Properties

Label 3332.53
Modulus $3332$
Conductor $833$
Order $168$
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(168))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,40,147]))
 
pari: [g,chi] = znchar(Mod(53,3332))
 

Basic properties

Modulus: \(3332\)
Conductor: \(833\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(168\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{833}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3332.cw

\(\chi_{3332}(9,\cdot)\) \(\chi_{3332}(25,\cdot)\) \(\chi_{3332}(53,\cdot)\) \(\chi_{3332}(93,\cdot)\) \(\chi_{3332}(121,\cdot)\) \(\chi_{3332}(389,\cdot)\) \(\chi_{3332}(417,\cdot)\) \(\chi_{3332}(457,\cdot)\) \(\chi_{3332}(485,\cdot)\) \(\chi_{3332}(501,\cdot)\) \(\chi_{3332}(529,\cdot)\) \(\chi_{3332}(597,\cdot)\) \(\chi_{3332}(865,\cdot)\) \(\chi_{3332}(893,\cdot)\) \(\chi_{3332}(933,\cdot)\) \(\chi_{3332}(977,\cdot)\) \(\chi_{3332}(1005,\cdot)\) \(\chi_{3332}(1045,\cdot)\) \(\chi_{3332}(1073,\cdot)\) \(\chi_{3332}(1369,\cdot)\) \(\chi_{3332}(1409,\cdot)\) \(\chi_{3332}(1437,\cdot)\) \(\chi_{3332}(1453,\cdot)\) \(\chi_{3332}(1481,\cdot)\) \(\chi_{3332}(1521,\cdot)\) \(\chi_{3332}(1817,\cdot)\) \(\chi_{3332}(1845,\cdot)\) \(\chi_{3332}(1885,\cdot)\) \(\chi_{3332}(1913,\cdot)\) \(\chi_{3332}(1957,\cdot)\) ...

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: $\Q(\zeta_{168})$
Fixed field: Number field defined by a degree 168 polynomial (not computed)

Values on generators

\((1667,885,785)\) → \((1,e\left(\frac{5}{21}\right),e\left(\frac{7}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 3332 }(53, a) \) \(1\)\(1\)\(e\left(\frac{19}{168}\right)\)\(e\left(\frac{47}{168}\right)\)\(e\left(\frac{19}{84}\right)\)\(e\left(\frac{109}{168}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{29}{168}\right)\)\(e\left(\frac{47}{84}\right)\)\(e\left(\frac{19}{56}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3332 }(53,a) \;\) at \(\;a = \) e.g. 2