Properties

Label 1215.191
Modulus 12151215
Conductor 243243
Order 162162
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1215, base_ring=CyclotomicField(162))
 
M = H._module
 
chi = DirichletCharacter(H, M([91,0]))
 
pari: [g,chi] = znchar(Mod(191,1215))
 

Basic properties

Modulus: 12151215
Conductor: 243243
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 162162
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from χ243(191,)\chi_{243}(191,\cdot)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1215.z

χ1215(11,)\chi_{1215}(11,\cdot) χ1215(41,)\chi_{1215}(41,\cdot) χ1215(56,)\chi_{1215}(56,\cdot) χ1215(86,)\chi_{1215}(86,\cdot) χ1215(101,)\chi_{1215}(101,\cdot) χ1215(131,)\chi_{1215}(131,\cdot) χ1215(146,)\chi_{1215}(146,\cdot) χ1215(176,)\chi_{1215}(176,\cdot) χ1215(191,)\chi_{1215}(191,\cdot) χ1215(221,)\chi_{1215}(221,\cdot) χ1215(236,)\chi_{1215}(236,\cdot) χ1215(266,)\chi_{1215}(266,\cdot) χ1215(281,)\chi_{1215}(281,\cdot) χ1215(311,)\chi_{1215}(311,\cdot) χ1215(326,)\chi_{1215}(326,\cdot) χ1215(356,)\chi_{1215}(356,\cdot) χ1215(371,)\chi_{1215}(371,\cdot) χ1215(401,)\chi_{1215}(401,\cdot) χ1215(416,)\chi_{1215}(416,\cdot) χ1215(446,)\chi_{1215}(446,\cdot) χ1215(461,)\chi_{1215}(461,\cdot) χ1215(491,)\chi_{1215}(491,\cdot) χ1215(506,)\chi_{1215}(506,\cdot) χ1215(536,)\chi_{1215}(536,\cdot) χ1215(551,)\chi_{1215}(551,\cdot) χ1215(581,)\chi_{1215}(581,\cdot) χ1215(596,)\chi_{1215}(596,\cdot) χ1215(626,)\chi_{1215}(626,\cdot) χ1215(641,)\chi_{1215}(641,\cdot) χ1215(671,)\chi_{1215}(671,\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(ζ81)\Q(\zeta_{81})
Fixed field: Number field defined by a degree 162 polynomial (not computed)

Values on generators

(731,487)(731,487)(e(91162),1)(e\left(\frac{91}{162}\right),1)

First values

aa 1-11122447788111113131414161617171919
χ1215(191,a) \chi_{ 1215 }(191, a) 1-111e(91162)e\left(\frac{91}{162}\right)e(1081)e\left(\frac{10}{81}\right)e(2681)e\left(\frac{26}{81}\right)e(3754)e\left(\frac{37}{54}\right)e(157162)e\left(\frac{157}{162}\right)e(4081)e\left(\frac{40}{81}\right)e(143162)e\left(\frac{143}{162}\right)e(2081)e\left(\frac{20}{81}\right)e(2954)e\left(\frac{29}{54}\right)e(1727)e\left(\frac{17}{27}\right)
sage: chi.jacobi_sum(n)
 
χ1215(191,a)   \chi_{ 1215 }(191,a) \; at   a=\;a = e.g. 2