Properties

Label 7168.111
Modulus $7168$
Conductor $1792$
Order $64$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7168, base_ring=CyclotomicField(64))
 
M = H._module
 
chi = DirichletCharacter(H, M([32,47,32]))
 
pari: [g,chi] = znchar(Mod(111,7168))
 

Basic properties

Modulus: \(7168\)
Conductor: \(1792\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1792}(755,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7168.bv

\(\chi_{7168}(111,\cdot)\) \(\chi_{7168}(335,\cdot)\) \(\chi_{7168}(559,\cdot)\) \(\chi_{7168}(783,\cdot)\) \(\chi_{7168}(1007,\cdot)\) \(\chi_{7168}(1231,\cdot)\) \(\chi_{7168}(1455,\cdot)\) \(\chi_{7168}(1679,\cdot)\) \(\chi_{7168}(1903,\cdot)\) \(\chi_{7168}(2127,\cdot)\) \(\chi_{7168}(2351,\cdot)\) \(\chi_{7168}(2575,\cdot)\) \(\chi_{7168}(2799,\cdot)\) \(\chi_{7168}(3023,\cdot)\) \(\chi_{7168}(3247,\cdot)\) \(\chi_{7168}(3471,\cdot)\) \(\chi_{7168}(3695,\cdot)\) \(\chi_{7168}(3919,\cdot)\) \(\chi_{7168}(4143,\cdot)\) \(\chi_{7168}(4367,\cdot)\) \(\chi_{7168}(4591,\cdot)\) \(\chi_{7168}(4815,\cdot)\) \(\chi_{7168}(5039,\cdot)\) \(\chi_{7168}(5263,\cdot)\) \(\chi_{7168}(5487,\cdot)\) \(\chi_{7168}(5711,\cdot)\) \(\chi_{7168}(5935,\cdot)\) \(\chi_{7168}(6159,\cdot)\) \(\chi_{7168}(6383,\cdot)\) \(\chi_{7168}(6607,\cdot)\) ...

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: $\Q(\zeta_{64})$
Fixed field: Number field defined by a degree 64 polynomial

Values on generators

\((1023,5125,1025)\) → \((-1,e\left(\frac{47}{64}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 7168 }(111, a) \) \(1\)\(1\)\(e\left(\frac{45}{64}\right)\)\(e\left(\frac{15}{64}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{59}{64}\right)\)\(e\left(\frac{1}{64}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{57}{64}\right)\)\(e\left(\frac{25}{32}\right)\)\(e\left(\frac{15}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7168 }(111,a) \;\) at \(\;a = \) e.g. 2