from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1476, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,40,39]))
pari: [g,chi] = znchar(Mod(43,1476))
Basic properties
| Modulus: | \(1476\) | |
| Conductor: | \(1476\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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 1476.ce
\(\chi_{1476}(43,\cdot)\) \(\chi_{1476}(103,\cdot)\) \(\chi_{1476}(115,\cdot)\) \(\chi_{1476}(295,\cdot)\) \(\chi_{1476}(367,\cdot)\) \(\chi_{1476}(535,\cdot)\) \(\chi_{1476}(607,\cdot)\) \(\chi_{1476}(787,\cdot)\) \(\chi_{1476}(799,\cdot)\) \(\chi_{1476}(859,\cdot)\) \(\chi_{1476}(907,\cdot)\) \(\chi_{1476}(979,\cdot)\) \(\chi_{1476}(1087,\cdot)\) \(\chi_{1476}(1291,\cdot)\) \(\chi_{1476}(1399,\cdot)\) \(\chi_{1476}(1471,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((739,821,1441)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{13}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1476 }(43, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) |
sage: chi.jacobi_sum(n)