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Properties

Label 1156.1001
Modulus $1156$
Conductor $17$
Order $8$
Real no
Primitive no
Minimal no
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1156, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([0,3]))
 
Copy content pari:[g,chi] = znchar(Mod(1001,1156))
 

Basic properties

Modulus: \(1156\)
Conductor: \(17\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(8\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{17}(15,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 1156.h

\(\chi_{1156}(733,\cdot)\) \(\chi_{1156}(757,\cdot)\) \(\chi_{1156}(977,\cdot)\) \(\chi_{1156}(1001,\cdot)\)

Copy content sage:chi.galois_orbit()
 
Copy content pari: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: \(\Q(\zeta_{17})^+\)

Values on generators

\((579,581)\) → \((1,e\left(\frac{3}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(19\)\(21\)\(23\)
\( \chi_{ 1156 }(1001, a) \) \(1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(e\left(\frac{5}{8}\right)\)\(-1\)\(i\)\(i\)\(-1\)\(e\left(\frac{5}{8}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1156 }(1001,a) \;\) at \(\;a = \) e.g. 2