from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8030, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,6,10]))
pari: [g,chi] = znchar(Mod(81,8030))
Basic properties
| Modulus: | \(8030\) | |
| Conductor: | \(803\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(15\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{803}(81,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8030.cf
\(\chi_{8030}(81,\cdot)\) \(\chi_{8030}(2181,\cdot)\) \(\chi_{8030}(2271,\cdot)\) \(\chi_{8030}(3001,\cdot)\) \(\chi_{8030}(4371,\cdot)\) \(\chi_{8030}(5921,\cdot)\) \(\chi_{8030}(6561,\cdot)\) \(\chi_{8030}(7291,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | Number field defined by a degree 15 polynomial |
Values on generators
\((1607,2191,881)\) → \((1,e\left(\frac{1}{5}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8030 }(81, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) |
sage: chi.jacobi_sum(n)