Basic properties
| sage: chi.conductor()
pari: znconreyconductor(g,chi)
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| Conductor | = | 4001 |
| sage: chi.multiplicative_order()
pari: charorder(g,chi)
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| Order | = | 2000 |
| 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)
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| Parity | = | Even |
| Orbit label | = | 4001.w |
| Orbit index | = | 23 |
Galois orbit
\(\chi_{4001}(9,\cdot)\) \(\chi_{4001}(18,\cdot)\) \(\chi_{4001}(29,\cdot)\) \(\chi_{4001}(31,\cdot)\) \(\chi_{4001}(33,\cdot)\) \(\chi_{4001}(36,\cdot)\) \(\chi_{4001}(45,\cdot)\) \(\chi_{4001}(47,\cdot)\) \(\chi_{4001}(61,\cdot)\) \(\chi_{4001}(62,\cdot)\) \(\chi_{4001}(66,\cdot)\) \(\chi_{4001}(72,\cdot)\) \(\chi_{4001}(83,\cdot)\) \(\chi_{4001}(89,\cdot)\) \(\chi_{4001}(90,\cdot)\) \(\chi_{4001}(94,\cdot)\) \(\chi_{4001}(97,\cdot)\) \(\chi_{4001}(111,\cdot)\) \(\chi_{4001}(116,\cdot)\) \(\chi_{4001}(117,\cdot)\) \(\chi_{4001}(118,\cdot)\) \(\chi_{4001}(122,\cdot)\) \(\chi_{4001}(124,\cdot)\) \(\chi_{4001}(126,\cdot)\) \(\chi_{4001}(137,\cdot)\) \(\chi_{4001}(138,\cdot)\) \(\chi_{4001}(145,\cdot)\) \(\chi_{4001}(149,\cdot)\) \(\chi_{4001}(155,\cdot)\) \(\chi_{4001}(165,\cdot)\) ...
Values on generators
\(3\) → \(e\left(\frac{1}{2000}\right)\)
Values
| -1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| \(1\) | \(1\) | \(e\left(\frac{23}{500}\right)\) | \(e\left(\frac{1}{2000}\right)\) | \(e\left(\frac{23}{250}\right)\) | \(e\left(\frac{59}{100}\right)\) | \(e\left(\frac{93}{2000}\right)\) | \(e\left(\frac{57}{500}\right)\) | \(e\left(\frac{69}{500}\right)\) | \(e\left(\frac{1}{1000}\right)\) | \(e\left(\frac{159}{250}\right)\) | \(e\left(\frac{57}{80}\right)\) |
Related number fields
| Field of values | \(\Q(\zeta_{2000})\) |