Properties

Label 9216.35
Modulus $9216$
Conductor $3072$
Order $256$
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(256))
 
M = H._module
 
chi = DirichletCharacter(H, M([128,203,128]))
 
pari: [g,chi] = znchar(Mod(35,9216))
 

Basic properties

Modulus: \(9216\)
Conductor: \(3072\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(256\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3072}(35,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9216.ck

\(\chi_{9216}(35,\cdot)\) \(\chi_{9216}(107,\cdot)\) \(\chi_{9216}(179,\cdot)\) \(\chi_{9216}(251,\cdot)\) \(\chi_{9216}(323,\cdot)\) \(\chi_{9216}(395,\cdot)\) \(\chi_{9216}(467,\cdot)\) \(\chi_{9216}(539,\cdot)\) \(\chi_{9216}(611,\cdot)\) \(\chi_{9216}(683,\cdot)\) \(\chi_{9216}(755,\cdot)\) \(\chi_{9216}(827,\cdot)\) \(\chi_{9216}(899,\cdot)\) \(\chi_{9216}(971,\cdot)\) \(\chi_{9216}(1043,\cdot)\) \(\chi_{9216}(1115,\cdot)\) \(\chi_{9216}(1187,\cdot)\) \(\chi_{9216}(1259,\cdot)\) \(\chi_{9216}(1331,\cdot)\) \(\chi_{9216}(1403,\cdot)\) \(\chi_{9216}(1475,\cdot)\) \(\chi_{9216}(1547,\cdot)\) \(\chi_{9216}(1619,\cdot)\) \(\chi_{9216}(1691,\cdot)\) \(\chi_{9216}(1763,\cdot)\) \(\chi_{9216}(1835,\cdot)\) \(\chi_{9216}(1907,\cdot)\) \(\chi_{9216}(1979,\cdot)\) \(\chi_{9216}(2051,\cdot)\) \(\chi_{9216}(2123,\cdot)\) ...

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

\((8191,2053,4097)\) → \((-1,e\left(\frac{203}{256}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 9216 }(35, a) \) \(1\)\(1\)\(e\left(\frac{75}{256}\right)\)\(e\left(\frac{87}{128}\right)\)\(e\left(\frac{231}{256}\right)\)\(e\left(\frac{133}{256}\right)\)\(e\left(\frac{13}{64}\right)\)\(e\left(\frac{61}{256}\right)\)\(e\left(\frac{77}{128}\right)\)\(e\left(\frac{75}{128}\right)\)\(e\left(\frac{9}{256}\right)\)\(e\left(\frac{11}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9216 }(35,a) \;\) at \(\;a = \) e.g. 2