Properties

Label 2015.17
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([15,10,14]))
 
pari: [g,chi] = znchar(Mod(17,2015))
 

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.gr

\(\chi_{2015}(17,\cdot)\) \(\chi_{2015}(43,\cdot)\) \(\chi_{2015}(303,\cdot)\) \(\chi_{2015}(478,\cdot)\) \(\chi_{2015}(517,\cdot)\) \(\chi_{2015}(673,\cdot)\) \(\chi_{2015}(823,\cdot)\) \(\chi_{2015}(972,\cdot)\) \(\chi_{2015}(1057,\cdot)\) \(\chi_{2015}(1252,\cdot)\) \(\chi_{2015}(1323,\cdot)\) \(\chi_{2015}(1512,\cdot)\) \(\chi_{2015}(1687,\cdot)\) \(\chi_{2015}(1778,\cdot)\) \(\chi_{2015}(1863,\cdot)\) \(\chi_{2015}(1882,\cdot)\)

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

\((807,1861,716)\) → \((i,e\left(\frac{1}{6}\right),e\left(\frac{7}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\( \chi_{ 2015 }(17, a) \) \(1\)\(1\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{19}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2015 }(17,a) \;\) at \(\;a = \) e.g. 2