from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(115, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,8]))
pari: [g,chi] = znchar(Mod(73,115))
Basic properties
| Modulus: | \(115\) | |
| Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 115.k
\(\chi_{115}(2,\cdot)\) \(\chi_{115}(3,\cdot)\) \(\chi_{115}(8,\cdot)\) \(\chi_{115}(12,\cdot)\) \(\chi_{115}(13,\cdot)\) \(\chi_{115}(18,\cdot)\) \(\chi_{115}(27,\cdot)\) \(\chi_{115}(32,\cdot)\) \(\chi_{115}(48,\cdot)\) \(\chi_{115}(52,\cdot)\) \(\chi_{115}(58,\cdot)\) \(\chi_{115}(62,\cdot)\) \(\chi_{115}(72,\cdot)\) \(\chi_{115}(73,\cdot)\) \(\chi_{115}(77,\cdot)\) \(\chi_{115}(78,\cdot)\) \(\chi_{115}(82,\cdot)\) \(\chi_{115}(87,\cdot)\) \(\chi_{115}(98,\cdot)\) \(\chi_{115}(108,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | 44.0.342865339180420288801608222738062084913425127327306009945459663867950439453125.1 |
Values on generators
\((47,51)\) → \((-i,e\left(\frac{2}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 115 }(73, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) |
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)