from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6010, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,22]))
pari: [g,chi] = znchar(Mod(619,6010))
Basic properties
| Modulus: | \(6010\) | |
| Conductor: | \(3005\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3005}(619,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6010.bo
\(\chi_{6010}(619,\cdot)\) \(\chi_{6010}(1369,\cdot)\) \(\chi_{6010}(2339,\cdot)\) \(\chi_{6010}(3329,\cdot)\) \(\chi_{6010}(3569,\cdot)\) \(\chi_{6010}(3849,\cdot)\) \(\chi_{6010}(4609,\cdot)\) \(\chi_{6010}(4959,\cdot)\)
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
\((3607,2411)\) → \((-1,e\left(\frac{11}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 6010 }(619, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)