Properties

Label 1157.12
Modulus $1157$
Conductor $1157$
Order $8$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1157, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([4,3]))
 
pari: [g,chi] = znchar(Mod(12,1157))
 

Basic properties

Modulus: \(1157\)
Conductor: \(1157\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
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 1157.o

\(\chi_{1157}(12,\cdot)\) \(\chi_{1157}(77,\cdot)\) \(\chi_{1157}(571,\cdot)\) \(\chi_{1157}(675,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.0.1263291155951203769.1

Values on generators

\((535,92)\) → \((-1,e\left(\frac{3}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1157 }(12, a) \) \(-1\)\(1\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(1\)\(-i\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(-1\)\(-i\)\(i\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1157 }(12,a) \;\) at \(\;a = \) e.g. 2