from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,27,6]))
pari: [g,chi] = znchar(Mod(112,2475))
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.gg
\(\chi_{2475}(13,\cdot)\) \(\chi_{2475}(112,\cdot)\) \(\chi_{2475}(547,\cdot)\) \(\chi_{2475}(733,\cdot)\) \(\chi_{2475}(778,\cdot)\) \(\chi_{2475}(952,\cdot)\) \(\chi_{2475}(1348,\cdot)\) \(\chi_{2475}(1372,\cdot)\) \(\chi_{2475}(1492,\cdot)\) \(\chi_{2475}(1663,\cdot)\) \(\chi_{2475}(1762,\cdot)\) \(\chi_{2475}(1777,\cdot)\) \(\chi_{2475}(2173,\cdot)\) \(\chi_{2475}(2317,\cdot)\) \(\chi_{2475}(2383,\cdot)\) \(\chi_{2475}(2428,\cdot)\)
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
\((551,2377,2026)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{9}{20}\right),e\left(\frac{1}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 2475 }(112, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(-i\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{37}{60}\right)\) |
sage: chi.jacobi_sum(n)