Properties

Label 8036.255
Modulus $8036$
Conductor $8036$
Order $84$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8036, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([42,26,63]))
 
pari: [g,chi] = znchar(Mod(255,8036))
 

Basic properties

Modulus: \(8036\)
Conductor: \(8036\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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 8036.dq

\(\chi_{8036}(255,\cdot)\) \(\chi_{8036}(647,\cdot)\) \(\chi_{8036}(747,\cdot)\) \(\chi_{8036}(1139,\cdot)\) \(\chi_{8036}(1895,\cdot)\) \(\chi_{8036}(2287,\cdot)\) \(\chi_{8036}(2551,\cdot)\) \(\chi_{8036}(2943,\cdot)\) \(\chi_{8036}(3043,\cdot)\) \(\chi_{8036}(3435,\cdot)\) \(\chi_{8036}(3699,\cdot)\) \(\chi_{8036}(4091,\cdot)\) \(\chi_{8036}(4191,\cdot)\) \(\chi_{8036}(4583,\cdot)\) \(\chi_{8036}(4847,\cdot)\) \(\chi_{8036}(5239,\cdot)\) \(\chi_{8036}(5339,\cdot)\) \(\chi_{8036}(5731,\cdot)\) \(\chi_{8036}(5995,\cdot)\) \(\chi_{8036}(6387,\cdot)\) \(\chi_{8036}(7143,\cdot)\) \(\chi_{8036}(7535,\cdot)\) \(\chi_{8036}(7635,\cdot)\) \(\chi_{8036}(8027,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((4019,493,785)\) → \((-1,e\left(\frac{13}{42}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 8036 }(255, a) \) \(1\)\(1\)\(e\left(\frac{5}{84}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{11}{84}\right)\)\(e\left(\frac{13}{28}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{41}{84}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{20}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8036 }(255,a) \;\) at \(\;a = \) e.g. 2