from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2156, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,25,27]))
pari: [g,chi] = znchar(Mod(215,2156))
Basic properties
| Modulus: | \(2156\) | |
| Conductor: | \(308\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{308}(215,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2156.bm
\(\chi_{2156}(19,\cdot)\) \(\chi_{2156}(215,\cdot)\) \(\chi_{2156}(227,\cdot)\) \(\chi_{2156}(607,\cdot)\) \(\chi_{2156}(1195,\cdot)\) \(\chi_{2156}(1207,\cdot)\) \(\chi_{2156}(1403,\cdot)\) \(\chi_{2156}(1795,\cdot)\)
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: | 30.0.18877886452453235195604373741566776206034848714286723760128.1 |
Values on generators
\((1079,1277,981)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{9}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 2156 }(215, a) \) | \(-1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) |
sage: chi.jacobi_sum(n)