Properties

Label 7168.29
Modulus $7168$
Conductor $1024$
Order $256$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7168, base_ring=CyclotomicField(256))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,123,0]))
 
pari: [g,chi] = znchar(Mod(29,7168))
 

Basic properties

Modulus: \(7168\)
Conductor: \(1024\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(256\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1024}(29,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7168.cl

\(\chi_{7168}(29,\cdot)\) \(\chi_{7168}(85,\cdot)\) \(\chi_{7168}(141,\cdot)\) \(\chi_{7168}(197,\cdot)\) \(\chi_{7168}(253,\cdot)\) \(\chi_{7168}(309,\cdot)\) \(\chi_{7168}(365,\cdot)\) \(\chi_{7168}(421,\cdot)\) \(\chi_{7168}(477,\cdot)\) \(\chi_{7168}(533,\cdot)\) \(\chi_{7168}(589,\cdot)\) \(\chi_{7168}(645,\cdot)\) \(\chi_{7168}(701,\cdot)\) \(\chi_{7168}(757,\cdot)\) \(\chi_{7168}(813,\cdot)\) \(\chi_{7168}(869,\cdot)\) \(\chi_{7168}(925,\cdot)\) \(\chi_{7168}(981,\cdot)\) \(\chi_{7168}(1037,\cdot)\) \(\chi_{7168}(1093,\cdot)\) \(\chi_{7168}(1149,\cdot)\) \(\chi_{7168}(1205,\cdot)\) \(\chi_{7168}(1261,\cdot)\) \(\chi_{7168}(1317,\cdot)\) \(\chi_{7168}(1373,\cdot)\) \(\chi_{7168}(1429,\cdot)\) \(\chi_{7168}(1485,\cdot)\) \(\chi_{7168}(1541,\cdot)\) \(\chi_{7168}(1597,\cdot)\) \(\chi_{7168}(1653,\cdot)\) ...

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

\((1023,5125,1025)\) → \((1,e\left(\frac{123}{256}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 7168 }(29, a) \) \(1\)\(1\)\(e\left(\frac{81}{256}\right)\)\(e\left(\frac{123}{256}\right)\)\(e\left(\frac{81}{128}\right)\)\(e\left(\frac{87}{256}\right)\)\(e\left(\frac{213}{256}\right)\)\(e\left(\frac{51}{64}\right)\)\(e\left(\frac{61}{64}\right)\)\(e\left(\frac{141}{256}\right)\)\(e\left(\frac{29}{128}\right)\)\(e\left(\frac{123}{128}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7168 }(29,a) \;\) at \(\;a = \) e.g. 2