from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1287, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,54,55]))
pari: [g,chi] = znchar(Mod(1073,1287))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1287.ed
\(\chi_{1287}(41,\cdot)\) \(\chi_{1287}(50,\cdot)\) \(\chi_{1287}(167,\cdot)\) \(\chi_{1287}(227,\cdot)\) \(\chi_{1287}(371,\cdot)\) \(\chi_{1287}(392,\cdot)\) \(\chi_{1287}(578,\cdot)\) \(\chi_{1287}(635,\cdot)\) \(\chi_{1287}(695,\cdot)\) \(\chi_{1287}(722,\cdot)\) \(\chi_{1287}(743,\cdot)\) \(\chi_{1287}(860,\cdot)\) \(\chi_{1287}(986,\cdot)\) \(\chi_{1287}(1073,\cdot)\) \(\chi_{1287}(1163,\cdot)\) \(\chi_{1287}(1190,\cdot)\)
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
\((1145,937,496)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{9}{10}\right),e\left(\frac{11}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1287 }(1073, a) \) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) |
sage: chi.jacobi_sum(n)