Properties

Label 4021.32
Modulus $4021$
Conductor $4021$
Order $804$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4021, base_ring=CyclotomicField(804))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(32,4021))
 

Basic properties

Modulus: \(4021\)
Conductor: \(4021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(804\)
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 4021.t

\(\chi_{4021}(28,\cdot)\) \(\chi_{4021}(29,\cdot)\) \(\chi_{4021}(32,\cdot)\) \(\chi_{4021}(57,\cdot)\) \(\chi_{4021}(95,\cdot)\) \(\chi_{4021}(99,\cdot)\) \(\chi_{4021}(113,\cdot)\) \(\chi_{4021}(129,\cdot)\) \(\chi_{4021}(146,\cdot)\) \(\chi_{4021}(165,\cdot)\) \(\chi_{4021}(202,\cdot)\) \(\chi_{4021}(211,\cdot)\) \(\chi_{4021}(215,\cdot)\) \(\chi_{4021}(229,\cdot)\) \(\chi_{4021}(262,\cdot)\) \(\chi_{4021}(275,\cdot)\) \(\chi_{4021}(281,\cdot)\) \(\chi_{4021}(302,\cdot)\) \(\chi_{4021}(318,\cdot)\) \(\chi_{4021}(322,\cdot)\) \(\chi_{4021}(364,\cdot)\) \(\chi_{4021}(368,\cdot)\) \(\chi_{4021}(369,\cdot)\) \(\chi_{4021}(377,\cdot)\) \(\chi_{4021}(416,\cdot)\) \(\chi_{4021}(427,\cdot)\) \(\chi_{4021}(446,\cdot)\) \(\chi_{4021}(457,\cdot)\) \(\chi_{4021}(462,\cdot)\) \(\chi_{4021}(471,\cdot)\) ...

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: $\Q(\zeta_{804})$
Fixed field: Number field defined by a degree 804 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{1}{804}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4021 }(32, a) \) \(-1\)\(1\)\(e\left(\frac{1}{804}\right)\)\(e\left(\frac{143}{201}\right)\)\(e\left(\frac{1}{402}\right)\)\(e\left(\frac{131}{201}\right)\)\(e\left(\frac{191}{268}\right)\)\(-i\)\(e\left(\frac{1}{268}\right)\)\(e\left(\frac{85}{201}\right)\)\(e\left(\frac{175}{268}\right)\)\(e\left(\frac{145}{804}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4021 }(32,a) \;\) at \(\;a = \) e.g. 2