Properties

Label 2667.64
Modulus $2667$
Conductor $127$
Order $7$
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(2667, base_ring=CyclotomicField(14))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,6]))
 
pari: [g,chi] = znchar(Mod(64,2667))
 

Basic properties

Modulus: \(2667\)
Conductor: \(127\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(7\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{127}(64,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2667.bo

\(\chi_{2667}(64,\cdot)\) \(\chi_{2667}(778,\cdot)\) \(\chi_{2667}(1429,\cdot)\) \(\chi_{2667}(1786,\cdot)\) \(\chi_{2667}(2290,\cdot)\) \(\chi_{2667}(2542,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: 7.7.4195872914689.1

Values on generators

\((890,1144,2416)\) → \((1,1,e\left(\frac{3}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 2667 }(64, a) \) \(1\)\(1\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2667 }(64,a) \;\) at \(\;a = \) e.g. 2