Properties

Label 6013.120
Modulus $6013$
Conductor $859$
Order $39$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6013, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,44]))
 
pari: [g,chi] = znchar(Mod(120,6013))
 

Basic properties

Modulus: \(6013\)
Conductor: \(859\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(39\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{859}(120,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6013.bi

\(\chi_{6013}(120,\cdot)\) \(\chi_{6013}(316,\cdot)\) \(\chi_{6013}(666,\cdot)\) \(\chi_{6013}(1135,\cdot)\) \(\chi_{6013}(1443,\cdot)\) \(\chi_{6013}(1695,\cdot)\) \(\chi_{6013}(1751,\cdot)\) \(\chi_{6013}(1891,\cdot)\) \(\chi_{6013}(1996,\cdot)\) \(\chi_{6013}(2374,\cdot)\) \(\chi_{6013}(2689,\cdot)\) \(\chi_{6013}(3095,\cdot)\) \(\chi_{6013}(3410,\cdot)\) \(\chi_{6013}(3424,\cdot)\) \(\chi_{6013}(3648,\cdot)\) \(\chi_{6013}(3893,\cdot)\) \(\chi_{6013}(4159,\cdot)\) \(\chi_{6013}(4439,\cdot)\) \(\chi_{6013}(4572,\cdot)\) \(\chi_{6013}(4607,\cdot)\) \(\chi_{6013}(4796,\cdot)\) \(\chi_{6013}(4824,\cdot)\) \(\chi_{6013}(4971,\cdot)\) \(\chi_{6013}(5384,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 39 polynomial

Values on generators

\((5155,3438)\) → \((1,e\left(\frac{22}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 6013 }(120, a) \) \(1\)\(1\)\(e\left(\frac{22}{39}\right)\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{8}{39}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{10}{39}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{10}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6013 }(120,a) \;\) at \(\;a = \) e.g. 2