Properties

Label 1156.747
Modulus $1156$
Conductor $1156$
Order $34$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1156, base_ring=CyclotomicField(34))
 
M = H._module
 
chi = DirichletCharacter(H, M([17,15]))
 
pari: [g,chi] = znchar(Mod(747,1156))
 

Basic properties

Modulus: \(1156\)
Conductor: \(1156\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(34\)
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 1156.l

\(\chi_{1156}(67,\cdot)\) \(\chi_{1156}(135,\cdot)\) \(\chi_{1156}(203,\cdot)\) \(\chi_{1156}(271,\cdot)\) \(\chi_{1156}(339,\cdot)\) \(\chi_{1156}(407,\cdot)\) \(\chi_{1156}(475,\cdot)\) \(\chi_{1156}(543,\cdot)\) \(\chi_{1156}(611,\cdot)\) \(\chi_{1156}(679,\cdot)\) \(\chi_{1156}(747,\cdot)\) \(\chi_{1156}(815,\cdot)\) \(\chi_{1156}(883,\cdot)\) \(\chi_{1156}(951,\cdot)\) \(\chi_{1156}(1019,\cdot)\) \(\chi_{1156}(1087,\cdot)\)

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: \(\Q(\zeta_{17})\)
Fixed field: 34.0.1637569504672609029759453328209845791289707218675094773643512138419836077449127814221529088.1

Values on generators

\((579,581)\) → \((-1,e\left(\frac{15}{34}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(19\)\(21\)\(23\)
\( \chi_{ 1156 }(747, a) \) \(-1\)\(1\)\(e\left(\frac{16}{17}\right)\)\(e\left(\frac{1}{34}\right)\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{8}{17}\right)\)\(e\left(\frac{33}{34}\right)\)\(e\left(\frac{23}{34}\right)\)\(e\left(\frac{14}{17}\right)\)\(e\left(\frac{15}{17}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1156 }(747,a) \;\) at \(\;a = \) e.g. 2