Properties

Label 6815.33
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([483,46,378]))
 
pari: [g,chi] = znchar(Mod(33,6815))
 

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.cz

\(\chi_{6815}(13,\cdot)\) \(\chi_{6815}(22,\cdot)\) \(\chi_{6815}(33,\cdot)\) \(\chi_{6815}(38,\cdot)\) \(\chi_{6815}(62,\cdot)\) \(\chi_{6815}(67,\cdot)\) \(\chi_{6815}(92,\cdot)\) \(\chi_{6815}(138,\cdot)\) \(\chi_{6815}(167,\cdot)\) \(\chi_{6815}(207,\cdot)\) \(\chi_{6815}(208,\cdot)\) \(\chi_{6815}(312,\cdot)\) \(\chi_{6815}(323,\cdot)\) \(\chi_{6815}(352,\cdot)\) \(\chi_{6815}(428,\cdot)\) \(\chi_{6815}(468,\cdot)\) \(\chi_{6815}(527,\cdot)\) \(\chi_{6815}(528,\cdot)\) \(\chi_{6815}(557,\cdot)\) \(\chi_{6815}(593,\cdot)\) \(\chi_{6815}(602,\cdot)\) \(\chi_{6815}(622,\cdot)\) \(\chi_{6815}(642,\cdot)\) \(\chi_{6815}(673,\cdot)\) \(\chi_{6815}(702,\cdot)\) \(\chi_{6815}(718,\cdot)\) \(\chi_{6815}(738,\cdot)\) \(\chi_{6815}(763,\cdot)\) \(\chi_{6815}(767,\cdot)\) \(\chi_{6815}(787,\cdot)\) ...

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

\((2727,2351,146)\) → \((-i,e\left(\frac{1}{14}\right),e\left(\frac{27}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 6815 }(33, a) \) \(1\)\(1\)\(e\left(\frac{249}{644}\right)\)\(e\left(\frac{223}{644}\right)\)\(e\left(\frac{249}{322}\right)\)\(e\left(\frac{118}{161}\right)\)\(e\left(\frac{251}{644}\right)\)\(e\left(\frac{103}{644}\right)\)\(e\left(\frac{223}{322}\right)\)\(e\left(\frac{144}{161}\right)\)\(e\left(\frac{11}{92}\right)\)\(e\left(\frac{639}{644}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6815 }(33,a) \;\) at \(\;a = \) e.g. 2