from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,20,9]))
pari: [g,chi] = znchar(Mod(283,1800))
Basic properties
| Modulus: | \(1800\) | |
| Conductor: | \(1800\) | 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 1800.dr
\(\chi_{1800}(67,\cdot)\) \(\chi_{1800}(187,\cdot)\) \(\chi_{1800}(283,\cdot)\) \(\chi_{1800}(403,\cdot)\) \(\chi_{1800}(427,\cdot)\) \(\chi_{1800}(547,\cdot)\) \(\chi_{1800}(763,\cdot)\) \(\chi_{1800}(787,\cdot)\) \(\chi_{1800}(1003,\cdot)\) \(\chi_{1800}(1123,\cdot)\) \(\chi_{1800}(1147,\cdot)\) \(\chi_{1800}(1267,\cdot)\) \(\chi_{1800}(1363,\cdot)\) \(\chi_{1800}(1483,\cdot)\) \(\chi_{1800}(1627,\cdot)\) \(\chi_{1800}(1723,\cdot)\)
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
\((1351,901,1001,577)\) → \((-1,-1,e\left(\frac{1}{3}\right),e\left(\frac{3}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1800 }(283, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{4}{15}\right)\) |
sage: chi.jacobi_sum(n)