Properties

Label 3072.13
Modulus $3072$
Conductor $1024$
Order $256$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3072, base_ring=CyclotomicField(256))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,239,0]))
 
pari: [g,chi] = znchar(Mod(13,3072))
 

Basic properties

Modulus: \(3072\)
Conductor: \(1024\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(256\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1024}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3072.bh

\(\chi_{3072}(13,\cdot)\) \(\chi_{3072}(37,\cdot)\) \(\chi_{3072}(61,\cdot)\) \(\chi_{3072}(85,\cdot)\) \(\chi_{3072}(109,\cdot)\) \(\chi_{3072}(133,\cdot)\) \(\chi_{3072}(157,\cdot)\) \(\chi_{3072}(181,\cdot)\) \(\chi_{3072}(205,\cdot)\) \(\chi_{3072}(229,\cdot)\) \(\chi_{3072}(253,\cdot)\) \(\chi_{3072}(277,\cdot)\) \(\chi_{3072}(301,\cdot)\) \(\chi_{3072}(325,\cdot)\) \(\chi_{3072}(349,\cdot)\) \(\chi_{3072}(373,\cdot)\) \(\chi_{3072}(397,\cdot)\) \(\chi_{3072}(421,\cdot)\) \(\chi_{3072}(445,\cdot)\) \(\chi_{3072}(469,\cdot)\) \(\chi_{3072}(493,\cdot)\) \(\chi_{3072}(517,\cdot)\) \(\chi_{3072}(541,\cdot)\) \(\chi_{3072}(565,\cdot)\) \(\chi_{3072}(589,\cdot)\) \(\chi_{3072}(613,\cdot)\) \(\chi_{3072}(637,\cdot)\) \(\chi_{3072}(661,\cdot)\) \(\chi_{3072}(685,\cdot)\) \(\chi_{3072}(709,\cdot)\) ...

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: $\Q(\zeta_{256})$
Fixed field: Number field defined by a degree 256 polynomial (not computed)

Values on generators

\((2047,2053,1025)\) → \((1,e\left(\frac{239}{256}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 3072 }(13, a) \) \(1\)\(1\)\(e\left(\frac{239}{256}\right)\)\(e\left(\frac{75}{128}\right)\)\(e\left(\frac{219}{256}\right)\)\(e\left(\frac{33}{256}\right)\)\(e\left(\frac{41}{64}\right)\)\(e\left(\frac{249}{256}\right)\)\(e\left(\frac{73}{128}\right)\)\(e\left(\frac{111}{128}\right)\)\(e\left(\frac{213}{256}\right)\)\(e\left(\frac{31}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3072 }(13,a) \;\) at \(\;a = \) e.g. 2