from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1480, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,18,27,7]))
pari: [g,chi] = znchar(Mod(1053,1480))
Basic properties
| Modulus: | \(1480\) | |
| Conductor: | \(1480\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | 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 1480.ed
\(\chi_{1480}(13,\cdot)\) \(\chi_{1480}(93,\cdot)\) \(\chi_{1480}(133,\cdot)\) \(\chi_{1480}(237,\cdot)\) \(\chi_{1480}(277,\cdot)\) \(\chi_{1480}(357,\cdot)\) \(\chi_{1480}(557,\cdot)\) \(\chi_{1480}(797,\cdot)\) \(\chi_{1480}(893,\cdot)\) \(\chi_{1480}(957,\cdot)\) \(\chi_{1480}(1053,\cdot)\) \(\chi_{1480}(1293,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((1111,741,297,1001)\) → \((1,-1,-i,e\left(\frac{7}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1480 }(1053, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)