from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1287, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,18,15]))
pari: [g,chi] = znchar(Mod(151,1287))
Basic properties
| Modulus: | \(1287\) | |
| Conductor: | \(1287\) | 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 1287.el
\(\chi_{1287}(112,\cdot)\) \(\chi_{1287}(151,\cdot)\) \(\chi_{1287}(304,\cdot)\) \(\chi_{1287}(382,\cdot)\) \(\chi_{1287}(502,\cdot)\) \(\chi_{1287}(580,\cdot)\) \(\chi_{1287}(655,\cdot)\) \(\chi_{1287}(733,\cdot)\) \(\chi_{1287}(772,\cdot)\) \(\chi_{1287}(853,\cdot)\) \(\chi_{1287}(931,\cdot)\) \(\chi_{1287}(970,\cdot)\) \(\chi_{1287}(1084,\cdot)\) \(\chi_{1287}(1201,\cdot)\) \(\chi_{1287}(1240,\cdot)\) \(\chi_{1287}(1282,\cdot)\)
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
\((1145,937,496)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{3}{10}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1287 }(151, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) |
sage: chi.jacobi_sum(n)