from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,36,40]))
pari: [g,chi] = znchar(Mod(221,2800))
Basic properties
| Modulus: | \(2800\) | |
| Conductor: | \(2800\) | 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 2800.fs
\(\chi_{2800}(221,\cdot)\) \(\chi_{2800}(261,\cdot)\) \(\chi_{2800}(541,\cdot)\) \(\chi_{2800}(781,\cdot)\) \(\chi_{2800}(821,\cdot)\) \(\chi_{2800}(1061,\cdot)\) \(\chi_{2800}(1341,\cdot)\) \(\chi_{2800}(1381,\cdot)\) \(\chi_{2800}(1621,\cdot)\) \(\chi_{2800}(1661,\cdot)\) \(\chi_{2800}(1941,\cdot)\) \(\chi_{2800}(2181,\cdot)\) \(\chi_{2800}(2221,\cdot)\) \(\chi_{2800}(2461,\cdot)\) \(\chi_{2800}(2741,\cdot)\) \(\chi_{2800}(2781,\cdot)\)
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
\((351,2101,2577,801)\) → \((1,-i,e\left(\frac{3}{5}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 2800 }(221, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) |
sage: chi.jacobi_sum(n)