from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10000, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([0,1,2]))
pari: [g,chi] = znchar(Mod(3749,10000))
Basic properties
| Modulus: | \(10000\) | |
| Conductor: | \(80\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{80}(69,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 10000.q
\(\chi_{10000}(3749,\cdot)\) \(\chi_{10000}(8749,\cdot)\)
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: | 4.4.51200.1 |
Values on generators
\((8751,2501,9377)\) → \((1,i,-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 10000 }(3749, a) \) | \(1\) | \(1\) | \(i\) | \(1\) | \(-1\) | \(i\) | \(i\) | \(-1\) | \(-i\) | \(i\) | \(1\) | \(-i\) |
sage: chi.jacobi_sum(n)