Properties

Label 3234.1355
Modulus $3234$
Conductor $1617$
Order $210$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3234, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([105,20,21]))
 
pari: [g,chi] = znchar(Mod(1355,3234))
 

Basic properties

Modulus: \(3234\)
Conductor: \(1617\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1617}(1355,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3234.ci

\(\chi_{3234}(95,\cdot)\) \(\chi_{3234}(107,\cdot)\) \(\chi_{3234}(149,\cdot)\) \(\chi_{3234}(233,\cdot)\) \(\chi_{3234}(305,\cdot)\) \(\chi_{3234}(347,\cdot)\) \(\chi_{3234}(359,\cdot)\) \(\chi_{3234}(431,\cdot)\) \(\chi_{3234}(611,\cdot)\) \(\chi_{3234}(695,\cdot)\) \(\chi_{3234}(767,\cdot)\) \(\chi_{3234}(809,\cdot)\) \(\chi_{3234}(821,\cdot)\) \(\chi_{3234}(893,\cdot)\) \(\chi_{3234}(1019,\cdot)\) \(\chi_{3234}(1031,\cdot)\) \(\chi_{3234}(1073,\cdot)\) \(\chi_{3234}(1229,\cdot)\) \(\chi_{3234}(1271,\cdot)\) \(\chi_{3234}(1283,\cdot)\) \(\chi_{3234}(1355,\cdot)\) \(\chi_{3234}(1481,\cdot)\) \(\chi_{3234}(1493,\cdot)\) \(\chi_{3234}(1535,\cdot)\) \(\chi_{3234}(1619,\cdot)\) \(\chi_{3234}(1691,\cdot)\) \(\chi_{3234}(1817,\cdot)\) \(\chi_{3234}(1943,\cdot)\) \(\chi_{3234}(1955,\cdot)\) \(\chi_{3234}(1997,\cdot)\) ...

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

\((1079,199,2059)\) → \((-1,e\left(\frac{2}{21}\right),e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3234 }(1355, a) \) \(1\)\(1\)\(e\left(\frac{139}{210}\right)\)\(e\left(\frac{17}{70}\right)\)\(e\left(\frac{82}{105}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{34}{105}\right)\)\(e\left(\frac{32}{35}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{26}{105}\right)\)\(e\left(\frac{8}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3234 }(1355,a) \;\) at \(\;a = \) e.g. 2