Properties

Label 243675.43
Modulus 243675243675
Conductor 4873548735
Order 684684
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(243675, base_ring=CyclotomicField(684))
 
M = H._module
 
chi = DirichletCharacter(H, M([152,513,104]))
 
pari: [g,chi] = znchar(Mod(43,243675))
 

Basic properties

Modulus: 243675243675
Conductor: 4873548735
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 684684
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from χ48735(43,)\chi_{48735}(43,\cdot)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 243675.yv

χ243675(43,)\chi_{243675}(43,\cdot) χ243675(682,)\chi_{243675}(682,\cdot) χ243675(1582,)\chi_{243675}(1582,\cdot) χ243675(6343,)\chi_{243675}(6343,\cdot) χ243675(7882,)\chi_{243675}(7882,\cdot) χ243675(8293,)\chi_{243675}(8293,\cdot) χ243675(8518,)\chi_{243675}(8518,\cdot) χ243675(9832,)\chi_{243675}(9832,\cdot) χ243675(10057,)\chi_{243675}(10057,\cdot) χ243675(10093,)\chi_{243675}(10093,\cdot) χ243675(11632,)\chi_{243675}(11632,\cdot) χ243675(11968,)\chi_{243675}(11968,\cdot) χ243675(12868,)\chi_{243675}(12868,\cdot) χ243675(13507,)\chi_{243675}(13507,\cdot) χ243675(14407,)\chi_{243675}(14407,\cdot) χ243675(19168,)\chi_{243675}(19168,\cdot) χ243675(20707,)\chi_{243675}(20707,\cdot) χ243675(21118,)\chi_{243675}(21118,\cdot) χ243675(21343,)\chi_{243675}(21343,\cdot) χ243675(22657,)\chi_{243675}(22657,\cdot) χ243675(22882,)\chi_{243675}(22882,\cdot) χ243675(22918,)\chi_{243675}(22918,\cdot) χ243675(24457,)\chi_{243675}(24457,\cdot) χ243675(26332,)\chi_{243675}(26332,\cdot) χ243675(27232,)\chi_{243675}(27232,\cdot) χ243675(31993,)\chi_{243675}(31993,\cdot) χ243675(33532,)\chi_{243675}(33532,\cdot) χ243675(33943,)\chi_{243675}(33943,\cdot) χ243675(35482,)\chi_{243675}(35482,\cdot) χ243675(35707,)\chi_{243675}(35707,\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(ζ684)\Q(\zeta_{684})
Fixed field: Number field defined by a degree 684 polynomial (not computed)

Values on generators

(36101,77977,129601)(36101,77977,129601)(e(29),i,e(26171))(e\left(\frac{2}{9}\right),-i,e\left(\frac{26}{171}\right))

First values

aa 1-11122447788111113131414161617172222
χ243675(43,a) \chi_{ 243675 }(43, a) 1-111e(85684)e\left(\frac{85}{684}\right)e(85342)e\left(\frac{85}{342}\right)e(77684)e\left(\frac{77}{684}\right)e(85228)e\left(\frac{85}{228}\right)e(68171)e\left(\frac{68}{171}\right)e(35684)e\left(\frac{35}{684}\right)e(938)e\left(\frac{9}{38}\right)e(85171)e\left(\frac{85}{171}\right)e(305684)e\left(\frac{305}{684}\right)e(119228)e\left(\frac{119}{228}\right)
sage: chi.jacobi_sum(n)
 
χ243675(43,a)   \chi_{ 243675 }(43,a) \; at   a=\;a = e.g. 2