Properties

Label 1021.50
Modulus $1021$
Conductor $1021$
Order $1020$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1021, base_ring=CyclotomicField(1020))
 
M = H._module
 
chi = DirichletCharacter(H, M([689]))
 
pari: [g,chi] = znchar(Mod(50,1021))
 

Basic properties

Modulus: \(1021\)
Conductor: \(1021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1020\)
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 1021.x

\(\chi_{1021}(10,\cdot)\) \(\chi_{1021}(22,\cdot)\) \(\chi_{1021}(30,\cdot)\) \(\chi_{1021}(31,\cdot)\) \(\chi_{1021}(34,\cdot)\) \(\chi_{1021}(35,\cdot)\) \(\chi_{1021}(37,\cdot)\) \(\chi_{1021}(40,\cdot)\) \(\chi_{1021}(43,\cdot)\) \(\chi_{1021}(46,\cdot)\) \(\chi_{1021}(50,\cdot)\) \(\chi_{1021}(53,\cdot)\) \(\chi_{1021}(59,\cdot)\) \(\chi_{1021}(65,\cdot)\) \(\chi_{1021}(66,\cdot)\) \(\chi_{1021}(76,\cdot)\) \(\chi_{1021}(77,\cdot)\) \(\chi_{1021}(82,\cdot)\) \(\chi_{1021}(90,\cdot)\) \(\chi_{1021}(93,\cdot)\) \(\chi_{1021}(94,\cdot)\) \(\chi_{1021}(95,\cdot)\) \(\chi_{1021}(102,\cdot)\) \(\chi_{1021}(103,\cdot)\) \(\chi_{1021}(105,\cdot)\) \(\chi_{1021}(109,\cdot)\) \(\chi_{1021}(111,\cdot)\) \(\chi_{1021}(119,\cdot)\) \(\chi_{1021}(120,\cdot)\) \(\chi_{1021}(122,\cdot)\) ...

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: $\Q(\zeta_{1020})$
Fixed field: Number field defined by a degree 1020 polynomial (not computed)

Values on generators

\(10\) → \(e\left(\frac{689}{1020}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1021 }(50, a) \) \(-1\)\(1\)\(e\left(\frac{319}{340}\right)\)\(e\left(\frac{13}{34}\right)\)\(e\left(\frac{149}{170}\right)\)\(e\left(\frac{188}{255}\right)\)\(e\left(\frac{109}{340}\right)\)\(e\left(\frac{49}{340}\right)\)\(e\left(\frac{277}{340}\right)\)\(e\left(\frac{13}{17}\right)\)\(e\left(\frac{689}{1020}\right)\)\(e\left(\frac{206}{255}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1021 }(50,a) \;\) at \(\;a = \) e.g. 2