Properties

Label 6864.335
Modulus $6864$
Conductor $1716$
Order $30$
Real no
Primitive no
Minimal yes
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([15,0,15,12,25]))
 
pari: [g,chi] = znchar(Mod(335,6864))
 

Basic properties

Modulus: \(6864\)
Conductor: \(1716\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1716}(335,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6864.jr

\(\chi_{6864}(335,\cdot)\) \(\chi_{6864}(719,\cdot)\) \(\chi_{6864}(1967,\cdot)\) \(\chi_{6864}(2831,\cdot)\) \(\chi_{6864}(3215,\cdot)\) \(\chi_{6864}(4079,\cdot)\) \(\chi_{6864}(5087,\cdot)\) \(\chi_{6864}(5327,\cdot)\)

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

\((2575,1717,4577,4369,2641)\) → \((-1,1,-1,e\left(\frac{2}{5}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)\(37\)
\( \chi_{ 6864 }(335, a) \) \(1\)\(1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{19}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6864 }(335,a) \;\) at \(\;a = \) e.g. 2