Properties

Label 2365.424
Modulus $2365$
Conductor $2365$
Order $30$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2365, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,27,5]))
 
pari: [g,chi] = znchar(Mod(424,2365))
 

Basic properties

Modulus: \(2365\)
Conductor: \(2365\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2365.ce

\(\chi_{2365}(424,\cdot)\) \(\chi_{2365}(854,\cdot)\) \(\chi_{2365}(1069,\cdot)\) \(\chi_{2365}(1284,\cdot)\) \(\chi_{2365}(1469,\cdot)\) \(\chi_{2365}(1899,\cdot)\) \(\chi_{2365}(2114,\cdot)\) \(\chi_{2365}(2329,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((947,431,1981)\) → \((-1,e\left(\frac{9}{10}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 2365 }(424, a) \) \(1\)\(1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{8}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2365 }(424,a) \;\) at \(\;a = \) e.g. 2