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Properties

Label 1248.157
Modulus $1248$
Conductor $32$
Order $8$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1248, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([0,3,0,0]))
 
Copy content pari:[g,chi] = znchar(Mod(157,1248))
 

Basic properties

Modulus: \(1248\)
Conductor: \(32\)
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 \(\chi_{32}(29,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 1248.ck

\(\chi_{1248}(157,\cdot)\) \(\chi_{1248}(469,\cdot)\) \(\chi_{1248}(781,\cdot)\) \(\chi_{1248}(1093,\cdot)\)

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: \(\Q(\zeta_{32})^+\)

Values on generators

\((703,1093,833,769)\) → \((1,e\left(\frac{3}{8}\right),1,1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 1248 }(157, a) \) \(1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(-i\)\(e\left(\frac{7}{8}\right)\)\(-1\)\(e\left(\frac{5}{8}\right)\)\(i\)\(-i\)\(e\left(\frac{1}{8}\right)\)\(1\)\(e\left(\frac{1}{8}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1248 }(157,a) \;\) at \(\;a = \) e.g. 2