Properties

Label 8015.408
Modulus $8015$
Conductor $8015$
Order $228$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8015, base_ring=CyclotomicField(228))
 
M = H._module
 
chi = DirichletCharacter(H, M([171,76,103]))
 
pari: [g,chi] = znchar(Mod(408,8015))
 

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.fz

\(\chi_{8015}(408,\cdot)\) \(\chi_{8015}(417,\cdot)\) \(\chi_{8015}(548,\cdot)\) \(\chi_{8015}(597,\cdot)\) \(\chi_{8015}(737,\cdot)\) \(\chi_{8015}(837,\cdot)\) \(\chi_{8015}(963,\cdot)\) \(\chi_{8015}(1068,\cdot)\) \(\chi_{8015}(1222,\cdot)\) \(\chi_{8015}(1327,\cdot)\) \(\chi_{8015}(1453,\cdot)\) \(\chi_{8015}(1873,\cdot)\) \(\chi_{8015}(1948,\cdot)\) \(\chi_{8015}(2067,\cdot)\) \(\chi_{8015}(2377,\cdot)\) \(\chi_{8015}(2447,\cdot)\) \(\chi_{8015}(2452,\cdot)\) \(\chi_{8015}(2543,\cdot)\) \(\chi_{8015}(2592,\cdot)\) \(\chi_{8015}(2643,\cdot)\) \(\chi_{8015}(3012,\cdot)\) \(\chi_{8015}(3168,\cdot)\) \(\chi_{8015}(3537,\cdot)\) \(\chi_{8015}(3572,\cdot)\) \(\chi_{8015}(3903,\cdot)\) \(\chi_{8015}(3922,\cdot)\) \(\chi_{8015}(3952,\cdot)\) \(\chi_{8015}(4048,\cdot)\) \(\chi_{8015}(4057,\cdot)\) \(\chi_{8015}(4083,\cdot)\) ...

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

\((3207,4581,4586)\) → \((-i,e\left(\frac{1}{3}\right),e\left(\frac{103}{228}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 8015 }(408, a) \) \(1\)\(1\)\(e\left(\frac{103}{114}\right)\)\(e\left(\frac{125}{228}\right)\)\(e\left(\frac{46}{57}\right)\)\(e\left(\frac{103}{228}\right)\)\(e\left(\frac{27}{38}\right)\)\(e\left(\frac{11}{114}\right)\)\(e\left(\frac{59}{114}\right)\)\(e\left(\frac{27}{76}\right)\)\(e\left(\frac{2}{19}\right)\)\(e\left(\frac{35}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8015 }(408,a) \;\) at \(\;a = \) e.g. 2