sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(403)
sage: chi = H[4]
pari: [g,chi] = znchar(Mod(4,403))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 403 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 30 |
| Real | = | No |
| sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
| ||
| Primitive | = | Yes |
| sage: chi.is_odd()
pari: zncharisodd(g,chi)
| ||
| Parity | = | Even |
| Orbit label | = | 403.bs |
| Orbit index | = | 45 |
Galois orbit
sage: chi.sage_character().galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\(\chi_{403}(4,\cdot)\) \(\chi_{403}(95,\cdot)\) \(\chi_{403}(101,\cdot)\) \(\chi_{403}(140,\cdot)\) \(\chi_{403}(225,\cdot)\) \(\chi_{403}(264,\cdot)\) \(\chi_{403}(283,\cdot)\) \(\chi_{403}(374,\cdot)\)
Values on generators
\((249,313)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{3}{5}\right))\)
Values
| -1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| \(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{15})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{403}(4,\cdot)) = \sum_{r\in \Z/403\Z} \chi_{403}(4,r) e\left(\frac{2r}{403}\right) = -18.5369941595+-7.7058320467i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{403}(4,\cdot),\chi_{403}(1,\cdot)) = \sum_{r\in \Z/403\Z} \chi_{403}(4,r) \chi_{403}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{403}(4,·))
= \sum_{r \in \Z/403\Z}
\chi_{403}(4,r) e\left(\frac{1 r + 2 r^{-1}}{403}\right)
= 1.7283499823+-8.1312473662i \)