from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(28900, base_ring=CyclotomicField(10))
M = H._module
chi = DirichletCharacter(H, M([0,3,5]))
pari: [g,chi] = znchar(Mod(2889,28900))
Basic properties
| Modulus: | \(28900\) | |
| Conductor: | \(425\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(10\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{425}(339,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 28900.be
\(\chi_{28900}(2889,\cdot)\) \(\chi_{28900}(8669,\cdot)\) \(\chi_{28900}(20229,\cdot)\) \(\chi_{28900}(26009,\cdot)\)
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: | 10.10.1083264923095703125.1 |
Values on generators
\((14451,24277,23701)\) → \((1,e\left(\frac{3}{10}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 28900 }(2889, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) |
sage: chi.jacobi_sum(n)