Properties

Label 2015.1322
Modulus $2015$
Conductor $2015$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

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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([15,40,16]))
 
pari: [g,chi] = znchar(Mod(1322,2015))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2015.gx

\(\chi_{2015}(107,\cdot)\) \(\chi_{2015}(113,\cdot)\) \(\chi_{2015}(412,\cdot)\) \(\chi_{2015}(568,\cdot)\) \(\chi_{2015}(607,\cdot)\) \(\chi_{2015}(627,\cdot)\) \(\chi_{2015}(887,\cdot)\) \(\chi_{2015}(913,\cdot)\) \(\chi_{2015}(1082,\cdot)\) \(\chi_{2015}(1218,\cdot)\) \(\chi_{2015}(1322,\cdot)\) \(\chi_{2015}(1413,\cdot)\) \(\chi_{2015}(1433,\cdot)\) \(\chi_{2015}(1693,\cdot)\) \(\chi_{2015}(1777,\cdot)\) \(\chi_{2015}(1888,\cdot)\)

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

\((807,1861,716)\) → \((i,e\left(\frac{2}{3}\right),e\left(\frac{4}{15}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\( \chi_{ 2015 }(1322, a) \) \(-1\)\(1\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{19}{30}\right)\)\(1\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{11}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2015 }(1322,a) \;\) at \(\;a = \) e.g. 2