from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(247, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([21,26]))
pari: [g,chi] = znchar(Mod(193,247))
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.br
\(\chi_{247}(2,\cdot)\) \(\chi_{247}(32,\cdot)\) \(\chi_{247}(41,\cdot)\) \(\chi_{247}(72,\cdot)\) \(\chi_{247}(110,\cdot)\) \(\chi_{247}(124,\cdot)\) \(\chi_{247}(128,\cdot)\) \(\chi_{247}(154,\cdot)\) \(\chi_{247}(162,\cdot)\) \(\chi_{247}(193,\cdot)\) \(\chi_{247}(223,\cdot)\) \(\chi_{247}(241,\cdot)\)
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: | 36.36.172883305869849387893002611628480390541620325819249742650713178237814353492521813.2 |
Values on generators
\((210,40)\) → \((e\left(\frac{7}{12}\right),e\left(\frac{13}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 247 }(193, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(-i\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(-i\) |
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)