Properties

Label 3200.57
Modulus $3200$
Conductor $320$
Order $16$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3200, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,13,4]))
 
pari: [g,chi] = znchar(Mod(57,3200))
 

Basic properties

Modulus: \(3200\)
Conductor: \(320\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{320}(277,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3200.bq

\(\chi_{3200}(57,\cdot)\) \(\chi_{3200}(393,\cdot)\) \(\chi_{3200}(857,\cdot)\) \(\chi_{3200}(1193,\cdot)\) \(\chi_{3200}(1657,\cdot)\) \(\chi_{3200}(1993,\cdot)\) \(\chi_{3200}(2457,\cdot)\) \(\chi_{3200}(2793,\cdot)\)

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: 16.0.147573952589676412928000000000000.2

Values on generators

\((1151,901,2177)\) → \((1,e\left(\frac{13}{16}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 3200 }(57, a) \) \(-1\)\(1\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(1\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{9}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3200 }(57,a) \;\) at \(\;a = \) e.g. 2