Properties

Label 8015.101
Modulus $8015$
Conductor $1603$
Order $228$
Real no
Primitive no
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([0,38,33]))
 
pari: [g,chi] = znchar(Mod(101,8015))
 

Basic properties

Modulus: \(8015\)
Conductor: \(1603\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(228\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1603}(101,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8015.gi

\(\chi_{8015}(101,\cdot)\) \(\chi_{8015}(136,\cdot)\) \(\chi_{8015}(551,\cdot)\) \(\chi_{8015}(586,\cdot)\) \(\chi_{8015}(796,\cdot)\) \(\chi_{8015}(801,\cdot)\) \(\chi_{8015}(1111,\cdot)\) \(\chi_{8015}(1251,\cdot)\) \(\chi_{8015}(1286,\cdot)\) \(\chi_{8015}(1361,\cdot)\) \(\chi_{8015}(1396,\cdot)\) \(\chi_{8015}(1426,\cdot)\) \(\chi_{8015}(1571,\cdot)\) \(\chi_{8015}(1601,\cdot)\) \(\chi_{8015}(1746,\cdot)\) \(\chi_{8015}(1811,\cdot)\) \(\chi_{8015}(1886,\cdot)\) \(\chi_{8015}(1916,\cdot)\) \(\chi_{8015}(2091,\cdot)\) \(\chi_{8015}(2236,\cdot)\) \(\chi_{8015}(2376,\cdot)\) \(\chi_{8015}(2511,\cdot)\) \(\chi_{8015}(2551,\cdot)\) \(\chi_{8015}(2726,\cdot)\) \(\chi_{8015}(2756,\cdot)\) \(\chi_{8015}(2761,\cdot)\) \(\chi_{8015}(3176,\cdot)\) \(\chi_{8015}(3321,\cdot)\) \(\chi_{8015}(3351,\cdot)\) \(\chi_{8015}(3456,\cdot)\) ...

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

\((3207,4581,4586)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{11}{76}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 8015 }(101, a) \) \(1\)\(1\)\(e\left(\frac{85}{228}\right)\)\(e\left(\frac{31}{114}\right)\)\(e\left(\frac{85}{114}\right)\)\(e\left(\frac{49}{76}\right)\)\(e\left(\frac{9}{76}\right)\)\(e\left(\frac{31}{57}\right)\)\(e\left(\frac{13}{114}\right)\)\(e\left(\frac{1}{57}\right)\)\(e\left(\frac{33}{76}\right)\)\(e\left(\frac{28}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8015 }(101,a) \;\) at \(\;a = \) e.g. 2