2