Properties

Label 1386.359
Modulus $1386$
Conductor $231$
Order $30$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,10,21]))
 
pari: [g,chi] = znchar(Mod(359,1386))
 

Basic properties

Modulus: \(1386\)
Conductor: \(231\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{231}(128,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1386.co

\(\chi_{1386}(107,\cdot)\) \(\chi_{1386}(233,\cdot)\) \(\chi_{1386}(305,\cdot)\) \(\chi_{1386}(359,\cdot)\) \(\chi_{1386}(431,\cdot)\) \(\chi_{1386}(557,\cdot)\) \(\chi_{1386}(611,\cdot)\) \(\chi_{1386}(809,\cdot)\)

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

\((155,199,1135)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1386 }(359, a) \) \(1\)\(1\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{3}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1386 }(359,a) \;\) at \(\;a = \) e.g. 2