Properties

Label 8512.5391
Modulus $8512$
Conductor $304$
Order $36$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8512, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,9,0,14]))
 
pari: [g,chi] = znchar(Mod(5391,8512))
 

Basic properties

Modulus: \(8512\)
Conductor: \(304\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{304}(299,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8512.it

\(\chi_{8512}(15,\cdot)\) \(\chi_{8512}(687,\cdot)\) \(\chi_{8512}(1135,\cdot)\) \(\chi_{8512}(1359,\cdot)\) \(\chi_{8512}(1807,\cdot)\) \(\chi_{8512}(2255,\cdot)\) \(\chi_{8512}(4271,\cdot)\) \(\chi_{8512}(4943,\cdot)\) \(\chi_{8512}(5391,\cdot)\) \(\chi_{8512}(5615,\cdot)\) \(\chi_{8512}(6063,\cdot)\) \(\chi_{8512}(6511,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.36.19036714782161565107424425435655777110146017378670996611401194085493506048.1

Values on generators

\((5055,6917,7297,3137)\) → \((-1,i,1,e\left(\frac{7}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\( \chi_{ 8512 }(5391, a) \) \(1\)\(1\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{11}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8512 }(5391,a) \;\) at \(\;a = \) e.g. 2