from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1530, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([8,24,39]))
pari: [g,chi] = znchar(Mod(29,1530))
Basic properties
| Modulus: | \(1530\) | |
| Conductor: | \(765\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{765}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1530.co
\(\chi_{1530}(29,\cdot)\) \(\chi_{1530}(209,\cdot)\) \(\chi_{1530}(299,\cdot)\) \(\chi_{1530}(329,\cdot)\) \(\chi_{1530}(419,\cdot)\) \(\chi_{1530}(479,\cdot)\) \(\chi_{1530}(779,\cdot)\) \(\chi_{1530}(839,\cdot)\) \(\chi_{1530}(929,\cdot)\) \(\chi_{1530}(959,\cdot)\) \(\chi_{1530}(1049,\cdot)\) \(\chi_{1530}(1229,\cdot)\) \(\chi_{1530}(1289,\cdot)\) \(\chi_{1530}(1319,\cdot)\) \(\chi_{1530}(1469,\cdot)\) \(\chi_{1530}(1499,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((1361,307,1261)\) → \((e\left(\frac{1}{6}\right),-1,e\left(\frac{13}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1530 }(29, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) |
sage: chi.jacobi_sum(n)