sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(1800)
sage: chi = H[1709]
pari: [g,chi] = znchar(Mod(1709,1800))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 600 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 10 |
| Real | = | No |
| sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
| ||
| Primitive | = | No |
| sage: chi.is_odd()
pari: zncharisodd(g,chi)
| ||
| Parity | = | Odd |
| Orbit label | = | 1800.bp |
| Orbit index | = | 42 |
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 ]
\(\chi_{1800}(269,\cdot)\) \(\chi_{1800}(629,\cdot)\) \(\chi_{1800}(989,\cdot)\) \(\chi_{1800}(1709,\cdot)\)
Inducing primitive character
Values on generators
\((1351,901,1001,577)\) → \((1,-1,-1,e\left(\frac{7}{10}\right))\)
Values
| -1 | 1 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 |
| \(-1\) | \(1\) | \(-1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{5})\) |