Properties

Label 2624.413
Modulus $2624$
Conductor $2624$
Order $16$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2624, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,11,6]))
 
pari: [g,chi] = znchar(Mod(413,2624))
 

Basic properties

Modulus: \(2624\)
Conductor: \(2624\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2624.cb

\(\chi_{2624}(413,\cdot)\) \(\chi_{2624}(629,\cdot)\) \(\chi_{2624}(653,\cdot)\) \(\chi_{2624}(1093,\cdot)\) \(\chi_{2624}(1725,\cdot)\) \(\chi_{2624}(1941,\cdot)\) \(\chi_{2624}(1965,\cdot)\) \(\chi_{2624}(2405,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: 16.0.22926811036981387084399571372388167248844423168.2

Values on generators

\((575,1477,129)\) → \((1,e\left(\frac{11}{16}\right),e\left(\frac{3}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 2624 }(413, a) \) \(-1\)\(1\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{3}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2624 }(413,a) \;\) at \(\;a = \) e.g. 2