Properties

Label 3584.181
Modulus $3584$
Conductor $3584$
Order $128$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3584, base_ring=CyclotomicField(128))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,101,64]))
 
pari: [g,chi] = znchar(Mod(181,3584))
 

Basic properties

Modulus: \(3584\)
Conductor: \(3584\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(128\)
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 3584.cd

\(\chi_{3584}(13,\cdot)\) \(\chi_{3584}(69,\cdot)\) \(\chi_{3584}(125,\cdot)\) \(\chi_{3584}(181,\cdot)\) \(\chi_{3584}(237,\cdot)\) \(\chi_{3584}(293,\cdot)\) \(\chi_{3584}(349,\cdot)\) \(\chi_{3584}(405,\cdot)\) \(\chi_{3584}(461,\cdot)\) \(\chi_{3584}(517,\cdot)\) \(\chi_{3584}(573,\cdot)\) \(\chi_{3584}(629,\cdot)\) \(\chi_{3584}(685,\cdot)\) \(\chi_{3584}(741,\cdot)\) \(\chi_{3584}(797,\cdot)\) \(\chi_{3584}(853,\cdot)\) \(\chi_{3584}(909,\cdot)\) \(\chi_{3584}(965,\cdot)\) \(\chi_{3584}(1021,\cdot)\) \(\chi_{3584}(1077,\cdot)\) \(\chi_{3584}(1133,\cdot)\) \(\chi_{3584}(1189,\cdot)\) \(\chi_{3584}(1245,\cdot)\) \(\chi_{3584}(1301,\cdot)\) \(\chi_{3584}(1357,\cdot)\) \(\chi_{3584}(1413,\cdot)\) \(\chi_{3584}(1469,\cdot)\) \(\chi_{3584}(1525,\cdot)\) \(\chi_{3584}(1581,\cdot)\) \(\chi_{3584}(1637,\cdot)\) ...

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: $\Q(\zeta_{128})$
Fixed field: Number field defined by a degree 128 polynomial (not computed)

Values on generators

\((1023,1541,1025)\) → \((1,e\left(\frac{101}{128}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 3584 }(181, a) \) \(-1\)\(1\)\(e\left(\frac{15}{128}\right)\)\(e\left(\frac{37}{128}\right)\)\(e\left(\frac{15}{64}\right)\)\(e\left(\frac{9}{128}\right)\)\(e\left(\frac{11}{128}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{19}{32}\right)\)\(e\left(\frac{83}{128}\right)\)\(e\left(\frac{3}{64}\right)\)\(e\left(\frac{37}{64}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3584 }(181,a) \;\) at \(\;a = \) e.g. 2