Properties

Label 8085.391
Modulus $8085$
Conductor $77$
Order $10$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8085, base_ring=CyclotomicField(10))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,5,9]))
 
pari: [g,chi] = znchar(Mod(391,8085))
 

Basic properties

Modulus: \(8085\)
Conductor: \(77\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{77}(6,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8085.bx

\(\chi_{8085}(391,\cdot)\) \(\chi_{8085}(1861,\cdot)\) \(\chi_{8085}(4066,\cdot)\) \(\chi_{8085}(7741,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 10.10.39630026842637.1

Values on generators

\((2696,4852,1816,3676)\) → \((1,1,-1,e\left(\frac{9}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(13\)\(16\)\(17\)\(19\)\(23\)\(26\)\(29\)
\( \chi_{ 8085 }(391, a) \) \(1\)\(1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(1\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8085 }(391,a) \;\) at \(\;a = \) e.g. 2