from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1375, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([1,40]))
pari: [g,chi] = znchar(Mod(1004,1375))
Basic properties
| Modulus: | \(1375\) | |
| Conductor: | \(1375\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1375.ci
\(\chi_{1375}(9,\cdot)\) \(\chi_{1375}(169,\cdot)\) \(\chi_{1375}(179,\cdot)\) \(\chi_{1375}(214,\cdot)\) \(\chi_{1375}(284,\cdot)\) \(\chi_{1375}(444,\cdot)\) \(\chi_{1375}(454,\cdot)\) \(\chi_{1375}(489,\cdot)\) \(\chi_{1375}(559,\cdot)\) \(\chi_{1375}(719,\cdot)\) \(\chi_{1375}(729,\cdot)\) \(\chi_{1375}(764,\cdot)\) \(\chi_{1375}(834,\cdot)\) \(\chi_{1375}(994,\cdot)\) \(\chi_{1375}(1004,\cdot)\) \(\chi_{1375}(1039,\cdot)\) \(\chi_{1375}(1109,\cdot)\) \(\chi_{1375}(1269,\cdot)\) \(\chi_{1375}(1279,\cdot)\) \(\chi_{1375}(1314,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
| Fixed field: | Number field defined by a degree 50 polynomial |
Values on generators
\((1002,376)\) → \((e\left(\frac{1}{50}\right),e\left(\frac{4}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 1375 }(1004, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{3}{25}\right)\) |
sage: chi.jacobi_sum(n)