from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1000, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,25,24]))
pari: [g,chi] = znchar(Mod(531,1000))
Basic properties
Modulus: | \(1000\) | |
Conductor: | \(1000\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(50\) | 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 1000.bc
\(\chi_{1000}(11,\cdot)\) \(\chi_{1000}(91,\cdot)\) \(\chi_{1000}(131,\cdot)\) \(\chi_{1000}(171,\cdot)\) \(\chi_{1000}(211,\cdot)\) \(\chi_{1000}(291,\cdot)\) \(\chi_{1000}(331,\cdot)\) \(\chi_{1000}(371,\cdot)\) \(\chi_{1000}(411,\cdot)\) \(\chi_{1000}(491,\cdot)\) \(\chi_{1000}(531,\cdot)\) \(\chi_{1000}(571,\cdot)\) \(\chi_{1000}(611,\cdot)\) \(\chi_{1000}(691,\cdot)\) \(\chi_{1000}(731,\cdot)\) \(\chi_{1000}(771,\cdot)\) \(\chi_{1000}(811,\cdot)\) \(\chi_{1000}(891,\cdot)\) \(\chi_{1000}(931,\cdot)\) \(\chi_{1000}(971,\cdot)\)
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
\((751,501,377)\) → \((-1,-1,e\left(\frac{12}{25}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 1000 }(531, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) |
sage: chi.jacobi_sum(n)