Properties

Label 3311.31
Modulus $3311$
Conductor $3311$
Order $210$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3311, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([35,126,170]))
 
pari: [g,chi] = znchar(Mod(31,3311))
 

Basic properties

Modulus: \(3311\)
Conductor: \(3311\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3311.fk

\(\chi_{3311}(31,\cdot)\) \(\chi_{3311}(38,\cdot)\) \(\chi_{3311}(103,\cdot)\) \(\chi_{3311}(229,\cdot)\) \(\chi_{3311}(311,\cdot)\) \(\chi_{3311}(339,\cdot)\) \(\chi_{3311}(367,\cdot)\) \(\chi_{3311}(488,\cdot)\) \(\chi_{3311}(740,\cdot)\) \(\chi_{3311}(873,\cdot)\) \(\chi_{3311}(927,\cdot)\) \(\chi_{3311}(971,\cdot)\) \(\chi_{3311}(1006,\cdot)\) \(\chi_{3311}(1158,\cdot)\) \(\chi_{3311}(1214,\cdot)\) \(\chi_{3311}(1235,\cdot)\) \(\chi_{3311}(1270,\cdot)\) \(\chi_{3311}(1307,\cdot)\) \(\chi_{3311}(1391,\cdot)\) \(\chi_{3311}(1433,\cdot)\) \(\chi_{3311}(1543,\cdot)\) \(\chi_{3311}(1571,\cdot)\) \(\chi_{3311}(1643,\cdot)\) \(\chi_{3311}(1692,\cdot)\) \(\chi_{3311}(1776,\cdot)\) \(\chi_{3311}(1830,\cdot)\) \(\chi_{3311}(1874,\cdot)\) \(\chi_{3311}(2061,\cdot)\) \(\chi_{3311}(2077,\cdot)\) \(\chi_{3311}(2117,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 210 polynomial (not computed)

Values on generators

\((1893,904,2927)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{3}{5}\right),e\left(\frac{17}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 3311 }(31, a) \) \(-1\)\(1\)\(e\left(\frac{83}{105}\right)\)\(e\left(\frac{163}{210}\right)\)\(e\left(\frac{61}{105}\right)\)\(e\left(\frac{33}{70}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{58}{105}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{1}{210}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3311 }(31,a) \;\) at \(\;a = \) e.g. 2