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Properties

Label 1600.199
Modulus $1600$
Conductor $160$
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(1600, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([4,5,4]))
 
Copy content pari:[g,chi] = znchar(Mod(199,1600))
 

Basic properties

Modulus: \(1600\)
Conductor: \(160\)
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 \(\chi_{160}(139,\cdot)\)
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 1600.z

\(\chi_{1600}(199,\cdot)\) \(\chi_{1600}(599,\cdot)\) \(\chi_{1600}(999,\cdot)\) \(\chi_{1600}(1399,\cdot)\)

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: 8.0.1342177280000.1

Values on generators

\((1151,901,577)\) → \((-1,e\left(\frac{5}{8}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1600 }(199, a) \) \(-1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(i\)\(-i\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(e\left(\frac{5}{8}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1600 }(199,a) \;\) at \(\;a = \) e.g. 2