from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,39,6]))
pari: [g,chi] = znchar(Mod(167,2475))
Basic properties
| Modulus: | \(2475\) | |
| Conductor: | \(2475\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2475.gl
\(\chi_{2475}(167,\cdot)\) \(\chi_{2475}(227,\cdot)\) \(\chi_{2475}(338,\cdot)\) \(\chi_{2475}(347,\cdot)\) \(\chi_{2475}(437,\cdot)\) \(\chi_{2475}(578,\cdot)\) \(\chi_{2475}(623,\cdot)\) \(\chi_{2475}(992,\cdot)\) \(\chi_{2475}(1058,\cdot)\) \(\chi_{2475}(1163,\cdot)\) \(\chi_{2475}(1172,\cdot)\) \(\chi_{2475}(1262,\cdot)\) \(\chi_{2475}(1877,\cdot)\) \(\chi_{2475}(1883,\cdot)\) \(\chi_{2475}(2228,\cdot)\) \(\chi_{2475}(2273,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((551,2377,2026)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{13}{20}\right),e\left(\frac{1}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 2475 }(167, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(-i\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(1\) | \(e\left(\frac{19}{60}\right)\) |
sage: chi.jacobi_sum(n)