Properties

Label 1021.64
Modulus $1021$
Conductor $1021$
Order $170$
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([163]))
 
pari: [g,chi] = znchar(Mod(64,1021))
 

Basic properties

Modulus: \(1021\)
Conductor: \(1021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(170\)
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.s

\(\chi_{1021}(4,\cdot)\) \(\chi_{1021}(36,\cdot)\) \(\chi_{1021}(42,\cdot)\) \(\chi_{1021}(48,\cdot)\) \(\chi_{1021}(49,\cdot)\) \(\chi_{1021}(56,\cdot)\) \(\chi_{1021}(64,\cdot)\) \(\chi_{1021}(79,\cdot)\) \(\chi_{1021}(85,\cdot)\) \(\chi_{1021}(187,\cdot)\) \(\chi_{1021}(202,\cdot)\) \(\chi_{1021}(213,\cdot)\) \(\chi_{1021}(234,\cdot)\) \(\chi_{1021}(273,\cdot)\) \(\chi_{1021}(278,\cdot)\) \(\chi_{1021}(312,\cdot)\) \(\chi_{1021}(314,\cdot)\) \(\chi_{1021}(319,\cdot)\) \(\chi_{1021}(324,\cdot)\) \(\chi_{1021}(339,\cdot)\) \(\chi_{1021}(364,\cdot)\) \(\chi_{1021}(375,\cdot)\) \(\chi_{1021}(378,\cdot)\) \(\chi_{1021}(382,\cdot)\) \(\chi_{1021}(415,\cdot)\) \(\chi_{1021}(441,\cdot)\) \(\chi_{1021}(452,\cdot)\) \(\chi_{1021}(460,\cdot)\) \(\chi_{1021}(504,\cdot)\) \(\chi_{1021}(530,\cdot)\) ...

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 170 polynomial (not computed)

Values on generators

\(10\) → \(e\left(\frac{163}{170}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1021 }(64, a) \) \(1\)\(1\)\(e\left(\frac{49}{170}\right)\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{49}{85}\right)\)\(e\left(\frac{57}{85}\right)\)\(e\left(\frac{29}{170}\right)\)\(e\left(\frac{169}{170}\right)\)\(e\left(\frac{147}{170}\right)\)\(e\left(\frac{13}{17}\right)\)\(e\left(\frac{163}{170}\right)\)\(e\left(\frac{29}{85}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1021 }(64,a) \;\) at \(\;a = \) e.g. 2