Properties

Label 8041.20
Modulus $8041$
Conductor $8041$
Order $1680$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8041, base_ring=CyclotomicField(1680))
 
M = H._module
 
chi = DirichletCharacter(H, M([1008,105,1480]))
 
pari: [g,chi] = znchar(Mod(20,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1680\)
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 8041.gd

\(\chi_{8041}(3,\cdot)\) \(\chi_{8041}(5,\cdot)\) \(\chi_{8041}(20,\cdot)\) \(\chi_{8041}(48,\cdot)\) \(\chi_{8041}(71,\cdot)\) \(\chi_{8041}(91,\cdot)\) \(\chi_{8041}(114,\cdot)\) \(\chi_{8041}(141,\cdot)\) \(\chi_{8041}(147,\cdot)\) \(\chi_{8041}(148,\cdot)\) \(\chi_{8041}(158,\cdot)\) \(\chi_{8041}(159,\cdot)\) \(\chi_{8041}(163,\cdot)\) \(\chi_{8041}(190,\cdot)\) \(\chi_{8041}(192,\cdot)\) \(\chi_{8041}(201,\cdot)\) \(\chi_{8041}(218,\cdot)\) \(\chi_{8041}(235,\cdot)\) \(\chi_{8041}(245,\cdot)\) \(\chi_{8041}(278,\cdot)\) \(\chi_{8041}(284,\cdot)\) \(\chi_{8041}(313,\cdot)\) \(\chi_{8041}(334,\cdot)\) \(\chi_{8041}(335,\cdot)\) \(\chi_{8041}(377,\cdot)\) \(\chi_{8041}(405,\cdot)\) \(\chi_{8041}(449,\cdot)\) \(\chi_{8041}(456,\cdot)\) \(\chi_{8041}(499,\cdot)\) \(\chi_{8041}(521,\cdot)\) ...

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: $\Q(\zeta_{1680})$
Fixed field: Number field defined by a degree 1680 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{1}{16}\right),e\left(\frac{37}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(20, a) \) \(1\)\(1\)\(e\left(\frac{73}{280}\right)\)\(e\left(\frac{1249}{1680}\right)\)\(e\left(\frac{73}{140}\right)\)\(e\left(\frac{1237}{1680}\right)\)\(e\left(\frac{1}{240}\right)\)\(e\left(\frac{173}{240}\right)\)\(e\left(\frac{219}{280}\right)\)\(e\left(\frac{409}{840}\right)\)\(e\left(\frac{335}{336}\right)\)\(e\left(\frac{89}{336}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(20,a) \;\) at \(\;a = \) e.g. 2