Properties

Label 2011.47
Modulus $2011$
Conductor $2011$
Order $670$
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(670))
 
M = H._module
 
chi = DirichletCharacter(H, M([549]))
 
pari: [g,chi] = znchar(Mod(47,2011))
 

Basic properties

Modulus: \(2011\)
Conductor: \(2011\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(670\)
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.n

\(\chi_{2011}(10,\cdot)\) \(\chi_{2011}(27,\cdot)\) \(\chi_{2011}(44,\cdot)\) \(\chi_{2011}(46,\cdot)\) \(\chi_{2011}(47,\cdot)\) \(\chi_{2011}(48,\cdot)\) \(\chi_{2011}(55,\cdot)\) \(\chi_{2011}(59,\cdot)\) \(\chi_{2011}(68,\cdot)\) \(\chi_{2011}(75,\cdot)\) \(\chi_{2011}(76,\cdot)\) \(\chi_{2011}(85,\cdot)\) \(\chi_{2011}(95,\cdot)\) \(\chi_{2011}(104,\cdot)\) \(\chi_{2011}(112,\cdot)\) \(\chi_{2011}(113,\cdot)\) \(\chi_{2011}(130,\cdot)\) \(\chi_{2011}(140,\cdot)\) \(\chi_{2011}(149,\cdot)\) \(\chi_{2011}(162,\cdot)\) \(\chi_{2011}(166,\cdot)\) \(\chi_{2011}(218,\cdot)\) \(\chi_{2011}(242,\cdot)\) \(\chi_{2011}(244,\cdot)\) \(\chi_{2011}(248,\cdot)\) \(\chi_{2011}(253,\cdot)\) \(\chi_{2011}(257,\cdot)\) \(\chi_{2011}(261,\cdot)\) \(\chi_{2011}(264,\cdot)\) \(\chi_{2011}(267,\cdot)\) ...

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: $\Q(\zeta_{335})$
Fixed field: Number field defined by a degree 670 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{549}{670}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2011 }(47, a) \) \(-1\)\(1\)\(e\left(\frac{101}{134}\right)\)\(e\left(\frac{549}{670}\right)\)\(e\left(\frac{34}{67}\right)\)\(e\left(\frac{23}{335}\right)\)\(e\left(\frac{192}{335}\right)\)\(e\left(\frac{443}{670}\right)\)\(e\left(\frac{35}{134}\right)\)\(e\left(\frac{214}{335}\right)\)\(e\left(\frac{551}{670}\right)\)\(e\left(\frac{483}{670}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2011 }(47,a) \;\) at \(\;a = \) e.g. 2