from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,3,14]))
pari: [g,chi] = znchar(Mod(458,2475))
Basic properties
| Modulus: | \(2475\) | |
| Conductor: | \(825\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{825}(458,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2475.dw
\(\chi_{2475}(458,\cdot)\) \(\chi_{2475}(503,\cdot)\) \(\chi_{2475}(1097,\cdot)\) \(\chi_{2475}(1502,\cdot)\) \(\chi_{2475}(1898,\cdot)\) \(\chi_{2475}(2042,\cdot)\) \(\chi_{2475}(2213,\cdot)\) \(\chi_{2475}(2312,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | 20.0.955500748952625950104466755874454975128173828125.4 |
Values on generators
\((551,2377,2026)\) → \((-1,e\left(\frac{3}{20}\right),e\left(\frac{7}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 2475 }(458, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(-i\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) |
sage: chi.jacobi_sum(n)