from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4275, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,9,20]))
pari: [g,chi] = znchar(Mod(83,4275))
Basic properties
| Modulus: | \(4275\) | |
| Conductor: | \(4275\) | 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 4275.ga
\(\chi_{4275}(83,\cdot)\) \(\chi_{4275}(752,\cdot)\) \(\chi_{4275}(767,\cdot)\) \(\chi_{4275}(923,\cdot)\) \(\chi_{4275}(938,\cdot)\) \(\chi_{4275}(1622,\cdot)\) \(\chi_{4275}(1778,\cdot)\) \(\chi_{4275}(2462,\cdot)\) \(\chi_{4275}(2477,\cdot)\) \(\chi_{4275}(2633,\cdot)\) \(\chi_{4275}(2648,\cdot)\) \(\chi_{4275}(3317,\cdot)\) \(\chi_{4275}(3488,\cdot)\) \(\chi_{4275}(3503,\cdot)\) \(\chi_{4275}(4172,\cdot)\) \(\chi_{4275}(4187,\cdot)\)
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
\((1901,1027,1351)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{3}{20}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
| \( \chi_{ 4275 }(83, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) |
sage: chi.jacobi_sum(n)