Properties

Label 2013.32
Modulus $2013$
Conductor $2013$
Order $12$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,6,1]))
 
pari: [g,chi] = znchar(Mod(32,2013))
 

Basic properties

Modulus: \(2013\)
Conductor: \(2013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
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 2013.cr

\(\chi_{2013}(32,\cdot)\) \(\chi_{2013}(395,\cdot)\) \(\chi_{2013}(692,\cdot)\) \(\chi_{2013}(1748,\cdot)\)

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: Number field defined by a degree 12 polynomial

Values on generators

\((1343,1465,1222)\) → \((-1,-1,e\left(\frac{1}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 2013 }(32, a) \) \(-1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(i\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2013 }(32,a) \;\) at \(\;a = \) e.g. 2