Properties

Label 2015.162
Modulus $2015$
Conductor $2015$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2015, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,25,56]))
 
pari: [g,chi] = znchar(Mod(162,2015))
 

Basic properties

Modulus: \(2015\)
Conductor: \(2015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 2015.hm

\(\chi_{2015}(162,\cdot)\) \(\chi_{2015}(258,\cdot)\) \(\chi_{2015}(293,\cdot)\) \(\chi_{2015}(392,\cdot)\) \(\chi_{2015}(422,\cdot)\) \(\chi_{2015}(617,\cdot)\) \(\chi_{2015}(813,\cdot)\) \(\chi_{2015}(847,\cdot)\) \(\chi_{2015}(908,\cdot)\) \(\chi_{2015}(1073,\cdot)\) \(\chi_{2015}(1103,\cdot)\) \(\chi_{2015}(1268,\cdot)\) \(\chi_{2015}(1497,\cdot)\) \(\chi_{2015}(1657,\cdot)\) \(\chi_{2015}(1692,\cdot)\) \(\chi_{2015}(1818,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((807,1861,716)\) → \((i,e\left(\frac{5}{12}\right),e\left(\frac{14}{15}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\( \chi_{ 2015 }(162, a) \) \(1\)\(1\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{1}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2015 }(162,a) \;\) at \(\;a = \) e.g. 2