from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6069, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,16,27]))
pari: [g,chi] = znchar(Mod(65,6069))
Basic properties
| Modulus: | \(6069\) | |
| Conductor: | \(357\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{357}(65,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6069.bu
\(\chi_{6069}(65,\cdot)\) \(\chi_{6069}(158,\cdot)\) \(\chi_{6069}(653,\cdot)\) \(\chi_{6069}(998,\cdot)\) \(\chi_{6069}(1892,\cdot)\) \(\chi_{6069}(2237,\cdot)\) \(\chi_{6069}(2732,\cdot)\) \(\chi_{6069}(2825,\cdot)\) \(\chi_{6069}(2930,\cdot)\) \(\chi_{6069}(3971,\cdot)\) \(\chi_{6069}(4295,\cdot)\) \(\chi_{6069}(4400,\cdot)\) \(\chi_{6069}(4559,\cdot)\) \(\chi_{6069}(4664,\cdot)\) \(\chi_{6069}(4988,\cdot)\) \(\chi_{6069}(6029,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((2024,4336,3760)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{9}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
| \( \chi_{ 6069 }(65, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(i\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{16}\right)\) |
sage: chi.jacobi_sum(n)