Properties

Label 3332.57
Modulus $3332$
Conductor $833$
Order $112$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(112))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,96,105]))
 
pari: [g,chi] = znchar(Mod(57,3332))
 

Basic properties

Modulus: \(3332\)
Conductor: \(833\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(112\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{833}(57,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3332.cs

\(\chi_{3332}(29,\cdot)\) \(\chi_{3332}(57,\cdot)\) \(\chi_{3332}(113,\cdot)\) \(\chi_{3332}(141,\cdot)\) \(\chi_{3332}(309,\cdot)\) \(\chi_{3332}(337,\cdot)\) \(\chi_{3332}(449,\cdot)\) \(\chi_{3332}(505,\cdot)\) \(\chi_{3332}(533,\cdot)\) \(\chi_{3332}(617,\cdot)\) \(\chi_{3332}(673,\cdot)\) \(\chi_{3332}(813,\cdot)\) \(\chi_{3332}(925,\cdot)\) \(\chi_{3332}(1009,\cdot)\) \(\chi_{3332}(1065,\cdot)\) \(\chi_{3332}(1093,\cdot)\) \(\chi_{3332}(1149,\cdot)\) \(\chi_{3332}(1261,\cdot)\) \(\chi_{3332}(1289,\cdot)\) \(\chi_{3332}(1401,\cdot)\) \(\chi_{3332}(1457,\cdot)\) \(\chi_{3332}(1485,\cdot)\) \(\chi_{3332}(1541,\cdot)\) \(\chi_{3332}(1625,\cdot)\) \(\chi_{3332}(1737,\cdot)\) \(\chi_{3332}(1877,\cdot)\) \(\chi_{3332}(1933,\cdot)\) \(\chi_{3332}(2017,\cdot)\) \(\chi_{3332}(2045,\cdot)\) \(\chi_{3332}(2101,\cdot)\) ...

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: $\Q(\zeta_{112})$
Fixed field: Number field defined by a degree 112 polynomial (not computed)

Values on generators

\((1667,885,785)\) → \((1,e\left(\frac{6}{7}\right),e\left(\frac{15}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 3332 }(57, a) \) \(-1\)\(1\)\(e\left(\frac{89}{112}\right)\)\(e\left(\frac{61}{112}\right)\)\(e\left(\frac{33}{56}\right)\)\(e\left(\frac{95}{112}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{19}{56}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{71}{112}\right)\)\(e\left(\frac{5}{56}\right)\)\(e\left(\frac{43}{112}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3332 }(57,a) \;\) at \(\;a = \) e.g. 2