Properties

Label 69696.2177
Modulus $69696$
Conductor $33$
Order $2$
Real yes
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69696, base_ring=CyclotomicField(2))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,1,1]))
 
pari: [g,chi] = znchar(Mod(2177,69696))
 

Basic properties

Modulus: \(69696\)
Conductor: \(33\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{33}(32,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 69696.b

\(\chi_{69696}(2177,\cdot)\)

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: \(\Q\)
Fixed field: \(\Q(\sqrt{33}) \)

Values on generators

\((67519,4357,54209,14401)\) → \((1,1,-1,-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 69696 }(2177, a) \) \(1\)\(1\)\(-1\)\(-1\)\(-1\)\(1\)\(-1\)\(-1\)\(1\)\(1\)\(1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 69696 }(2177,a) \;\) at \(\;a = \) e.g. 2