Properties

Label 1096.811
Modulus $1096$
Conductor $1096$
Order $68$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1096, base_ring=CyclotomicField(68))
 
M = H._module
 
chi = DirichletCharacter(H, M([34,34,27]))
 
pari: [g,chi] = znchar(Mod(811,1096))
 

Basic properties

Modulus: \(1096\)
Conductor: \(1096\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(68\)
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 1096.z

\(\chi_{1096}(11,\cdot)\) \(\chi_{1096}(19,\cdot)\) \(\chi_{1096}(107,\cdot)\) \(\chi_{1096}(139,\cdot)\) \(\chi_{1096}(235,\cdot)\) \(\chi_{1096}(267,\cdot)\) \(\chi_{1096}(283,\cdot)\) \(\chi_{1096}(291,\cdot)\) \(\chi_{1096}(299,\cdot)\) \(\chi_{1096}(379,\cdot)\) \(\chi_{1096}(403,\cdot)\) \(\chi_{1096}(419,\cdot)\) \(\chi_{1096}(443,\cdot)\) \(\chi_{1096}(523,\cdot)\) \(\chi_{1096}(531,\cdot)\) \(\chi_{1096}(539,\cdot)\) \(\chi_{1096}(555,\cdot)\) \(\chi_{1096}(587,\cdot)\) \(\chi_{1096}(683,\cdot)\) \(\chi_{1096}(715,\cdot)\) \(\chi_{1096}(803,\cdot)\) \(\chi_{1096}(811,\cdot)\) \(\chi_{1096}(883,\cdot)\) \(\chi_{1096}(891,\cdot)\) \(\chi_{1096}(915,\cdot)\) \(\chi_{1096}(923,\cdot)\) \(\chi_{1096}(931,\cdot)\) \(\chi_{1096}(987,\cdot)\) \(\chi_{1096}(995,\cdot)\) \(\chi_{1096}(1003,\cdot)\) ...

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: $\Q(\zeta_{68})$
Fixed field: Number field defined by a degree 68 polynomial

Values on generators

\((823,549,825)\) → \((-1,-1,e\left(\frac{27}{68}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1096 }(811, a) \) \(-1\)\(1\)\(e\left(\frac{27}{68}\right)\)\(e\left(\frac{19}{68}\right)\)\(e\left(\frac{3}{17}\right)\)\(e\left(\frac{27}{34}\right)\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{29}{68}\right)\)\(e\left(\frac{23}{34}\right)\)\(e\left(\frac{3}{34}\right)\)\(e\left(\frac{9}{34}\right)\)\(e\left(\frac{39}{68}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1096 }(811,a) \;\) at \(\;a = \) e.g. 2