Properties

Label 2011.143
Modulus $2011$
Conductor $2011$
Order $402$
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(402))
 
M = H._module
 
chi = DirichletCharacter(H, M([193]))
 
pari: [g,chi] = znchar(Mod(143,2011))
 

Basic properties

Modulus: \(2011\)
Conductor: \(2011\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(402\)
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.m

\(\chi_{2011}(2,\cdot)\) \(\chi_{2011}(15,\cdot)\) \(\chi_{2011}(32,\cdot)\) \(\chi_{2011}(37,\cdot)\) \(\chi_{2011}(67,\cdot)\) \(\chi_{2011}(91,\cdot)\) \(\chi_{2011}(128,\cdot)\) \(\chi_{2011}(139,\cdot)\) \(\chi_{2011}(143,\cdot)\) \(\chi_{2011}(148,\cdot)\) \(\chi_{2011}(163,\cdot)\) \(\chi_{2011}(171,\cdot)\) \(\chi_{2011}(217,\cdot)\) \(\chi_{2011}(221,\cdot)\) \(\chi_{2011}(230,\cdot)\) \(\chi_{2011}(231,\cdot)\) \(\chi_{2011}(234,\cdot)\) \(\chi_{2011}(240,\cdot)\) \(\chi_{2011}(243,\cdot)\) \(\chi_{2011}(266,\cdot)\) \(\chi_{2011}(275,\cdot)\) \(\chi_{2011}(277,\cdot)\) \(\chi_{2011}(281,\cdot)\) \(\chi_{2011}(341,\cdot)\) \(\chi_{2011}(357,\cdot)\) \(\chi_{2011}(363,\cdot)\) \(\chi_{2011}(364,\cdot)\) \(\chi_{2011}(365,\cdot)\) \(\chi_{2011}(377,\cdot)\) \(\chi_{2011}(388,\cdot)\) ...

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: $\Q(\zeta_{201})$
Fixed field: Number field defined by a degree 402 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{193}{402}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2011 }(143, a) \) \(-1\)\(1\)\(e\left(\frac{251}{402}\right)\)\(e\left(\frac{193}{402}\right)\)\(e\left(\frac{50}{201}\right)\)\(e\left(\frac{125}{201}\right)\)\(e\left(\frac{7}{67}\right)\)\(e\left(\frac{217}{402}\right)\)\(e\left(\frac{117}{134}\right)\)\(e\left(\frac{193}{201}\right)\)\(e\left(\frac{33}{134}\right)\)\(e\left(\frac{347}{402}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2011 }(143,a) \;\) at \(\;a = \) e.g. 2