from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6010, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,37]))
pari: [g,chi] = znchar(Mod(289,6010))
Basic properties
| Modulus: | \(6010\) | |
| Conductor: | \(3005\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3005}(289,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6010.by
\(\chi_{6010}(289,\cdot)\) \(\chi_{6010}(969,\cdot)\) \(\chi_{6010}(1019,\cdot)\) \(\chi_{6010}(1559,\cdot)\) \(\chi_{6010}(1569,\cdot)\) \(\chi_{6010}(1649,\cdot)\) \(\chi_{6010}(2079,\cdot)\) \(\chi_{6010}(2729,\cdot)\) \(\chi_{6010}(3159,\cdot)\) \(\chi_{6010}(3239,\cdot)\) \(\chi_{6010}(3249,\cdot)\) \(\chi_{6010}(3789,\cdot)\) \(\chi_{6010}(3839,\cdot)\) \(\chi_{6010}(4519,\cdot)\) \(\chi_{6010}(5249,\cdot)\) \(\chi_{6010}(5569,\cdot)\)
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
\((3607,2411)\) → \((-1,e\left(\frac{37}{60}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 6010 }(289, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) |
sage: chi.jacobi_sum(n)