from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2511, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([5,12]))
pari: [g,chi] = znchar(Mod(1403,2511))
Basic properties
| Modulus: | \(2511\) | |
| Conductor: | \(279\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{279}(101,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2511.bu
\(\chi_{2511}(188,\cdot)\) \(\chi_{2511}(512,\cdot)\) \(\chi_{2511}(593,\cdot)\) \(\chi_{2511}(1025,\cdot)\) \(\chi_{2511}(1349,\cdot)\) \(\chi_{2511}(1403,\cdot)\) \(\chi_{2511}(1430,\cdot)\) \(\chi_{2511}(2240,\cdot)\)
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
\((1055,406)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{2}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2511 }(1403, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) |
sage: chi.jacobi_sum(n)