from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1800, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,15,10,18]))
pari: [g,chi] = znchar(Mod(1021,1800))
Basic properties
| Modulus: | \(1800\) | |
| Conductor: | \(1800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | 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.cv
\(\chi_{1800}(61,\cdot)\) \(\chi_{1800}(421,\cdot)\) \(\chi_{1800}(661,\cdot)\) \(\chi_{1800}(781,\cdot)\) \(\chi_{1800}(1021,\cdot)\) \(\chi_{1800}(1141,\cdot)\) \(\chi_{1800}(1381,\cdot)\) \(\chi_{1800}(1741,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((1351,901,1001,577)\) → \((1,-1,e\left(\frac{1}{3}\right),e\left(\frac{3}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1800 }(1021, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{15}\right)\) |
sage: chi.jacobi_sum(n)