Properties

Label 3234.1867
Modulus $3234$
Conductor $539$
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([0,145,63]))
 
pari: [g,chi] = znchar(Mod(1867,3234))
 

Basic properties

Modulus: \(3234\)
Conductor: \(539\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{539}(250,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3234.ck

\(\chi_{3234}(61,\cdot)\) \(\chi_{3234}(73,\cdot)\) \(\chi_{3234}(145,\cdot)\) \(\chi_{3234}(271,\cdot)\) \(\chi_{3234}(283,\cdot)\) \(\chi_{3234}(409,\cdot)\) \(\chi_{3234}(481,\cdot)\) \(\chi_{3234}(523,\cdot)\) \(\chi_{3234}(535,\cdot)\) \(\chi_{3234}(733,\cdot)\) \(\chi_{3234}(745,\cdot)\) \(\chi_{3234}(787,\cdot)\) \(\chi_{3234}(871,\cdot)\) \(\chi_{3234}(943,\cdot)\) \(\chi_{3234}(985,\cdot)\) \(\chi_{3234}(997,\cdot)\) \(\chi_{3234}(1069,\cdot)\) \(\chi_{3234}(1249,\cdot)\) \(\chi_{3234}(1333,\cdot)\) \(\chi_{3234}(1405,\cdot)\) \(\chi_{3234}(1447,\cdot)\) \(\chi_{3234}(1459,\cdot)\) \(\chi_{3234}(1531,\cdot)\) \(\chi_{3234}(1657,\cdot)\) \(\chi_{3234}(1669,\cdot)\) \(\chi_{3234}(1711,\cdot)\) \(\chi_{3234}(1867,\cdot)\) \(\chi_{3234}(1909,\cdot)\) \(\chi_{3234}(1921,\cdot)\) \(\chi_{3234}(1993,\cdot)\) ...

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

\((1079,199,2059)\) → \((1,e\left(\frac{29}{42}\right),e\left(\frac{3}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3234 }(1867, a) \) \(1\)\(1\)\(e\left(\frac{47}{210}\right)\)\(e\left(\frac{3}{35}\right)\)\(e\left(\frac{101}{105}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{47}{105}\right)\)\(e\left(\frac{37}{70}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{73}{105}\right)\)\(e\left(\frac{9}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3234 }(1867,a) \;\) at \(\;a = \) e.g. 2