from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(187, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([12,35]))
pari: [g,chi] = znchar(Mod(19,187))
Basic properties
| Modulus: | \(187\) | |
| Conductor: | \(187\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | 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 187.q
\(\chi_{187}(2,\cdot)\) \(\chi_{187}(8,\cdot)\) \(\chi_{187}(19,\cdot)\) \(\chi_{187}(83,\cdot)\) \(\chi_{187}(94,\cdot)\) \(\chi_{187}(117,\cdot)\) \(\chi_{187}(127,\cdot)\) \(\chi_{187}(128,\cdot)\) \(\chi_{187}(134,\cdot)\) \(\chi_{187}(138,\cdot)\) \(\chi_{187}(145,\cdot)\) \(\chi_{187}(151,\cdot)\) \(\chi_{187}(161,\cdot)\) \(\chi_{187}(162,\cdot)\) \(\chi_{187}(172,\cdot)\) \(\chi_{187}(178,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.0.359624204259227998212313764863527746816862563620018205460931204658277030572367073.1 |
Values on generators
\((35,122)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{7}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 187 }(19, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) |
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)