Properties

Label 3311.47
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([175,168,60]))
 
pari: [g,chi] = znchar(Mod(47,3311))
 

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.fn

\(\chi_{3311}(47,\cdot)\) \(\chi_{3311}(59,\cdot)\) \(\chi_{3311}(213,\cdot)\) \(\chi_{3311}(236,\cdot)\) \(\chi_{3311}(262,\cdot)\) \(\chi_{3311}(269,\cdot)\) \(\chi_{3311}(355,\cdot)\) \(\chi_{3311}(465,\cdot)\) \(\chi_{3311}(537,\cdot)\) \(\chi_{3311}(570,\cdot)\) \(\chi_{3311}(752,\cdot)\) \(\chi_{3311}(852,\cdot)\) \(\chi_{3311}(950,\cdot)\) \(\chi_{3311}(962,\cdot)\) \(\chi_{3311}(1048,\cdot)\) \(\chi_{3311}(1116,\cdot)\) \(\chi_{3311}(1153,\cdot)\) \(\chi_{3311}(1202,\cdot)\) \(\chi_{3311}(1258,\cdot)\) \(\chi_{3311}(1263,\cdot)\) \(\chi_{3311}(1368,\cdot)\) \(\chi_{3311}(1417,\cdot)\) \(\chi_{3311}(1466,\cdot)\) \(\chi_{3311}(1655,\cdot)\) \(\chi_{3311}(1741,\cdot)\) \(\chi_{3311}(1774,\cdot)\) \(\chi_{3311}(1853,\cdot)\) \(\chi_{3311}(1951,\cdot)\) \(\chi_{3311}(1956,\cdot)\) \(\chi_{3311}(2105,\cdot)\) ...

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

\((1893,904,2927)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{4}{5}\right),e\left(\frac{2}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 3311 }(47, a) \) \(-1\)\(1\)\(e\left(\frac{19}{105}\right)\)\(e\left(\frac{109}{210}\right)\)\(e\left(\frac{38}{105}\right)\)\(e\left(\frac{107}{210}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{19}{35}\right)\)\(e\left(\frac{4}{105}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{31}{70}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3311 }(47,a) \;\) at \(\;a = \) e.g. 2