from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1375, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([28,20]))
pari: [g,chi] = znchar(Mod(1061,1375))
Basic properties
| Modulus: | \(1375\) | |
| Conductor: | \(1375\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(25\) | 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.bs
\(\chi_{1375}(81,\cdot)\) \(\chi_{1375}(141,\cdot)\) \(\chi_{1375}(146,\cdot)\) \(\chi_{1375}(236,\cdot)\) \(\chi_{1375}(356,\cdot)\) \(\chi_{1375}(416,\cdot)\) \(\chi_{1375}(421,\cdot)\) \(\chi_{1375}(511,\cdot)\) \(\chi_{1375}(631,\cdot)\) \(\chi_{1375}(691,\cdot)\) \(\chi_{1375}(696,\cdot)\) \(\chi_{1375}(786,\cdot)\) \(\chi_{1375}(906,\cdot)\) \(\chi_{1375}(966,\cdot)\) \(\chi_{1375}(971,\cdot)\) \(\chi_{1375}(1061,\cdot)\) \(\chi_{1375}(1181,\cdot)\) \(\chi_{1375}(1241,\cdot)\) \(\chi_{1375}(1246,\cdot)\) \(\chi_{1375}(1336,\cdot)\)
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 25 polynomial |
Values on generators
\((1002,376)\) → \((e\left(\frac{14}{25}\right),e\left(\frac{2}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 1375 }(1061, a) \) | \(1\) | \(1\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) |
sage: chi.jacobi_sum(n)