Properties

Label 2057.89
Modulus $2057$
Conductor $2057$
Order $44$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2057, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([24,33]))
 
pari: [g,chi] = znchar(Mod(89,2057))
 

Basic properties

Modulus: \(2057\)
Conductor: \(2057\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2057.x

\(\chi_{2057}(89,\cdot)\) \(\chi_{2057}(166,\cdot)\) \(\chi_{2057}(276,\cdot)\) \(\chi_{2057}(353,\cdot)\) \(\chi_{2057}(463,\cdot)\) \(\chi_{2057}(540,\cdot)\) \(\chi_{2057}(650,\cdot)\) \(\chi_{2057}(837,\cdot)\) \(\chi_{2057}(914,\cdot)\) \(\chi_{2057}(1024,\cdot)\) \(\chi_{2057}(1101,\cdot)\) \(\chi_{2057}(1288,\cdot)\) \(\chi_{2057}(1398,\cdot)\) \(\chi_{2057}(1475,\cdot)\) \(\chi_{2057}(1585,\cdot)\) \(\chi_{2057}(1662,\cdot)\) \(\chi_{2057}(1772,\cdot)\) \(\chi_{2057}(1849,\cdot)\) \(\chi_{2057}(1959,\cdot)\) \(\chi_{2057}(2036,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((970,122)\) → \((e\left(\frac{6}{11}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 2057 }(89, a) \) \(1\)\(1\)\(e\left(\frac{1}{22}\right)\)\(-i\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{3}{44}\right)\)\(e\left(\frac{3}{22}\right)\)\(-1\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{37}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2057 }(89,a) \;\) at \(\;a = \) e.g. 2