Properties

Label 1156.155
Modulus $1156$
Conductor $68$
Order $8$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1156, base_ring=CyclotomicField(8))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4,7]))
 
pari: [g,chi] = znchar(Mod(155,1156))
 

Basic properties

Modulus: \(1156\)
Conductor: \(68\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{68}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1156.g

\(\chi_{1156}(155,\cdot)\) \(\chi_{1156}(179,\cdot)\) \(\chi_{1156}(399,\cdot)\) \(\chi_{1156}(423,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{8})\)
Fixed field: 8.0.105046700288.1

Values on generators

\((579,581)\) → \((-1,e\left(\frac{7}{8}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(19\)\(21\)\(23\)
\( \chi_{ 1156 }(155, a) \) \(-1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(e\left(\frac{5}{8}\right)\)\(-1\)\(-i\)\(-i\)\(-1\)\(e\left(\frac{5}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1156 }(155,a) \;\) at \(\;a = \) e.g. 2