from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(247, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([21,22]))
pari: [g,chi] = znchar(Mod(167,247))
Basic properties
| Modulus: | \(247\) | |
| Conductor: | \(247\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | 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 247.bq
\(\chi_{247}(15,\cdot)\) \(\chi_{247}(33,\cdot)\) \(\chi_{247}(59,\cdot)\) \(\chi_{247}(67,\cdot)\) \(\chi_{247}(71,\cdot)\) \(\chi_{247}(89,\cdot)\) \(\chi_{247}(97,\cdot)\) \(\chi_{247}(98,\cdot)\) \(\chi_{247}(136,\cdot)\) \(\chi_{247}(167,\cdot)\) \(\chi_{247}(184,\cdot)\) \(\chi_{247}(219,\cdot)\)
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: | 36.36.172883305869849387893002611628480390541620325819249742650713178237814353492521813.1 |
Values on generators
\((210,40)\) → \((e\left(\frac{7}{12}\right),e\left(\frac{11}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 247 }(167, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)