from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3751, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([24,17]))
pari: [g,chi] = znchar(Mod(487,3751))
Basic properties
| Modulus: | \(3751\) | |
| Conductor: | \(341\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{341}(146,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3751.bu
\(\chi_{3751}(487,\cdot)\) \(\chi_{3751}(995,\cdot)\) \(\chi_{3751}(1654,\cdot)\) \(\chi_{3751}(1842,\cdot)\) \(\chi_{3751}(1939,\cdot)\) \(\chi_{3751}(2380,\cdot)\) \(\chi_{3751}(3276,\cdot)\) \(\chi_{3751}(3639,\cdot)\)
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: | 30.0.174949846185675996887411683488270911769488970043898517884513025302911.2 |
Values on generators
\((2543,2421)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{17}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 3751 }(487, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) |
sage: chi.jacobi_sum(n)