from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1480, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,18,27,4]))
pari: [g,chi] = znchar(Mod(53,1480))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1480.ej
\(\chi_{1480}(53,\cdot)\) \(\chi_{1480}(157,\cdot)\) \(\chi_{1480}(197,\cdot)\) \(\chi_{1480}(293,\cdot)\) \(\chi_{1480}(453,\cdot)\) \(\chi_{1480}(477,\cdot)\) \(\chi_{1480}(493,\cdot)\) \(\chi_{1480}(773,\cdot)\) \(\chi_{1480}(1117,\cdot)\) \(\chi_{1480}(1237,\cdot)\) \(\chi_{1480}(1413,\cdot)\) \(\chi_{1480}(1477,\cdot)\)
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
\((1111,741,297,1001)\) → \((1,-1,-i,e\left(\frac{1}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1480 }(53, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) |
sage: chi.jacobi_sum(n)