from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([20,18,18]))
pari: [g,chi] = znchar(Mod(196,2475))
Basic properties
| Modulus: | \(2475\) | |
| Conductor: | \(2475\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(15\) | 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.cw
\(\chi_{2475}(196,\cdot)\) \(\chi_{2475}(466,\cdot)\) \(\chi_{2475}(1006,\cdot)\) \(\chi_{2475}(1021,\cdot)\) \(\chi_{2475}(1291,\cdot)\) \(\chi_{2475}(1411,\cdot)\) \(\chi_{2475}(1831,\cdot)\) \(\chi_{2475}(2236,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | Number field defined by a degree 15 polynomial |
Values on generators
\((551,2377,2026)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{3}{5}\right),e\left(\frac{3}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 2475 }(196, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) |
sage: chi.jacobi_sum(n)