from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9240, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,15,0,15,25,3]))
pari: [g,chi] = znchar(Mod(1069,9240))
Basic properties
| Modulus: | \(9240\) | |
| Conductor: | \(3080\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3080}(1069,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9240.li
\(\chi_{9240}(1069,\cdot)\) \(\chi_{9240}(1669,\cdot)\) \(\chi_{9240}(1909,\cdot)\) \(\chi_{9240}(4429,\cdot)\) \(\chi_{9240}(5029,\cdot)\) \(\chi_{9240}(5869,\cdot)\) \(\chi_{9240}(6949,\cdot)\) \(\chi_{9240}(8389,\cdot)\)
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 30 polynomial |
Values on generators
\((2311,4621,6161,3697,5281,2521)\) → \((1,-1,1,-1,e\left(\frac{5}{6}\right),e\left(\frac{1}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 9240 }(1069, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(-1\) | \(e\left(\frac{7}{15}\right)\) |
sage: chi.jacobi_sum(n)