Properties

Label 2015.838
Modulus $2015$
Conductor $65$
Order $12$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2015, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,5,0]))
 
pari: [g,chi] = znchar(Mod(838,2015))
 

Basic properties

Modulus: \(2015\)
Conductor: \(65\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{65}(58,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2015.dz

\(\chi_{2015}(838,\cdot)\) \(\chi_{2015}(1272,\cdot)\) \(\chi_{2015}(1458,\cdot)\) \(\chi_{2015}(1892,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.12.3500313269603515625.1

Values on generators

\((807,1861,716)\) → \((-i,e\left(\frac{5}{12}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\( \chi_{ 2015 }(838, a) \) \(1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(i\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2015 }(838,a) \;\) at \(\;a = \) e.g. 2