sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(75)
sage: chi = H[22]
pari: [g,chi] = znchar(Mod(22,75))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 25 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 20 |
| Real | = | no |
| sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
| ||
| Primitive | = | no |
| Minimal | = | yes |
| sage: chi.is_odd()
pari: zncharisodd(g,chi)
| ||
| Parity | = | odd |
| Orbit label | = | 75.k |
| Orbit index | = | 11 |
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 ]
\(\chi_{75}(13,\cdot)\) \(\chi_{75}(22,\cdot)\) \(\chi_{75}(28,\cdot)\) \(\chi_{75}(37,\cdot)\) \(\chi_{75}(52,\cdot)\) \(\chi_{75}(58,\cdot)\) \(\chi_{75}(67,\cdot)\) \(\chi_{75}(73,\cdot)\)
Values on generators
\((26,52)\) → \((1,e\left(\frac{17}{20}\right))\)
Values
| -1 | 1 | 2 | 4 | 7 | 8 | 11 | 13 | 14 | 16 | 17 | 19 |
| \(-1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{20})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{75}(22,\cdot)) = \sum_{r\in \Z/75\Z} \chi_{75}(22,r) e\left(\frac{2r}{75}\right) = -3.1871199487+-3.8525662139i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{75}(22,\cdot),\chi_{75}(1,\cdot)) = \sum_{r\in \Z/75\Z} \chi_{75}(22,r) \chi_{75}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{75}(22,·))
= \sum_{r \in \Z/75\Z}
\chi_{75}(22,r) e\left(\frac{1 r + 2 r^{-1}}{75}\right)
= -0.0 \)