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Properties

Label 2890.399
Modulus $2890$
Conductor $85$
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(2890, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([4,5]))
 
Copy content pari:[g,chi] = znchar(Mod(399,2890))
 

Basic properties

Modulus: \(2890\)
Conductor: \(85\)
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 \(\chi_{85}(59,\cdot)\)
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 2890.n

\(\chi_{2890}(179,\cdot)\) \(\chi_{2890}(399,\cdot)\) \(\chi_{2890}(1579,\cdot)\) \(\chi_{2890}(1889,\cdot)\)

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: 8.8.256461670625.1

Values on generators

\((1157,581)\) → \((-1,e\left(\frac{5}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 2890 }(399, a) \) \(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(i\)\(e\left(\frac{3}{8}\right)\)\(1\)\(-i\)\(-1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{8}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 2890 }(399,a) \;\) at \(\;a = \) e.g. 2