Properties

Label 6864.361
Modulus $6864$
Conductor $1144$
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(6864, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,15,0,18,25]))
 
pari: [g,chi] = znchar(Mod(361,6864))
 

Basic properties

Modulus: \(6864\)
Conductor: \(1144\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1144}(933,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6864.jn

\(\chi_{6864}(361,\cdot)\) \(\chi_{6864}(1369,\cdot)\) \(\chi_{6864}(1609,\cdot)\) \(\chi_{6864}(3481,\cdot)\) \(\chi_{6864}(3865,\cdot)\) \(\chi_{6864}(5113,\cdot)\) \(\chi_{6864}(5977,\cdot)\) \(\chi_{6864}(6361,\cdot)\)

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

\((2575,1717,4577,4369,2641)\) → \((1,-1,1,e\left(\frac{3}{5}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)\(37\)
\( \chi_{ 6864 }(361, a) \) \(1\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{8}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6864 }(361,a) \;\) at \(\;a = \) e.g. 2