from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,57,48]))
pari: [g,chi] = znchar(Mod(2038,2475))
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.gi
\(\chi_{2475}(202,\cdot)\) \(\chi_{2475}(247,\cdot)\) \(\chi_{2475}(592,\cdot)\) \(\chi_{2475}(598,\cdot)\) \(\chi_{2475}(1213,\cdot)\) \(\chi_{2475}(1303,\cdot)\) \(\chi_{2475}(1312,\cdot)\) \(\chi_{2475}(1417,\cdot)\) \(\chi_{2475}(1483,\cdot)\) \(\chi_{2475}(1852,\cdot)\) \(\chi_{2475}(1897,\cdot)\) \(\chi_{2475}(2038,\cdot)\) \(\chi_{2475}(2128,\cdot)\) \(\chi_{2475}(2137,\cdot)\) \(\chi_{2475}(2248,\cdot)\) \(\chi_{2475}(2308,\cdot)\)
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
\((551,2377,2026)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{19}{20}\right),e\left(\frac{4}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
\( \chi_{ 2475 }(2038, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(i\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(-1\) | \(e\left(\frac{7}{60}\right)\) |
sage: chi.jacobi_sum(n)