sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(8041)
sage: chi = H[123]
pari: [g,chi] = znchar(Mod(123,8041))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 8041 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 60 |
| 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 | = | 8041.dn |
| Orbit index | = | 92 |
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 ]
\(\chi_{8041}(123,\cdot)\) \(\chi_{8041}(480,\cdot)\) \(\chi_{8041}(523,\cdot)\) \(\chi_{8041}(854,\cdot)\) \(\chi_{8041}(897,\cdot)\) \(\chi_{8041}(1942,\cdot)\) \(\chi_{8041}(2316,\cdot)\) \(\chi_{8041}(5640,\cdot)\) \(\chi_{8041}(6014,\cdot)\) \(\chi_{8041}(6371,\cdot)\) \(\chi_{8041}(6745,\cdot)\) \(\chi_{8041}(7059,\cdot)\) \(\chi_{8041}(7102,\cdot)\) \(\chi_{8041}(7433,\cdot)\) \(\chi_{8041}(7476,\cdot)\) \(\chi_{8041}(7790,\cdot)\)
Values on generators
\((6580,2366,562)\) → \((e\left(\frac{1}{10}\right),-i,e\left(\frac{1}{6}\right))\)
Values
| -1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12 |
| \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{60})\) |