Properties

Label 2688.965
Modulus $2688$
Conductor $2688$
Order $32$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2688, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,17,16,16]))
 
pari: [g,chi] = znchar(Mod(965,2688))
 

Basic properties

Modulus: \(2688\)
Conductor: \(2688\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2688.db

\(\chi_{2688}(125,\cdot)\) \(\chi_{2688}(293,\cdot)\) \(\chi_{2688}(461,\cdot)\) \(\chi_{2688}(629,\cdot)\) \(\chi_{2688}(797,\cdot)\) \(\chi_{2688}(965,\cdot)\) \(\chi_{2688}(1133,\cdot)\) \(\chi_{2688}(1301,\cdot)\) \(\chi_{2688}(1469,\cdot)\) \(\chi_{2688}(1637,\cdot)\) \(\chi_{2688}(1805,\cdot)\) \(\chi_{2688}(1973,\cdot)\) \(\chi_{2688}(2141,\cdot)\) \(\chi_{2688}(2309,\cdot)\) \(\chi_{2688}(2477,\cdot)\) \(\chi_{2688}(2645,\cdot)\)

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: \(\Q(\zeta_{32})\)
Fixed field: 32.32.4489912604053908534055314729400632754872833954383027744299245049859706934263808.1

Values on generators

\((127,2437,1793,1921)\) → \((1,e\left(\frac{17}{32}\right),-1,-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 2688 }(965, a) \) \(1\)\(1\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{21}{32}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{23}{32}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{27}{32}\right)\)\(-i\)\(e\left(\frac{9}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2688 }(965,a) \;\) at \(\;a = \) e.g. 2