from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1305, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([0,3,2]))
pari: [g,chi] = znchar(Mod(28,1305))
Basic properties
| Modulus: | \(1305\) | |
| Conductor: | \(145\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{145}(28,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1305.o
\(\chi_{1305}(28,\cdot)\) \(\chi_{1305}(1072,\cdot)\)
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: | \(\mathbb{Q}(i)\) |
| Fixed field: | 4.0.105125.2 |
Values on generators
\((146,262,901)\) → \((1,-i,-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1305 }(28, a) \) | \(-1\) | \(1\) | \(i\) | \(-1\) | \(-i\) | \(-i\) | \(-1\) | \(i\) | \(1\) | \(1\) | \(i\) | \(1\) |
sage: chi.jacobi_sum(n)