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)