Properties

Label 2304.737
Modulus $2304$
Conductor $96$
Order $8$
Real no
Primitive no
Minimal no
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2304, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([0,1,4]))
 
Copy content pari:[g,chi] = znchar(Mod(737,2304))
 

Basic properties

Modulus: \(2304\)
Conductor: \(96\)
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 \(\chi_{96}(5,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 2304.x

\(\chi_{2304}(161,\cdot)\) \(\chi_{2304}(737,\cdot)\) \(\chi_{2304}(1313,\cdot)\) \(\chi_{2304}(1889,\cdot)\)

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: 8.0.173946175488.1

Values on generators

\((1279,2053,1793)\) → \((1,e\left(\frac{1}{8}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 2304 }(737, a) \) \(-1\)\(1\)\(e\left(\frac{5}{8}\right)\)\(i\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(1\)\(e\left(\frac{7}{8}\right)\)\(i\)\(i\)\(e\left(\frac{7}{8}\right)\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 2304 }(737,a) \;\) at \(\;a = \) e.g. 2