from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(151, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([31]))
pari: [g,chi] = znchar(Mod(27,151))
Basic properties
| Modulus: | \(151\) | |
| Conductor: | \(151\) | 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 151.j
\(\chi_{151}(3,\cdot)\) \(\chi_{151}(24,\cdot)\) \(\chi_{151}(26,\cdot)\) \(\chi_{151}(27,\cdot)\) \(\chi_{151}(28,\cdot)\) \(\chi_{151}(41,\cdot)\) \(\chi_{151}(53,\cdot)\) \(\chi_{151}(57,\cdot)\) \(\chi_{151}(60,\cdot)\) \(\chi_{151}(65,\cdot)\) \(\chi_{151}(67,\cdot)\) \(\chi_{151}(70,\cdot)\) \(\chi_{151}(73,\cdot)\) \(\chi_{151}(79,\cdot)\) \(\chi_{151}(83,\cdot)\) \(\chi_{151}(101,\cdot)\) \(\chi_{151}(107,\cdot)\) \(\chi_{151}(122,\cdot)\) \(\chi_{151}(131,\cdot)\) \(\chi_{151}(142,\cdot)\)
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
\(6\) → \(e\left(\frac{31}{50}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 151 }(27, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) |
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)