from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4000, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,25,49]))
pari: [g,chi] = znchar(Mod(719,4000))
Basic properties
| Modulus: | \(4000\) | |
| Conductor: | \(1000\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1000}(219,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4000.cg
\(\chi_{4000}(79,\cdot)\) \(\chi_{4000}(239,\cdot)\) \(\chi_{4000}(559,\cdot)\) \(\chi_{4000}(719,\cdot)\) \(\chi_{4000}(879,\cdot)\) \(\chi_{4000}(1039,\cdot)\) \(\chi_{4000}(1359,\cdot)\) \(\chi_{4000}(1519,\cdot)\) \(\chi_{4000}(1679,\cdot)\) \(\chi_{4000}(1839,\cdot)\) \(\chi_{4000}(2159,\cdot)\) \(\chi_{4000}(2319,\cdot)\) \(\chi_{4000}(2479,\cdot)\) \(\chi_{4000}(2639,\cdot)\) \(\chi_{4000}(2959,\cdot)\) \(\chi_{4000}(3119,\cdot)\) \(\chi_{4000}(3279,\cdot)\) \(\chi_{4000}(3439,\cdot)\) \(\chi_{4000}(3759,\cdot)\) \(\chi_{4000}(3919,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
| Fixed field: | Number field defined by a degree 50 polynomial |
Values on generators
\((2751,2501,1377)\) → \((-1,-1,e\left(\frac{49}{50}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 4000 }(719, a) \) | \(-1\) | \(1\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{29}{50}\right)\) |
sage: chi.jacobi_sum(n)