Properties

Label 2013.386
Modulus $2013$
Conductor $183$
Order $10$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, base_ring=CyclotomicField(10))
 
M = H._module
 
chi = DirichletCharacter(H, M([5,0,4]))
 
pari: [g,chi] = znchar(Mod(386,2013))
 

Basic properties

Modulus: \(2013\)
Conductor: \(183\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{183}(20,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2013.cg

\(\chi_{2013}(386,\cdot)\) \(\chi_{2013}(485,\cdot)\) \(\chi_{2013}(1046,\cdot)\) \(\chi_{2013}(1376,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 10.0.46584877058339283.1

Values on generators

\((1343,1465,1222)\) → \((-1,1,e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 2013 }(386, a) \) \(-1\)\(1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(1\)\(-1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{3}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2013 }(386,a) \;\) at \(\;a = \) e.g. 2