from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1815, base_ring=CyclotomicField(10))
M = H._module
chi = DirichletCharacter(H, M([5,5,6]))
pari: [g,chi] = znchar(Mod(614,1815))
Basic properties
| Modulus: | \(1815\) | |
| Conductor: | \(165\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(10\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{165}(119,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1815.o
\(\chi_{1815}(269,\cdot)\) \(\chi_{1815}(614,\cdot)\) \(\chi_{1815}(1049,\cdot)\) \(\chi_{1815}(1334,\cdot)\)
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: | \(\Q(\zeta_{5})\) |
| Fixed field: | 10.0.162778775259375.1 |
Values on generators
\((1211,727,1696)\) → \((-1,-1,e\left(\frac{3}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 1815 }(614, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)