Basic properties
| Modulus: | \(689\) | |
| Conductor: | \(689\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(39\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 689.y
\(\chi_{689}(16,\cdot)\) \(\chi_{689}(42,\cdot)\) \(\chi_{689}(68,\cdot)\) \(\chi_{689}(81,\cdot)\) \(\chi_{689}(100,\cdot)\) \(\chi_{689}(152,\cdot)\) \(\chi_{689}(172,\cdot)\) \(\chi_{689}(256,\cdot)\) \(\chi_{689}(289,\cdot)\) \(\chi_{689}(328,\cdot)\) \(\chi_{689}(334,\cdot)\) \(\chi_{689}(354,\cdot)\) \(\chi_{689}(360,\cdot)\) \(\chi_{689}(367,\cdot)\) \(\chi_{689}(386,\cdot)\) \(\chi_{689}(399,\cdot)\) \(\chi_{689}(471,\cdot)\) \(\chi_{689}(490,\cdot)\) \(\chi_{689}(523,\cdot)\) \(\chi_{689}(607,\cdot)\) \(\chi_{689}(627,\cdot)\) \(\chi_{689}(646,\cdot)\) \(\chi_{689}(672,\cdot)\) \(\chi_{689}(685,\cdot)\)
Related number fields
| Field of values: | $\Q(\zeta_{39})$ |
| Fixed field: | Number field defined by a degree 39 polynomial |
Values on generators
\((54,638)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{10}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 689 }(523, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) |