from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1081, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([0,28]))
pari: [g,chi] = znchar(Mod(24,1081))
Basic properties
| Modulus: | \(1081\) | |
| Conductor: | \(47\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(23\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{47}(24,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1081.i
\(\chi_{1081}(24,\cdot)\) \(\chi_{1081}(162,\cdot)\) \(\chi_{1081}(277,\cdot)\) \(\chi_{1081}(300,\cdot)\) \(\chi_{1081}(346,\cdot)\) \(\chi_{1081}(392,\cdot)\) \(\chi_{1081}(484,\cdot)\) \(\chi_{1081}(507,\cdot)\) \(\chi_{1081}(553,\cdot)\) \(\chi_{1081}(576,\cdot)\) \(\chi_{1081}(645,\cdot)\) \(\chi_{1081}(714,\cdot)\) \(\chi_{1081}(737,\cdot)\) \(\chi_{1081}(760,\cdot)\) \(\chi_{1081}(806,\cdot)\) \(\chi_{1081}(852,\cdot)\) \(\chi_{1081}(921,\cdot)\) \(\chi_{1081}(944,\cdot)\) \(\chi_{1081}(967,\cdot)\) \(\chi_{1081}(990,\cdot)\) \(\chi_{1081}(1036,\cdot)\) \(\chi_{1081}(1059,\cdot)\)
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: | \(\Q(\zeta_{23})\) |
| Fixed field: | Number field defined by a degree 23 polynomial |
Values on generators
\((189,898)\) → \((1,e\left(\frac{14}{23}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1081 }(24, a) \) | \(1\) | \(1\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) |
sage: chi.jacobi_sum(n)