from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8030, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([10,16,5]))
pari: [g,chi] = znchar(Mod(4317,8030))
Basic properties
| Modulus: | \(8030\) | |
| Conductor: | \(4015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{4015}(302,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8030.eh
\(\chi_{8030}(533,\cdot)\) \(\chi_{8030}(647,\cdot)\) \(\chi_{8030}(1263,\cdot)\) \(\chi_{8030}(2127,\cdot)\) \(\chi_{8030}(2533,\cdot)\) \(\chi_{8030}(3567,\cdot)\) \(\chi_{8030}(4183,\cdot)\) \(\chi_{8030}(4317,\cdot)\) \(\chi_{8030}(4723,\cdot)\) \(\chi_{8030}(5047,\cdot)\) \(\chi_{8030}(5757,\cdot)\) \(\chi_{8030}(6373,\cdot)\) \(\chi_{8030}(6913,\cdot)\) \(\chi_{8030}(7643,\cdot)\) \(\chi_{8030}(7947,\cdot)\) \(\chi_{8030}(7967,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((1607,2191,881)\) → \((i,e\left(\frac{2}{5}\right),e\left(\frac{1}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8030 }(4317, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{27}{40}\right)\) |
sage: chi.jacobi_sum(n)