from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8281, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([24,7]))
pari: [g,chi] = znchar(Mod(316,8281))
Basic properties
| Modulus: | \(8281\) | |
| Conductor: | \(637\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{637}(316,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8281.cb
\(\chi_{8281}(316,\cdot)\) \(\chi_{8281}(1037,\cdot)\) \(\chi_{8281}(1499,\cdot)\) \(\chi_{8281}(2220,\cdot)\) \(\chi_{8281}(2682,\cdot)\) \(\chi_{8281}(3403,\cdot)\) \(\chi_{8281}(3865,\cdot)\) \(\chi_{8281}(4586,\cdot)\) \(\chi_{8281}(5769,\cdot)\) \(\chi_{8281}(6231,\cdot)\) \(\chi_{8281}(6952,\cdot)\) \(\chi_{8281}(7414,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((1522,3382)\) → \((e\left(\frac{4}{7}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 8281 }(316, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) |
sage: chi.jacobi_sum(n)