Properties

Label 8015.144
Modulus $8015$
Conductor $8015$
Order $114$
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(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([57,76,22]))
 
pari: [g,chi] = znchar(Mod(144,8015))
 

Basic properties

Modulus: \(8015\)
Conductor: \(8015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(114\)
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.fc

\(\chi_{8015}(144,\cdot)\) \(\chi_{8015}(184,\cdot)\) \(\chi_{8015}(249,\cdot)\) \(\chi_{8015}(254,\cdot)\) \(\chi_{8015}(284,\cdot)\) \(\chi_{8015}(919,\cdot)\) \(\chi_{8015}(1089,\cdot)\) \(\chi_{8015}(1159,\cdot)\) \(\chi_{8015}(1304,\cdot)\) \(\chi_{8015}(1369,\cdot)\) \(\chi_{8015}(1654,\cdot)\) \(\chi_{8015}(1999,\cdot)\) \(\chi_{8015}(2109,\cdot)\) \(\chi_{8015}(2144,\cdot)\) \(\chi_{8015}(2594,\cdot)\) \(\chi_{8015}(2839,\cdot)\) \(\chi_{8015}(3194,\cdot)\) \(\chi_{8015}(3364,\cdot)\) \(\chi_{8015}(3399,\cdot)\) \(\chi_{8015}(4204,\cdot)\) \(\chi_{8015}(4589,\cdot)\) \(\chi_{8015}(4729,\cdot)\) \(\chi_{8015}(4939,\cdot)\) \(\chi_{8015}(5119,\cdot)\) \(\chi_{8015}(5149,\cdot)\) \(\chi_{8015}(5189,\cdot)\) \(\chi_{8015}(5399,\cdot)\) \(\chi_{8015}(5679,\cdot)\) \(\chi_{8015}(5744,\cdot)\) \(\chi_{8015}(5854,\cdot)\) ...

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 114 polynomial (not computed)

Values on generators

\((3207,4581,4586)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{11}{57}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 8015 }(144, a) \) \(1\)\(1\)\(e\left(\frac{101}{114}\right)\)\(e\left(\frac{35}{114}\right)\)\(e\left(\frac{44}{57}\right)\)\(e\left(\frac{11}{57}\right)\)\(e\left(\frac{25}{38}\right)\)\(e\left(\frac{35}{57}\right)\)\(e\left(\frac{53}{57}\right)\)\(e\left(\frac{3}{38}\right)\)\(e\left(\frac{3}{38}\right)\)\(e\left(\frac{31}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8015 }(144,a) \;\) at \(\;a = \) e.g. 2