Properties

Label 4608.67
Modulus $4608$
Conductor $4608$
Order $384$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4608, base_ring=CyclotomicField(384))
 
M = H._module
 
chi = DirichletCharacter(H, M([192,345,128]))
 
pari: [g,chi] = znchar(Mod(67,4608))
 

Basic properties

Modulus: \(4608\)
Conductor: \(4608\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(384\)
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 4608.cl

\(\chi_{4608}(43,\cdot)\) \(\chi_{4608}(67,\cdot)\) \(\chi_{4608}(115,\cdot)\) \(\chi_{4608}(139,\cdot)\) \(\chi_{4608}(187,\cdot)\) \(\chi_{4608}(211,\cdot)\) \(\chi_{4608}(259,\cdot)\) \(\chi_{4608}(283,\cdot)\) \(\chi_{4608}(331,\cdot)\) \(\chi_{4608}(355,\cdot)\) \(\chi_{4608}(403,\cdot)\) \(\chi_{4608}(427,\cdot)\) \(\chi_{4608}(475,\cdot)\) \(\chi_{4608}(499,\cdot)\) \(\chi_{4608}(547,\cdot)\) \(\chi_{4608}(571,\cdot)\) \(\chi_{4608}(619,\cdot)\) \(\chi_{4608}(643,\cdot)\) \(\chi_{4608}(691,\cdot)\) \(\chi_{4608}(715,\cdot)\) \(\chi_{4608}(763,\cdot)\) \(\chi_{4608}(787,\cdot)\) \(\chi_{4608}(835,\cdot)\) \(\chi_{4608}(859,\cdot)\) \(\chi_{4608}(907,\cdot)\) \(\chi_{4608}(931,\cdot)\) \(\chi_{4608}(979,\cdot)\) \(\chi_{4608}(1003,\cdot)\) \(\chi_{4608}(1051,\cdot)\) \(\chi_{4608}(1075,\cdot)\) ...

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: $\Q(\zeta_{384})$
Fixed field: Number field defined by a degree 384 polynomial (not computed)

Values on generators

\((3583,2053,4097)\) → \((-1,e\left(\frac{115}{128}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 4608 }(67, a) \) \(-1\)\(1\)\(e\left(\frac{217}{384}\right)\)\(e\left(\frac{61}{192}\right)\)\(e\left(\frac{77}{384}\right)\)\(e\left(\frac{151}{384}\right)\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{21}{128}\right)\)\(e\left(\frac{143}{192}\right)\)\(e\left(\frac{25}{192}\right)\)\(e\left(\frac{323}{384}\right)\)\(e\left(\frac{17}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4608 }(67,a) \;\) at \(\;a = \) e.g. 2