Properties

Label 8281.188
Modulus $8281$
Conductor $637$
Order $84$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,35]))
 
pari: [g,chi] = znchar(Mod(188,8281))
 

Basic properties

Modulus: \(8281\)
Conductor: \(637\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{637}(188,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8281.cx

\(\chi_{8281}(188,\cdot)\) \(\chi_{8281}(258,\cdot)\) \(\chi_{8281}(657,\cdot)\) \(\chi_{8281}(1441,\cdot)\) \(\chi_{8281}(1770,\cdot)\) \(\chi_{8281}(1840,\cdot)\) \(\chi_{8281}(2554,\cdot)\) \(\chi_{8281}(2624,\cdot)\) \(\chi_{8281}(2953,\cdot)\) \(\chi_{8281}(3023,\cdot)\) \(\chi_{8281}(3737,\cdot)\) \(\chi_{8281}(3807,\cdot)\) \(\chi_{8281}(4136,\cdot)\) \(\chi_{8281}(4206,\cdot)\) \(\chi_{8281}(4920,\cdot)\) \(\chi_{8281}(4990,\cdot)\) \(\chi_{8281}(5319,\cdot)\) \(\chi_{8281}(6103,\cdot)\) \(\chi_{8281}(6502,\cdot)\) \(\chi_{8281}(6572,\cdot)\) \(\chi_{8281}(7286,\cdot)\) \(\chi_{8281}(7356,\cdot)\) \(\chi_{8281}(7685,\cdot)\) \(\chi_{8281}(7755,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1522,3382)\) → \((e\left(\frac{5}{14}\right),e\left(\frac{5}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 8281 }(188, a) \) \(1\)\(1\)\(e\left(\frac{59}{84}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{61}{84}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{17}{84}\right)\)\(e\left(\frac{3}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8281 }(188,a) \;\) at \(\;a = \) e.g. 2