Properties

Label 3311.57
Modulus $3311$
Conductor $473$
Order $210$
Real no
Primitive no
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([0,21,100]))
 
pari: [g,chi] = znchar(Mod(57,3311))
 

Basic properties

Modulus: \(3311\)
Conductor: \(473\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{473}(57,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3311.fd

\(\chi_{3311}(57,\cdot)\) \(\chi_{3311}(239,\cdot)\) \(\chi_{3311}(281,\cdot)\) \(\chi_{3311}(316,\cdot)\) \(\chi_{3311}(358,\cdot)\) \(\chi_{3311}(470,\cdot)\) \(\chi_{3311}(547,\cdot)\) \(\chi_{3311}(568,\cdot)\) \(\chi_{3311}(701,\cdot)\) \(\chi_{3311}(799,\cdot)\) \(\chi_{3311}(827,\cdot)\) \(\chi_{3311}(855,\cdot)\) \(\chi_{3311}(1128,\cdot)\) \(\chi_{3311}(1135,\cdot)\) \(\chi_{3311}(1184,\cdot)\) \(\chi_{3311}(1261,\cdot)\) \(\chi_{3311}(1436,\cdot)\) \(\chi_{3311}(1443,\cdot)\) \(\chi_{3311}(1471,\cdot)\) \(\chi_{3311}(1520,\cdot)\) \(\chi_{3311}(1674,\cdot)\) \(\chi_{3311}(1702,\cdot)\) \(\chi_{3311}(1744,\cdot)\) \(\chi_{3311}(1751,\cdot)\) \(\chi_{3311}(1821,\cdot)\) \(\chi_{3311}(1905,\cdot)\) \(\chi_{3311}(1975,\cdot)\) \(\chi_{3311}(2031,\cdot)\) \(\chi_{3311}(2052,\cdot)\) \(\chi_{3311}(2059,\cdot)\) ...

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

\((1893,904,2927)\) → \((1,e\left(\frac{1}{10}\right),e\left(\frac{10}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 3311 }(57, a) \) \(-1\)\(1\)\(e\left(\frac{67}{70}\right)\)\(e\left(\frac{29}{105}\right)\)\(e\left(\frac{32}{35}\right)\)\(e\left(\frac{32}{105}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{61}{70}\right)\)\(e\left(\frac{58}{105}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{71}{210}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3311 }(57,a) \;\) at \(\;a = \) e.g. 2