Properties

Label 1296.19
Modulus 12961296
Conductor 432432
Order 3636
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,27,32]))
 
pari: [g,chi] = znchar(Mod(19,1296))
 

Basic properties

Modulus: 12961296
Conductor: 432432
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 3636
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from χ432(115,)\chi_{432}(115,\cdot)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1296.bi

χ1296(19,)\chi_{1296}(19,\cdot) χ1296(91,)\chi_{1296}(91,\cdot) χ1296(235,)\chi_{1296}(235,\cdot) χ1296(307,)\chi_{1296}(307,\cdot) χ1296(451,)\chi_{1296}(451,\cdot) χ1296(523,)\chi_{1296}(523,\cdot) χ1296(667,)\chi_{1296}(667,\cdot) χ1296(739,)\chi_{1296}(739,\cdot) χ1296(883,)\chi_{1296}(883,\cdot) χ1296(955,)\chi_{1296}(955,\cdot) χ1296(1099,)\chi_{1296}(1099,\cdot) χ1296(1171,)\chi_{1296}(1171,\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(ζ36)\Q(\zeta_{36})
Fixed field: 36.0.614667125325361522818798575155151578949632894783197825857500612833312768.1

Values on generators

(1135,325,1217)(1135,325,1217)(1,i,e(89))(-1,-i,e\left(\frac{8}{9}\right))

First values

aa 1-111557711111313171719192323252529293131
χ1296(19,a) \chi_{ 1296 }(19, a) 1-111e(736)e\left(\frac{7}{36}\right)e(29)e\left(\frac{2}{9}\right)e(2936)e\left(\frac{29}{36}\right)e(1336)e\left(\frac{13}{36}\right)e(13)e\left(\frac{1}{3}\right)e(512)e\left(\frac{5}{12}\right)e(79)e\left(\frac{7}{9}\right)e(718)e\left(\frac{7}{18}\right)e(536)e\left(\frac{5}{36}\right)e(518)e\left(\frac{5}{18}\right)
sage: chi.jacobi_sum(n)
 
χ1296(19,a)   \chi_{ 1296 }(19,a) \; at   a=\;a = e.g. 2