from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8624, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,20,21]))
pari: [g,chi] = znchar(Mod(65,8624))
Basic properties
| Modulus: | \(8624\) | |
| Conductor: | \(539\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{539}(65,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8624.ex
\(\chi_{8624}(65,\cdot)\) \(\chi_{8624}(417,\cdot)\) \(\chi_{8624}(1297,\cdot)\) \(\chi_{8624}(1649,\cdot)\) \(\chi_{8624}(2881,\cdot)\) \(\chi_{8624}(3761,\cdot)\) \(\chi_{8624}(4113,\cdot)\) \(\chi_{8624}(4993,\cdot)\) \(\chi_{8624}(5345,\cdot)\) \(\chi_{8624}(6225,\cdot)\) \(\chi_{8624}(6577,\cdot)\) \(\chi_{8624}(7457,\cdot)\)
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
\((5391,6469,7745,3137)\) → \((1,1,e\left(\frac{10}{21}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 8624 }(65, a) \) | \(-1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) |
sage: chi.jacobi_sum(n)