Properties

Label 8041.515
Modulus $8041$
Conductor $8041$
Order $80$
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(80))
 
M = H._module
 
chi = DirichletCharacter(H, M([48,25,40]))
 
pari: [g,chi] = znchar(Mod(515,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
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.dy

\(\chi_{8041}(214,\cdot)\) \(\chi_{8041}(300,\cdot)\) \(\chi_{8041}(515,\cdot)\) \(\chi_{8041}(687,\cdot)\) \(\chi_{8041}(1246,\cdot)\) \(\chi_{8041}(1676,\cdot)\) \(\chi_{8041}(2149,\cdot)\) \(\chi_{8041}(2407,\cdot)\) \(\chi_{8041}(2579,\cdot)\) \(\chi_{8041}(2880,\cdot)\) \(\chi_{8041}(3138,\cdot)\) \(\chi_{8041}(3525,\cdot)\) \(\chi_{8041}(3611,\cdot)\) \(\chi_{8041}(3998,\cdot)\) \(\chi_{8041}(4041,\cdot)\) \(\chi_{8041}(4772,\cdot)\) \(\chi_{8041}(4944,\cdot)\) \(\chi_{8041}(4987,\cdot)\) \(\chi_{8041}(5417,\cdot)\) \(\chi_{8041}(5460,\cdot)\) \(\chi_{8041}(5503,\cdot)\) \(\chi_{8041}(5718,\cdot)\) \(\chi_{8041}(6191,\cdot)\) \(\chi_{8041}(6363,\cdot)\) \(\chi_{8041}(6406,\cdot)\) \(\chi_{8041}(6449,\cdot)\) \(\chi_{8041}(6879,\cdot)\) \(\chi_{8041}(6922,\cdot)\) \(\chi_{8041}(7137,\cdot)\) \(\chi_{8041}(7610,\cdot)\) ...

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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(515, a) \) \(1\)\(1\)\(e\left(\frac{19}{40}\right)\)\(e\left(\frac{49}{80}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{37}{80}\right)\)\(e\left(\frac{7}{80}\right)\)\(e\left(\frac{11}{80}\right)\)\(e\left(\frac{17}{40}\right)\)\(e\left(\frac{9}{40}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{9}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(515,a) \;\) at \(\;a = \) e.g. 2