from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2000, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,0,24]))
pari: [g,chi] = znchar(Mod(31,2000))
Basic properties
| Modulus: | \(2000\) | |
| Conductor: | \(500\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{500}(31,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2000.bv
\(\chi_{2000}(31,\cdot)\) \(\chi_{2000}(111,\cdot)\) \(\chi_{2000}(191,\cdot)\) \(\chi_{2000}(271,\cdot)\) \(\chi_{2000}(431,\cdot)\) \(\chi_{2000}(511,\cdot)\) \(\chi_{2000}(591,\cdot)\) \(\chi_{2000}(671,\cdot)\) \(\chi_{2000}(831,\cdot)\) \(\chi_{2000}(911,\cdot)\) \(\chi_{2000}(991,\cdot)\) \(\chi_{2000}(1071,\cdot)\) \(\chi_{2000}(1231,\cdot)\) \(\chi_{2000}(1311,\cdot)\) \(\chi_{2000}(1391,\cdot)\) \(\chi_{2000}(1471,\cdot)\) \(\chi_{2000}(1631,\cdot)\) \(\chi_{2000}(1711,\cdot)\) \(\chi_{2000}(1791,\cdot)\) \(\chi_{2000}(1871,\cdot)\)
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
\((751,501,1377)\) → \((-1,1,e\left(\frac{12}{25}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2000 }(31, a) \) | \(-1\) | \(1\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{29}{50}\right)\) |
sage: chi.jacobi_sum(n)