from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3100, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,15,19]))
pari: [g,chi] = znchar(Mod(849,3100))
Basic properties
| Modulus: | \(3100\) | |
| Conductor: | \(155\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{155}(74,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3100.ex
\(\chi_{3100}(549,\cdot)\) \(\chi_{3100}(849,\cdot)\) \(\chi_{3100}(1749,\cdot)\) \(\chi_{3100}(2049,\cdot)\) \(\chi_{3100}(2249,\cdot)\) \(\chi_{3100}(2349,\cdot)\) \(\chi_{3100}(2749,\cdot)\) \(\chi_{3100}(3049,\cdot)\)
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: | 30.0.542049797152523742060051576582522667353340789794921875.1 |
Values on generators
\((1551,2977,1801)\) → \((1,-1,e\left(\frac{19}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 3100 }(849, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) |
sage: chi.jacobi_sum(n)