Properties

Label 8044.27
Modulus $8044$
Conductor $8044$
Order $670$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8044, base_ring=CyclotomicField(670))
 
M = H._module
 
chi = DirichletCharacter(H, M([335,1]))
 
pari: [g,chi] = znchar(Mod(27,8044))
 

Basic properties

Modulus: \(8044\)
Conductor: \(8044\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(670\)
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 8044.bb

\(\chi_{8044}(27,\cdot)\) \(\chi_{8044}(47,\cdot)\) \(\chi_{8044}(55,\cdot)\) \(\chi_{8044}(59,\cdot)\) \(\chi_{8044}(75,\cdot)\) \(\chi_{8044}(95,\cdot)\) \(\chi_{8044}(267,\cdot)\) \(\chi_{8044}(335,\cdot)\) \(\chi_{8044}(343,\cdot)\) \(\chi_{8044}(351,\cdot)\) \(\chi_{8044}(391,\cdot)\) \(\chi_{8044}(439,\cdot)\) \(\chi_{8044}(463,\cdot)\) \(\chi_{8044}(535,\cdot)\) \(\chi_{8044}(623,\cdot)\) \(\chi_{8044}(711,\cdot)\) \(\chi_{8044}(715,\cdot)\) \(\chi_{8044}(819,\cdot)\) \(\chi_{8044}(891,\cdot)\) \(\chi_{8044}(919,\cdot)\) \(\chi_{8044}(971,\cdot)\) \(\chi_{8044}(975,\cdot)\) \(\chi_{8044}(1107,\cdot)\) \(\chi_{8044}(1115,\cdot)\) \(\chi_{8044}(1135,\cdot)\) \(\chi_{8044}(1179,\cdot)\) \(\chi_{8044}(1199,\cdot)\) \(\chi_{8044}(1215,\cdot)\) \(\chi_{8044}(1251,\cdot)\) \(\chi_{8044}(1331,\cdot)\) ...

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: $\Q(\zeta_{335})$
Fixed field: Number field defined by a degree 670 polynomial (not computed)

Values on generators

\((4023,4025)\) → \((-1,e\left(\frac{1}{670}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8044 }(27, a) \) \(1\)\(1\)\(e\left(\frac{168}{335}\right)\)\(e\left(\frac{177}{335}\right)\)\(e\left(\frac{66}{335}\right)\)\(e\left(\frac{1}{335}\right)\)\(e\left(\frac{16}{335}\right)\)\(e\left(\frac{299}{335}\right)\)\(e\left(\frac{2}{67}\right)\)\(e\left(\frac{347}{670}\right)\)\(e\left(\frac{79}{335}\right)\)\(e\left(\frac{234}{335}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8044 }(27,a) \;\) at \(\;a = \) e.g. 2