from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(287, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,3]))
pari: [g,chi] = znchar(Mod(200,287))
Basic properties
| Modulus: | \(287\) | |
| Conductor: | \(287\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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 287.bc
\(\chi_{287}(2,\cdot)\) \(\chi_{287}(39,\cdot)\) \(\chi_{287}(46,\cdot)\) \(\chi_{287}(74,\cdot)\) \(\chi_{287}(102,\cdot)\) \(\chi_{287}(121,\cdot)\) \(\chi_{287}(128,\cdot)\) \(\chi_{287}(144,\cdot)\) \(\chi_{287}(156,\cdot)\) \(\chi_{287}(172,\cdot)\) \(\chi_{287}(184,\cdot)\) \(\chi_{287}(200,\cdot)\) \(\chi_{287}(207,\cdot)\) \(\chi_{287}(226,\cdot)\) \(\chi_{287}(254,\cdot)\) \(\chi_{287}(282,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((206,211)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 287 }(200, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{41}{60}\right)\) |
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)