Properties

Label 6815.32
Modulus $6815$
Conductor $6815$
Order $644$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6815, base_ring=CyclotomicField(644))
 
M = H._module
 
chi = DirichletCharacter(H, M([161,115,616]))
 
pari: [g,chi] = znchar(Mod(32,6815))
 

Basic properties

Modulus: \(6815\)
Conductor: \(6815\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(644\)
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 6815.da

\(\chi_{6815}(2,\cdot)\) \(\chi_{6815}(8,\cdot)\) \(\chi_{6815}(18,\cdot)\) \(\chi_{6815}(32,\cdot)\) \(\chi_{6815}(68,\cdot)\) \(\chi_{6815}(72,\cdot)\) \(\chi_{6815}(143,\cdot)\) \(\chi_{6815}(147,\cdot)\) \(\chi_{6815}(153,\cdot)\) \(\chi_{6815}(177,\cdot)\) \(\chi_{6815}(213,\cdot)\) \(\chi_{6815}(222,\cdot)\) \(\chi_{6815}(272,\cdot)\) \(\chi_{6815}(288,\cdot)\) \(\chi_{6815}(298,\cdot)\) \(\chi_{6815}(363,\cdot)\) \(\chi_{6815}(403,\cdot)\) \(\chi_{6815}(427,\cdot)\) \(\chi_{6815}(437,\cdot)\) \(\chi_{6815}(507,\cdot)\) \(\chi_{6815}(512,\cdot)\) \(\chi_{6815}(572,\cdot)\) \(\chi_{6815}(578,\cdot)\) \(\chi_{6815}(582,\cdot)\) \(\chi_{6815}(588,\cdot)\) \(\chi_{6815}(598,\cdot)\) \(\chi_{6815}(648,\cdot)\) \(\chi_{6815}(653,\cdot)\) \(\chi_{6815}(707,\cdot)\) \(\chi_{6815}(717,\cdot)\) ...

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: $\Q(\zeta_{644})$
Fixed field: Number field defined by a degree 644 polynomial (not computed)

Values on generators

\((2727,2351,146)\) → \((i,e\left(\frac{5}{28}\right),e\left(\frac{22}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 6815 }(32, a) \) \(1\)\(1\)\(e\left(\frac{104}{161}\right)\)\(e\left(\frac{249}{322}\right)\)\(e\left(\frac{47}{161}\right)\)\(e\left(\frac{135}{322}\right)\)\(e\left(\frac{1}{644}\right)\)\(e\left(\frac{151}{161}\right)\)\(e\left(\frac{88}{161}\right)\)\(e\left(\frac{103}{644}\right)\)\(e\left(\frac{3}{46}\right)\)\(e\left(\frac{313}{644}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6815 }(32,a) \;\) at \(\;a = \) e.g. 2