Properties

Label 3234.907
Modulus $3234$
Conductor $539$
Order $105$
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([0,80,84]))
 
pari: [g,chi] = znchar(Mod(907,3234))
 

Basic properties

Modulus: \(3234\)
Conductor: \(539\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(105\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{539}(368,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3234.ce

\(\chi_{3234}(25,\cdot)\) \(\chi_{3234}(37,\cdot)\) \(\chi_{3234}(163,\cdot)\) \(\chi_{3234}(235,\cdot)\) \(\chi_{3234}(247,\cdot)\) \(\chi_{3234}(289,\cdot)\) \(\chi_{3234}(445,\cdot)\) \(\chi_{3234}(487,\cdot)\) \(\chi_{3234}(499,\cdot)\) \(\chi_{3234}(625,\cdot)\) \(\chi_{3234}(697,\cdot)\) \(\chi_{3234}(709,\cdot)\) \(\chi_{3234}(751,\cdot)\) \(\chi_{3234}(823,\cdot)\) \(\chi_{3234}(907,\cdot)\) \(\chi_{3234}(1087,\cdot)\) \(\chi_{3234}(1159,\cdot)\) \(\chi_{3234}(1171,\cdot)\) \(\chi_{3234}(1213,\cdot)\) \(\chi_{3234}(1285,\cdot)\) \(\chi_{3234}(1369,\cdot)\) \(\chi_{3234}(1411,\cdot)\) \(\chi_{3234}(1423,\cdot)\) \(\chi_{3234}(1621,\cdot)\) \(\chi_{3234}(1633,\cdot)\) \(\chi_{3234}(1675,\cdot)\) \(\chi_{3234}(1747,\cdot)\) \(\chi_{3234}(1873,\cdot)\) \(\chi_{3234}(1885,\cdot)\) \(\chi_{3234}(2011,\cdot)\) ...

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 105 polynomial (not computed)

Values on generators

\((1079,199,2059)\) → \((1,e\left(\frac{8}{21}\right),e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3234 }(907, a) \) \(1\)\(1\)\(e\left(\frac{68}{105}\right)\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{13}{105}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{31}{105}\right)\)\(e\left(\frac{23}{35}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{104}{105}\right)\)\(e\left(\frac{32}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3234 }(907,a) \;\) at \(\;a = \) e.g. 2