from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(586971, base_ring=CyclotomicField(8470))
M = H._module
chi = DirichletCharacter(H, M([4235,7260,21]))
chi.galois_orbit()
[g,chi] = znchar(Mod(8,586971))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(586971\) | |
| Conductor: | \(195657\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8470\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 195657.gz | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{4235})$ |
| Fixed field: | Number field defined by a degree 8470 polynomial (not computed) |
First 9 of 2640 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{586971}(8,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3338}{4235}\right)\) | \(e\left(\frac{2441}{4235}\right)\) | \(e\left(\frac{6889}{8470}\right)\) | \(e\left(\frac{1544}{4235}\right)\) | \(e\left(\frac{1019}{1694}\right)\) | \(e\left(\frac{2231}{8470}\right)\) | \(e\left(\frac{647}{4235}\right)\) | \(e\left(\frac{1367}{4235}\right)\) | \(e\left(\frac{799}{1210}\right)\) | \(e\left(\frac{3301}{8470}\right)\) |
| \(\chi_{586971}(134,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3516}{4235}\right)\) | \(e\left(\frac{2797}{4235}\right)\) | \(e\left(\frac{5843}{8470}\right)\) | \(e\left(\frac{2078}{4235}\right)\) | \(e\left(\frac{881}{1694}\right)\) | \(e\left(\frac{2657}{8470}\right)\) | \(e\left(\frac{1359}{4235}\right)\) | \(e\left(\frac{24}{4235}\right)\) | \(e\left(\frac{573}{1210}\right)\) | \(e\left(\frac{2967}{8470}\right)\) |
| \(\chi_{586971}(260,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{922}{4235}\right)\) | \(e\left(\frac{1844}{4235}\right)\) | \(e\left(\frac{1101}{8470}\right)\) | \(e\left(\frac{2766}{4235}\right)\) | \(e\left(\frac{589}{1694}\right)\) | \(e\left(\frac{2159}{8470}\right)\) | \(e\left(\frac{3688}{4235}\right)\) | \(e\left(\frac{2608}{4235}\right)\) | \(e\left(\frac{1161}{1210}\right)\) | \(e\left(\frac{4789}{8470}\right)\) |
| \(\chi_{586971}(512,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1544}{4235}\right)\) | \(e\left(\frac{3088}{4235}\right)\) | \(e\left(\frac{3727}{8470}\right)\) | \(e\left(\frac{397}{4235}\right)\) | \(e\left(\frac{1363}{1694}\right)\) | \(e\left(\frac{6693}{8470}\right)\) | \(e\left(\frac{1941}{4235}\right)\) | \(e\left(\frac{4101}{4235}\right)\) | \(e\left(\frac{1187}{1210}\right)\) | \(e\left(\frac{1433}{8470}\right)\) |
| \(\chi_{586971}(701,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1013}{4235}\right)\) | \(e\left(\frac{2026}{4235}\right)\) | \(e\left(\frac{709}{8470}\right)\) | \(e\left(\frac{3039}{4235}\right)\) | \(e\left(\frac{547}{1694}\right)\) | \(e\left(\frac{521}{8470}\right)\) | \(e\left(\frac{4052}{4235}\right)\) | \(e\left(\frac{137}{4235}\right)\) | \(e\left(\frac{19}{1210}\right)\) | \(e\left(\frac{4761}{8470}\right)\) |
| \(\chi_{586971}(827,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{981}{4235}\right)\) | \(e\left(\frac{1962}{4235}\right)\) | \(e\left(\frac{7083}{8470}\right)\) | \(e\left(\frac{2943}{4235}\right)\) | \(e\left(\frac{115}{1694}\right)\) | \(e\left(\frac{4727}{8470}\right)\) | \(e\left(\frac{3924}{4235}\right)\) | \(e\left(\frac{3519}{4235}\right)\) | \(e\left(\frac{753}{1210}\right)\) | \(e\left(\frac{2537}{8470}\right)\) |
| \(\chi_{586971}(953,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3812}{4235}\right)\) | \(e\left(\frac{3389}{4235}\right)\) | \(e\left(\frac{8291}{8470}\right)\) | \(e\left(\frac{2966}{4235}\right)\) | \(e\left(\frac{1489}{1694}\right)\) | \(e\left(\frac{8219}{8470}\right)\) | \(e\left(\frac{2543}{4235}\right)\) | \(e\left(\frac{503}{4235}\right)\) | \(e\left(\frac{741}{1210}\right)\) | \(e\left(\frac{6599}{8470}\right)\) |
| \(\chi_{586971}(1205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2789}{4235}\right)\) | \(e\left(\frac{1343}{4235}\right)\) | \(e\left(\frac{6927}{8470}\right)\) | \(e\left(\frac{4132}{4235}\right)\) | \(e\left(\frac{807}{1694}\right)\) | \(e\left(\frac{13}{8470}\right)\) | \(e\left(\frac{2686}{4235}\right)\) | \(e\left(\frac{3011}{4235}\right)\) | \(e\left(\frac{1027}{1210}\right)\) | \(e\left(\frac{1143}{8470}\right)\) |
| \(\chi_{586971}(1394,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3693}{4235}\right)\) | \(e\left(\frac{3151}{4235}\right)\) | \(e\left(\frac{6849}{8470}\right)\) | \(e\left(\frac{2609}{4235}\right)\) | \(e\left(\frac{1153}{1694}\right)\) | \(e\left(\frac{1891}{8470}\right)\) | \(e\left(\frac{2067}{4235}\right)\) | \(e\left(\frac{2757}{4235}\right)\) | \(e\left(\frac{559}{1210}\right)\) | \(e\left(\frac{4681}{8470}\right)\) |