Properties

Label 8015.51
Modulus $8015$
Conductor $1603$
Order $57$
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(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,38,20]))
 
pari: [g,chi] = znchar(Mod(51,8015))
 

Basic properties

Modulus: \(8015\)
Conductor: \(1603\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(57\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1603}(51,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8015.eh

\(\chi_{8015}(51,\cdot)\) \(\chi_{8015}(396,\cdot)\) \(\chi_{8015}(506,\cdot)\) \(\chi_{8015}(541,\cdot)\) \(\chi_{8015}(991,\cdot)\) \(\chi_{8015}(1236,\cdot)\) \(\chi_{8015}(1591,\cdot)\) \(\chi_{8015}(1761,\cdot)\) \(\chi_{8015}(1796,\cdot)\) \(\chi_{8015}(2601,\cdot)\) \(\chi_{8015}(2986,\cdot)\) \(\chi_{8015}(3126,\cdot)\) \(\chi_{8015}(3336,\cdot)\) \(\chi_{8015}(3516,\cdot)\) \(\chi_{8015}(3546,\cdot)\) \(\chi_{8015}(3586,\cdot)\) \(\chi_{8015}(3796,\cdot)\) \(\chi_{8015}(4076,\cdot)\) \(\chi_{8015}(4141,\cdot)\) \(\chi_{8015}(4251,\cdot)\) \(\chi_{8015}(4531,\cdot)\) \(\chi_{8015}(4706,\cdot)\) \(\chi_{8015}(4776,\cdot)\) \(\chi_{8015}(4846,\cdot)\) \(\chi_{8015}(5191,\cdot)\) \(\chi_{8015}(5896,\cdot)\) \(\chi_{8015}(6556,\cdot)\) \(\chi_{8015}(6596,\cdot)\) \(\chi_{8015}(6661,\cdot)\) \(\chi_{8015}(6666,\cdot)\) ...

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: $\Q(\zeta_{57})$
Fixed field: Number field defined by a degree 57 polynomial

Values on generators

\((3207,4581,4586)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{10}{57}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 8015 }(51, a) \) \(1\)\(1\)\(e\left(\frac{20}{57}\right)\)\(e\left(\frac{47}{57}\right)\)\(e\left(\frac{40}{57}\right)\)\(e\left(\frac{10}{57}\right)\)\(e\left(\frac{1}{19}\right)\)\(e\left(\frac{37}{57}\right)\)\(e\left(\frac{43}{57}\right)\)\(e\left(\frac{10}{19}\right)\)\(e\left(\frac{10}{19}\right)\)\(e\left(\frac{23}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8015 }(51,a) \;\) at \(\;a = \) e.g. 2