↑

Properties

Label 2394.5
Modulus $2394$
Conductor $1197$
Order $18$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2394, base_ring=CyclotomicField(18)) M = H._module chi = DirichletCharacter(H, M([15,15,16]))
 
Copy content pari:[g,chi] = znchar(Mod(5,2394))
 

Basic properties

Modulus: \(2394\)
Conductor: \(1197\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(18\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1197}(5,\cdot)\)
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 2394.gj

\(\chi_{2394}(5,\cdot)\) \(\chi_{2394}(101,\cdot)\) \(\chi_{2394}(131,\cdot)\) \(\chi_{2394}(479,\cdot)\) \(\chi_{2394}(731,\cdot)\) \(\chi_{2394}(1517,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: Number field defined by a degree 18 polynomial

Values on generators

\((533,1711,1009)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{5}{6}\right),e\left(\frac{8}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 2394 }(5, a) \) \(1\)\(1\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{9}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 2394 }(5,a) \;\) at \(\;a = \) e.g. 2