from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2385, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([4,3,0]))
pari: [g,chi] = znchar(Mod(2227,2385))
Basic properties
| Modulus: | \(2385\) | |
| Conductor: | \(45\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{45}(22,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2385.bh
\(\chi_{2385}(637,\cdot)\) \(\chi_{2385}(1273,\cdot)\) \(\chi_{2385}(2068,\cdot)\) \(\chi_{2385}(2227,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
| Fixed field: | 12.0.84075626953125.1 |
Values on generators
\((1856,1432,1486)\) → \((e\left(\frac{1}{3}\right),i,1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 2385 }(2227, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(-1\) |
sage: chi.jacobi_sum(n)