Properties

Label 9216.47
Modulus $9216$
Conductor $2304$
Order $192$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9216, base_ring=CyclotomicField(192))
 
M = H._module
 
chi = DirichletCharacter(H, M([96,129,32]))
 
pari: [g,chi] = znchar(Mod(47,9216))
 

Basic properties

Modulus: \(9216\)
Conductor: \(2304\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(192\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2304}(1955,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9216.ch

\(\chi_{9216}(47,\cdot)\) \(\chi_{9216}(239,\cdot)\) \(\chi_{9216}(335,\cdot)\) \(\chi_{9216}(527,\cdot)\) \(\chi_{9216}(623,\cdot)\) \(\chi_{9216}(815,\cdot)\) \(\chi_{9216}(911,\cdot)\) \(\chi_{9216}(1103,\cdot)\) \(\chi_{9216}(1199,\cdot)\) \(\chi_{9216}(1391,\cdot)\) \(\chi_{9216}(1487,\cdot)\) \(\chi_{9216}(1679,\cdot)\) \(\chi_{9216}(1775,\cdot)\) \(\chi_{9216}(1967,\cdot)\) \(\chi_{9216}(2063,\cdot)\) \(\chi_{9216}(2255,\cdot)\) \(\chi_{9216}(2351,\cdot)\) \(\chi_{9216}(2543,\cdot)\) \(\chi_{9216}(2639,\cdot)\) \(\chi_{9216}(2831,\cdot)\) \(\chi_{9216}(2927,\cdot)\) \(\chi_{9216}(3119,\cdot)\) \(\chi_{9216}(3215,\cdot)\) \(\chi_{9216}(3407,\cdot)\) \(\chi_{9216}(3503,\cdot)\) \(\chi_{9216}(3695,\cdot)\) \(\chi_{9216}(3791,\cdot)\) \(\chi_{9216}(3983,\cdot)\) \(\chi_{9216}(4079,\cdot)\) \(\chi_{9216}(4271,\cdot)\) ...

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: $\Q(\zeta_{192})$
Fixed field: Number field defined by a degree 192 polynomial (not computed)

Values on generators

\((8191,2053,4097)\) → \((-1,e\left(\frac{43}{64}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 9216 }(47, a) \) \(1\)\(1\)\(e\left(\frac{97}{192}\right)\)\(e\left(\frac{85}{96}\right)\)\(e\left(\frac{149}{192}\right)\)\(e\left(\frac{175}{192}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{61}{64}\right)\)\(e\left(\frac{71}{96}\right)\)\(e\left(\frac{1}{96}\right)\)\(e\left(\frac{155}{192}\right)\)\(e\left(\frac{5}{24}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9216 }(47,a) \;\) at \(\;a = \) e.g. 2