sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(4033)
sage: chi = H[3]
pari: [g,chi] = znchar(Mod(3,4033))
Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
| Conductor | = | 4033 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
| ||
| Order | = | 54 |
| 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 | = | 4033.hw |
| Orbit index | = | 205 |
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 ]
\(\chi_{4033}(3,\cdot)\) \(\chi_{4033}(243,\cdot)\) \(\chi_{4033}(798,\cdot)\) \(\chi_{4033}(844,\cdot)\) \(\chi_{4033}(990,\cdot)\) \(\chi_{4033}(1029,\cdot)\) \(\chi_{4033}(1288,\cdot)\) \(\chi_{4033}(1575,\cdot)\) \(\chi_{4033}(2076,\cdot)\) \(\chi_{4033}(2187,\cdot)\) \(\chi_{4033}(2260,\cdot)\) \(\chi_{4033}(2689,\cdot)\) \(\chi_{4033}(2803,\cdot)\) \(\chi_{4033}(3149,\cdot)\) \(\chi_{4033}(3296,\cdot)\) \(\chi_{4033}(3503,\cdot)\) \(\chi_{4033}(3728,\cdot)\) \(\chi_{4033}(3836,\cdot)\)
Values on generators
\((1963,2295)\) → \((e\left(\frac{13}{18}\right),e\left(\frac{13}{27}\right))\)
Values
| -1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(-1\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{27})\) |