from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,3,18]))
pari: [g,chi] = znchar(Mod(52,2475))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2475.fr
\(\chi_{2475}(52,\cdot)\) \(\chi_{2475}(178,\cdot)\) \(\chi_{2475}(238,\cdot)\) \(\chi_{2475}(292,\cdot)\) \(\chi_{2475}(337,\cdot)\) \(\chi_{2475}(358,\cdot)\) \(\chi_{2475}(448,\cdot)\) \(\chi_{2475}(772,\cdot)\) \(\chi_{2475}(877,\cdot)\) \(\chi_{2475}(1003,\cdot)\) \(\chi_{2475}(1183,\cdot)\) \(\chi_{2475}(1273,\cdot)\) \(\chi_{2475}(1597,\cdot)\) \(\chi_{2475}(1888,\cdot)\) \(\chi_{2475}(1942,\cdot)\) \(\chi_{2475}(1987,\cdot)\)
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
\((551,2377,2026)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{20}\right),e\left(\frac{3}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 2475 }(52, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{53}{60}\right)\) |
sage: chi.jacobi_sum(n)