from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(125, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([2]))
pari: [g,chi] = znchar(Mod(16,125))
Basic properties
| Modulus: | \(125\) | |
| Conductor: | \(125\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(25\) | 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 125.g
\(\chi_{125}(6,\cdot)\) \(\chi_{125}(11,\cdot)\) \(\chi_{125}(16,\cdot)\) \(\chi_{125}(21,\cdot)\) \(\chi_{125}(31,\cdot)\) \(\chi_{125}(36,\cdot)\) \(\chi_{125}(41,\cdot)\) \(\chi_{125}(46,\cdot)\) \(\chi_{125}(56,\cdot)\) \(\chi_{125}(61,\cdot)\) \(\chi_{125}(66,\cdot)\) \(\chi_{125}(71,\cdot)\) \(\chi_{125}(81,\cdot)\) \(\chi_{125}(86,\cdot)\) \(\chi_{125}(91,\cdot)\) \(\chi_{125}(96,\cdot)\) \(\chi_{125}(106,\cdot)\) \(\chi_{125}(111,\cdot)\) \(\chi_{125}(116,\cdot)\) \(\chi_{125}(121,\cdot)\)
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 25 polynomial |
Values on generators
\(2\) → \(e\left(\frac{1}{25}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 125 }(16, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)