Properties

Label 1080.7
Modulus $1080$
Conductor $540$
Order $36$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,0,32,9]))
 
pari: [g,chi] = znchar(Mod(7,1080))
 

Basic properties

Modulus: \(1080\)
Conductor: \(540\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{540}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1080.cp

\(\chi_{1080}(7,\cdot)\) \(\chi_{1080}(103,\cdot)\) \(\chi_{1080}(223,\cdot)\) \(\chi_{1080}(247,\cdot)\) \(\chi_{1080}(367,\cdot)\) \(\chi_{1080}(463,\cdot)\) \(\chi_{1080}(583,\cdot)\) \(\chi_{1080}(607,\cdot)\) \(\chi_{1080}(727,\cdot)\) \(\chi_{1080}(823,\cdot)\) \(\chi_{1080}(943,\cdot)\) \(\chi_{1080}(967,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: Number field defined by a degree 36 polynomial

Values on generators

\((271,541,1001,217)\) → \((-1,1,e\left(\frac{8}{9}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1080 }(7, a) \) \(1\)\(1\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1080 }(7,a) \;\) at \(\;a = \) e.g. 2