Properties

Label 1122.331
Modulus $1122$
Conductor $17$
Order $8$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1122, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,5]))
 
pari: [g,chi] = znchar(Mod(331,1122))
 

Basic properties

Modulus: \(1122\)
Conductor: \(17\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{17}(8,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1122.o

\(\chi_{1122}(331,\cdot)\) \(\chi_{1122}(529,\cdot)\) \(\chi_{1122}(661,\cdot)\) \(\chi_{1122}(859,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: \(\Q(\zeta_{17})^+\)

Values on generators

\((749,409,1057)\) → \((1,1,e\left(\frac{5}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)\(37\)
\( \chi_{ 1122 }(331, a) \) \(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(-1\)\(-i\)\(e\left(\frac{3}{8}\right)\)\(i\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(1\)\(e\left(\frac{5}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1122 }(331,a) \;\) at \(\;a = \) e.g. 2