Properties

Label 1800.493
Modulus $1800$
Conductor $360$
Order $12$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,6,8,9]))
 
pari: [g,chi] = znchar(Mod(493,1800))
 

Basic properties

Modulus: \(1800\)
Conductor: \(360\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{360}(133,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1800.ck

\(\chi_{1800}(157,\cdot)\) \(\chi_{1800}(493,\cdot)\) \(\chi_{1800}(1093,\cdot)\) \(\chi_{1800}(1357,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.0.22039921152000000000.1

Values on generators

\((1351,901,1001,577)\) → \((1,-1,e\left(\frac{2}{3}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1800 }(493, a) \) \(-1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{12}\right)\)\(-i\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(i\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1800 }(493,a) \;\) at \(\;a = \) e.g. 2