from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(243675, base_ring=CyclotomicField(3420))
M = H._module
chi = DirichletCharacter(H, M([1710,1197,830]))
chi.galois_orbit()
[g,chi] = znchar(Mod(53,243675))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(243675\) | |
| Conductor: | \(27075\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(3420\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 27075.fn | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{3420})$ |
| Fixed field: | Number field defined by a degree 3420 polynomial (not computed) |
First 4 of 864 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{243675}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{317}{3420}\right)\) | \(e\left(\frac{317}{1710}\right)\) | \(e\left(\frac{35}{228}\right)\) | \(e\left(\frac{317}{1140}\right)\) | \(e\left(\frac{487}{570}\right)\) | \(e\left(\frac{1693}{3420}\right)\) | \(e\left(\frac{421}{1710}\right)\) | \(e\left(\frac{317}{855}\right)\) | \(e\left(\frac{3071}{3420}\right)\) | \(e\left(\frac{3239}{3420}\right)\) |
| \(\chi_{243675}(242,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2563}{3420}\right)\) | \(e\left(\frac{853}{1710}\right)\) | \(e\left(\frac{37}{228}\right)\) | \(e\left(\frac{283}{1140}\right)\) | \(e\left(\frac{23}{570}\right)\) | \(e\left(\frac{1907}{3420}\right)\) | \(e\left(\frac{1559}{1710}\right)\) | \(e\left(\frac{853}{855}\right)\) | \(e\left(\frac{1429}{3420}\right)\) | \(e\left(\frac{2701}{3420}\right)\) |
| \(\chi_{243675}(458,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2953}{3420}\right)\) | \(e\left(\frac{1243}{1710}\right)\) | \(e\left(\frac{175}{228}\right)\) | \(e\left(\frac{673}{1140}\right)\) | \(e\left(\frac{383}{570}\right)\) | \(e\left(\frac{257}{3420}\right)\) | \(e\left(\frac{1079}{1710}\right)\) | \(e\left(\frac{388}{855}\right)\) | \(e\left(\frac{2359}{3420}\right)\) | \(e\left(\frac{1831}{3420}\right)\) |
| \(\chi_{243675}(998,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3041}{3420}\right)\) | \(e\left(\frac{1331}{1710}\right)\) | \(e\left(\frac{143}{228}\right)\) | \(e\left(\frac{761}{1140}\right)\) | \(e\left(\frac{511}{570}\right)\) | \(e\left(\frac{1849}{3420}\right)\) | \(e\left(\frac{883}{1710}\right)\) | \(e\left(\frac{476}{855}\right)\) | \(e\left(\frac{1043}{3420}\right)\) | \(e\left(\frac{2687}{3420}\right)\) |