Properties

Label 2011.611
Modulus $2011$
Conductor $2011$
Order $134$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2011, base_ring=CyclotomicField(134))
 
M = H._module
 
chi = DirichletCharacter(H, M([105]))
 
pari: [g,chi] = znchar(Mod(611,2011))
 

Basic properties

Modulus: \(2011\)
Conductor: \(2011\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(134\)
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 2011.j

\(\chi_{2011}(8,\cdot)\) \(\chi_{2011}(60,\cdot)\) \(\chi_{2011}(97,\cdot)\) \(\chi_{2011}(147,\cdot)\) \(\chi_{2011}(175,\cdot)\) \(\chi_{2011}(213,\cdot)\) \(\chi_{2011}(233,\cdot)\) \(\chi_{2011}(268,\cdot)\) \(\chi_{2011}(307,\cdot)\) \(\chi_{2011}(378,\cdot)\) \(\chi_{2011}(386,\cdot)\) \(\chi_{2011}(410,\cdot)\) \(\chi_{2011}(418,\cdot)\) \(\chi_{2011}(422,\cdot)\) \(\chi_{2011}(450,\cdot)\) \(\chi_{2011}(512,\cdot)\) \(\chi_{2011}(557,\cdot)\) \(\chi_{2011}(572,\cdot)\) \(\chi_{2011}(592,\cdot)\) \(\chi_{2011}(597,\cdot)\) \(\chi_{2011}(609,\cdot)\) \(\chi_{2011}(611,\cdot)\) \(\chi_{2011}(646,\cdot)\) \(\chi_{2011}(678,\cdot)\) \(\chi_{2011}(725,\cdot)\) \(\chi_{2011}(742,\cdot)\) \(\chi_{2011}(767,\cdot)\) \(\chi_{2011}(823,\cdot)\) \(\chi_{2011}(824,\cdot)\) \(\chi_{2011}(835,\cdot)\) ...

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: $\Q(\zeta_{67})$
Fixed field: Number field defined by a degree 134 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{105}{134}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2011 }(611, a) \) \(-1\)\(1\)\(e\left(\frac{131}{134}\right)\)\(e\left(\frac{105}{134}\right)\)\(e\left(\frac{64}{67}\right)\)\(e\left(\frac{26}{67}\right)\)\(e\left(\frac{51}{67}\right)\)\(e\left(\frac{125}{134}\right)\)\(e\left(\frac{125}{134}\right)\)\(e\left(\frac{38}{67}\right)\)\(e\left(\frac{49}{134}\right)\)\(e\left(\frac{77}{134}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2011 }(611,a) \;\) at \(\;a = \) e.g. 2