Properties

Label 8015.2
Modulus $8015$
Conductor $8015$
Order $228$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8015, base_ring=CyclotomicField(228))
 
M = H._module
 
chi = DirichletCharacter(H, M([57,76,21]))
 
pari: [g,chi] = znchar(Mod(2,8015))
 

Basic properties

Modulus: \(8015\)
Conductor: \(8015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(228\)
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 8015.fy

\(\chi_{8015}(2,\cdot)\) \(\chi_{8015}(32,\cdot)\) \(\chi_{8015}(123,\cdot)\) \(\chi_{8015}(128,\cdot)\) \(\chi_{8015}(177,\cdot)\) \(\chi_{8015}(207,\cdot)\) \(\chi_{8015}(242,\cdot)\) \(\chi_{8015}(317,\cdot)\) \(\chi_{8015}(338,\cdot)\) \(\chi_{8015}(492,\cdot)\) \(\chi_{8015}(653,\cdot)\) \(\chi_{8015}(807,\cdot)\) \(\chi_{8015}(823,\cdot)\) \(\chi_{8015}(828,\cdot)\) \(\chi_{8015}(968,\cdot)\) \(\chi_{8015}(1017,\cdot)\) \(\chi_{8015}(1143,\cdot)\) \(\chi_{8015}(1467,\cdot)\) \(\chi_{8015}(1633,\cdot)\) \(\chi_{8015}(1717,\cdot)\) \(\chi_{8015}(2048,\cdot)\) \(\chi_{8015}(2083,\cdot)\) \(\chi_{8015}(2167,\cdot)\) \(\chi_{8015}(2258,\cdot)\) \(\chi_{8015}(2298,\cdot)\) \(\chi_{8015}(2433,\cdot)\) \(\chi_{8015}(2727,\cdot)\) \(\chi_{8015}(2802,\cdot)\) \(\chi_{8015}(2832,\cdot)\) \(\chi_{8015}(2893,\cdot)\) ...

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: $\Q(\zeta_{228})$
Fixed field: Number field defined by a degree 228 polynomial (not computed)

Values on generators

\((3207,4581,4586)\) → \((i,e\left(\frac{1}{3}\right),e\left(\frac{7}{76}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 8015 }(2, a) \) \(1\)\(1\)\(e\left(\frac{97}{114}\right)\)\(e\left(\frac{55}{228}\right)\)\(e\left(\frac{40}{57}\right)\)\(e\left(\frac{7}{76}\right)\)\(e\left(\frac{21}{38}\right)\)\(e\left(\frac{55}{114}\right)\)\(e\left(\frac{29}{114}\right)\)\(e\left(\frac{215}{228}\right)\)\(e\left(\frac{10}{19}\right)\)\(e\left(\frac{23}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8015 }(2,a) \;\) at \(\;a = \) e.g. 2