from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8030, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,24,55]))
pari: [g,chi] = znchar(Mod(49,8030))
Basic properties
| Modulus: | \(8030\) | |
| Conductor: | \(4015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{4015}(49,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8030.ej
\(\chi_{8030}(49,\cdot)\) \(\chi_{8030}(289,\cdot)\) \(\chi_{8030}(389,\cdot)\) \(\chi_{8030}(779,\cdot)\) \(\chi_{8030}(1609,\cdot)\) \(\chi_{8030}(2479,\cdot)\) \(\chi_{8030}(2579,\cdot)\) \(\chi_{8030}(3309,\cdot)\) \(\chi_{8030}(3699,\cdot)\) \(\chi_{8030}(3799,\cdot)\) \(\chi_{8030}(4669,\cdot)\) \(\chi_{8030}(5399,\cdot)\) \(\chi_{8030}(5889,\cdot)\) \(\chi_{8030}(5989,\cdot)\) \(\chi_{8030}(6229,\cdot)\) \(\chi_{8030}(6719,\cdot)\)
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
\((1607,2191,881)\) → \((-1,e\left(\frac{2}{5}\right),e\left(\frac{11}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8030 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(-i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{53}{60}\right)\) |
sage: chi.jacobi_sum(n)