Properties

Label 4021.27
Modulus $4021$
Conductor $4021$
Order $335$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4021, base_ring=CyclotomicField(670))
 
M = H._module
 
chi = DirichletCharacter(H, M([152]))
 
pari: [g,chi] = znchar(Mod(27,4021))
 

Basic properties

Modulus: \(4021\)
Conductor: \(4021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(335\)
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 4021.q

\(\chi_{4021}(27,\cdot)\) \(\chi_{4021}(38,\cdot)\) \(\chi_{4021}(42,\cdot)\) \(\chi_{4021}(45,\cdot)\) \(\chi_{4021}(48,\cdot)\) \(\chi_{4021}(70,\cdot)\) \(\chi_{4021}(71,\cdot)\) \(\chi_{4021}(75,\cdot)\) \(\chi_{4021}(80,\cdot)\) \(\chi_{4021}(86,\cdot)\) \(\chi_{4021}(94,\cdot)\) \(\chi_{4021}(101,\cdot)\) \(\chi_{4021}(125,\cdot)\) \(\chi_{4021}(209,\cdot)\) \(\chi_{4021}(221,\cdot)\) \(\chi_{4021}(231,\cdot)\) \(\chi_{4021}(244,\cdot)\) \(\chi_{4021}(264,\cdot)\) \(\chi_{4021}(269,\cdot)\) \(\chi_{4021}(289,\cdot)\) \(\chi_{4021}(333,\cdot)\) \(\chi_{4021}(346,\cdot)\) \(\chi_{4021}(356,\cdot)\) \(\chi_{4021}(359,\cdot)\) \(\chi_{4021}(385,\cdot)\) \(\chi_{4021}(440,\cdot)\) \(\chi_{4021}(473,\cdot)\) \(\chi_{4021}(491,\cdot)\) \(\chi_{4021}(517,\cdot)\) \(\chi_{4021}(518,\cdot)\) ...

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: $\Q(\zeta_{335})$
Fixed field: Number field defined by a degree 335 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{76}{335}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4021 }(27, a) \) \(1\)\(1\)\(e\left(\frac{76}{335}\right)\)\(e\left(\frac{324}{335}\right)\)\(e\left(\frac{152}{335}\right)\)\(e\left(\frac{294}{335}\right)\)\(e\left(\frac{13}{67}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{228}{335}\right)\)\(e\left(\frac{313}{335}\right)\)\(e\left(\frac{7}{67}\right)\)\(e\left(\frac{32}{335}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4021 }(27,a) \;\) at \(\;a = \) e.g. 2