Properties

Label 1021.126
Modulus $1021$
Conductor $1021$
Order $85$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1021, base_ring=CyclotomicField(170))
 
M = H._module
 
chi = DirichletCharacter(H, M([16]))
 
pari: [g,chi] = znchar(Mod(126,1021))
 

Basic properties

Modulus: \(1021\)
Conductor: \(1021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(85\)
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 1021.q

\(\chi_{1021}(12,\cdot)\) \(\chi_{1021}(14,\cdot)\) \(\chi_{1021}(16,\cdot)\) \(\chi_{1021}(71,\cdot)\) \(\chi_{1021}(73,\cdot)\) \(\chi_{1021}(78,\cdot)\) \(\chi_{1021}(91,\cdot)\) \(\chi_{1021}(104,\cdot)\) \(\chi_{1021}(108,\cdot)\) \(\chi_{1021}(113,\cdot)\) \(\chi_{1021}(115,\cdot)\) \(\chi_{1021}(125,\cdot)\) \(\chi_{1021}(126,\cdot)\) \(\chi_{1021}(144,\cdot)\) \(\chi_{1021}(147,\cdot)\) \(\chi_{1021}(168,\cdot)\) \(\chi_{1021}(192,\cdot)\) \(\chi_{1021}(196,\cdot)\) \(\chi_{1021}(224,\cdot)\) \(\chi_{1021}(235,\cdot)\) \(\chi_{1021}(237,\cdot)\) \(\chi_{1021}(253,\cdot)\) \(\chi_{1021}(255,\cdot)\) \(\chi_{1021}(256,\cdot)\) \(\chi_{1021}(262,\cdot)\) \(\chi_{1021}(275,\cdot)\) \(\chi_{1021}(302,\cdot)\) \(\chi_{1021}(310,\cdot)\) \(\chi_{1021}(316,\cdot)\) \(\chi_{1021}(335,\cdot)\) ...

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: $\Q(\zeta_{85})$
Fixed field: Number field defined by a degree 85 polynomial

Values on generators

\(10\) → \(e\left(\frac{8}{85}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1021 }(126, a) \) \(1\)\(1\)\(e\left(\frac{29}{85}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{58}{85}\right)\)\(e\left(\frac{64}{85}\right)\)\(e\left(\frac{64}{85}\right)\)\(e\left(\frac{74}{85}\right)\)\(e\left(\frac{2}{85}\right)\)\(e\left(\frac{14}{17}\right)\)\(e\left(\frac{8}{85}\right)\)\(e\left(\frac{43}{85}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1021 }(126,a) \;\) at \(\;a = \) e.g. 2