Properties

Label 8041.1885
Modulus $8041$
Conductor $8041$
Order $120$
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(120))
 
M = H._module
 
chi = DirichletCharacter(H, M([24,45,40]))
 
pari: [g,chi] = znchar(Mod(1885,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(120\)
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.ek

\(\chi_{8041}(36,\cdot)\) \(\chi_{8041}(49,\cdot)\) \(\chi_{8041}(423,\cdot)\) \(\chi_{8041}(995,\cdot)\) \(\chi_{8041}(1369,\cdot)\) \(\chi_{8041}(1511,\cdot)\) \(\chi_{8041}(1885,\cdot)\) \(\chi_{8041}(2242,\cdot)\) \(\chi_{8041}(2457,\cdot)\) \(\chi_{8041}(2616,\cdot)\) \(\chi_{8041}(2831,\cdot)\) \(\chi_{8041}(2973,\cdot)\) \(\chi_{8041}(3188,\cdot)\) \(\chi_{8041}(3347,\cdot)\) \(\chi_{8041}(3562,\cdot)\) \(\chi_{8041}(3833,\cdot)\) \(\chi_{8041}(3919,\cdot)\) \(\chi_{8041}(4207,\cdot)\) \(\chi_{8041}(4293,\cdot)\) \(\chi_{8041}(4779,\cdot)\) \(\chi_{8041}(5153,\cdot)\) \(\chi_{8041}(5295,\cdot)\) \(\chi_{8041}(5669,\cdot)\) \(\chi_{8041}(6026,\cdot)\) \(\chi_{8041}(6241,\cdot)\) \(\chi_{8041}(6400,\cdot)\) \(\chi_{8041}(6615,\cdot)\) \(\chi_{8041}(6757,\cdot)\) \(\chi_{8041}(6972,\cdot)\) \(\chi_{8041}(7131,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{3}{8}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(1885, a) \) \(1\)\(1\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{37}{120}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{120}\right)\)\(e\left(\frac{91}{120}\right)\)\(e\left(\frac{23}{120}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{5}{24}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(1885,a) \;\) at \(\;a = \) e.g. 2