Properties

Label 8041.639
Modulus $8041$
Conductor $731$
Order $48$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8041, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,8]))
 
pari: [g,chi] = znchar(Mod(639,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(731\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{731}(639,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8041.dg

\(\chi_{8041}(265,\cdot)\) \(\chi_{8041}(639,\cdot)\) \(\chi_{8041}(738,\cdot)\) \(\chi_{8041}(1112,\cdot)\) \(\chi_{8041}(2630,\cdot)\) \(\chi_{8041}(3004,\cdot)\) \(\chi_{8041}(3576,\cdot)\) \(\chi_{8041}(3950,\cdot)\) \(\chi_{8041}(4049,\cdot)\) \(\chi_{8041}(4423,\cdot)\) \(\chi_{8041}(4995,\cdot)\) \(\chi_{8041}(5369,\cdot)\) \(\chi_{8041}(5468,\cdot)\) \(\chi_{8041}(5842,\cdot)\) \(\chi_{8041}(6414,\cdot)\) \(\chi_{8041}(6788,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((6580,2366,562)\) → \((1,e\left(\frac{3}{16}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(639, a) \) \(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{17}{48}\right)\)\(i\)\(e\left(\frac{5}{48}\right)\)\(e\left(\frac{23}{48}\right)\)\(e\left(\frac{43}{48}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{29}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(639,a) \;\) at \(\;a = \) e.g. 2