from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1000, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,25,31]))
pari: [g,chi] = znchar(Mod(29,1000))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1000.bd
\(\chi_{1000}(29,\cdot)\) \(\chi_{1000}(69,\cdot)\) \(\chi_{1000}(109,\cdot)\) \(\chi_{1000}(189,\cdot)\) \(\chi_{1000}(229,\cdot)\) \(\chi_{1000}(269,\cdot)\) \(\chi_{1000}(309,\cdot)\) \(\chi_{1000}(389,\cdot)\) \(\chi_{1000}(429,\cdot)\) \(\chi_{1000}(469,\cdot)\) \(\chi_{1000}(509,\cdot)\) \(\chi_{1000}(589,\cdot)\) \(\chi_{1000}(629,\cdot)\) \(\chi_{1000}(669,\cdot)\) \(\chi_{1000}(709,\cdot)\) \(\chi_{1000}(789,\cdot)\) \(\chi_{1000}(829,\cdot)\) \(\chi_{1000}(869,\cdot)\) \(\chi_{1000}(909,\cdot)\) \(\chi_{1000}(989,\cdot)\)
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
\((751,501,377)\) → \((1,-1,e\left(\frac{31}{50}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1000 }(29, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{13}{25}\right)\) |
sage: chi.jacobi_sum(n)