from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(287, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,7]))
pari: [g,chi] = znchar(Mod(111,287))
Basic properties
| Modulus: | \(287\) | |
| Conductor: | \(287\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 287.bb
\(\chi_{287}(6,\cdot)\) \(\chi_{287}(13,\cdot)\) \(\chi_{287}(34,\cdot)\) \(\chi_{287}(48,\cdot)\) \(\chi_{287}(69,\cdot)\) \(\chi_{287}(76,\cdot)\) \(\chi_{287}(97,\cdot)\) \(\chi_{287}(104,\cdot)\) \(\chi_{287}(111,\cdot)\) \(\chi_{287}(153,\cdot)\) \(\chi_{287}(181,\cdot)\) \(\chi_{287}(188,\cdot)\) \(\chi_{287}(216,\cdot)\) \(\chi_{287}(258,\cdot)\) \(\chi_{287}(265,\cdot)\) \(\chi_{287}(272,\cdot)\)
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: | 40.40.63172957949423116502957480067191906200305068755882825968063357506461803384975161.1 |
Values on generators
\((206,211)\) → \((-1,e\left(\frac{7}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 287 }(111, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(i\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) |
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)