Properties

Label 2011.11
Modulus $2011$
Conductor $2011$
Order $2010$
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(2010))
 
M = H._module
 
chi = DirichletCharacter(H, M([1037]))
 
pari: [g,chi] = znchar(Mod(11,2011))
 

Basic properties

Modulus: \(2011\)
Conductor: \(2011\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2010\)
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.p

\(\chi_{2011}(3,\cdot)\) \(\chi_{2011}(7,\cdot)\) \(\chi_{2011}(11,\cdot)\) \(\chi_{2011}(12,\cdot)\) \(\chi_{2011}(17,\cdot)\) \(\chi_{2011}(18,\cdot)\) \(\chi_{2011}(19,\cdot)\) \(\chi_{2011}(26,\cdot)\) \(\chi_{2011}(28,\cdot)\) \(\chi_{2011}(29,\cdot)\) \(\chi_{2011}(35,\cdot)\) \(\chi_{2011}(39,\cdot)\) \(\chi_{2011}(40,\cdot)\) \(\chi_{2011}(42,\cdot)\) \(\chi_{2011}(50,\cdot)\) \(\chi_{2011}(61,\cdot)\) \(\chi_{2011}(62,\cdot)\) \(\chi_{2011}(66,\cdot)\) \(\chi_{2011}(69,\cdot)\) \(\chi_{2011}(73,\cdot)\) \(\chi_{2011}(79,\cdot)\) \(\chi_{2011}(82,\cdot)\) \(\chi_{2011}(86,\cdot)\) \(\chi_{2011}(90,\cdot)\) \(\chi_{2011}(93,\cdot)\) \(\chi_{2011}(98,\cdot)\) \(\chi_{2011}(99,\cdot)\) \(\chi_{2011}(102,\cdot)\) \(\chi_{2011}(107,\cdot)\) \(\chi_{2011}(108,\cdot)\) ...

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: $\Q(\zeta_{1005})$
Fixed field: Number field defined by a degree 2010 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{1037}{2010}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2011 }(11, a) \) \(-1\)\(1\)\(e\left(\frac{161}{402}\right)\)\(e\left(\frac{1037}{2010}\right)\)\(e\left(\frac{161}{201}\right)\)\(e\left(\frac{304}{1005}\right)\)\(e\left(\frac{307}{335}\right)\)\(e\left(\frac{539}{2010}\right)\)\(e\left(\frac{27}{134}\right)\)\(e\left(\frac{32}{1005}\right)\)\(e\left(\frac{471}{670}\right)\)\(e\left(\frac{19}{2010}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2011 }(11,a) \;\) at \(\;a = \) e.g. 2