Properties

Label 8085.361
Modulus $8085$
Conductor $77$
Order $15$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8085, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,20,18]))
 
pari: [g,chi] = znchar(Mod(361,8085))
 

Basic properties

Modulus: \(8085\)
Conductor: \(77\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{77}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8085.dc

\(\chi_{8085}(361,\cdot)\) \(\chi_{8085}(961,\cdot)\) \(\chi_{8085}(1831,\cdot)\) \(\chi_{8085}(2566,\cdot)\) \(\chi_{8085}(3166,\cdot)\) \(\chi_{8085}(4636,\cdot)\) \(\chi_{8085}(5371,\cdot)\) \(\chi_{8085}(6241,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 15.15.886528337182930278529.1

Values on generators

\((2696,4852,1816,3676)\) → \((1,1,e\left(\frac{2}{3}\right),e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(13\)\(16\)\(17\)\(19\)\(23\)\(26\)\(29\)
\( \chi_{ 8085 }(361, a) \) \(1\)\(1\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{1}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8085 }(361,a) \;\) at \(\;a = \) e.g. 2