Properties

Label 4608.37
Modulus $4608$
Conductor $512$
Order $128$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4608, base_ring=CyclotomicField(128))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,25,0]))
 
pari: [g,chi] = znchar(Mod(37,4608))
 

Basic properties

Modulus: \(4608\)
Conductor: \(512\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(128\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{512}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4608.cb

\(\chi_{4608}(37,\cdot)\) \(\chi_{4608}(109,\cdot)\) \(\chi_{4608}(181,\cdot)\) \(\chi_{4608}(253,\cdot)\) \(\chi_{4608}(325,\cdot)\) \(\chi_{4608}(397,\cdot)\) \(\chi_{4608}(469,\cdot)\) \(\chi_{4608}(541,\cdot)\) \(\chi_{4608}(613,\cdot)\) \(\chi_{4608}(685,\cdot)\) \(\chi_{4608}(757,\cdot)\) \(\chi_{4608}(829,\cdot)\) \(\chi_{4608}(901,\cdot)\) \(\chi_{4608}(973,\cdot)\) \(\chi_{4608}(1045,\cdot)\) \(\chi_{4608}(1117,\cdot)\) \(\chi_{4608}(1189,\cdot)\) \(\chi_{4608}(1261,\cdot)\) \(\chi_{4608}(1333,\cdot)\) \(\chi_{4608}(1405,\cdot)\) \(\chi_{4608}(1477,\cdot)\) \(\chi_{4608}(1549,\cdot)\) \(\chi_{4608}(1621,\cdot)\) \(\chi_{4608}(1693,\cdot)\) \(\chi_{4608}(1765,\cdot)\) \(\chi_{4608}(1837,\cdot)\) \(\chi_{4608}(1909,\cdot)\) \(\chi_{4608}(1981,\cdot)\) \(\chi_{4608}(2053,\cdot)\) \(\chi_{4608}(2125,\cdot)\) ...

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

\((3583,2053,4097)\) → \((1,e\left(\frac{25}{128}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 4608 }(37, a) \) \(1\)\(1\)\(e\left(\frac{25}{128}\right)\)\(e\left(\frac{29}{64}\right)\)\(e\left(\frac{77}{128}\right)\)\(e\left(\frac{87}{128}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{63}{128}\right)\)\(e\left(\frac{47}{64}\right)\)\(e\left(\frac{25}{64}\right)\)\(e\left(\frac{3}{128}\right)\)\(e\left(\frac{9}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4608 }(37,a) \;\) at \(\;a = \) e.g. 2