from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6027, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,20,51]))
pari: [g,chi] = znchar(Mod(4489,6027))
Basic properties
| Modulus: | \(6027\) | |
| Conductor: | \(287\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{287}(184,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6027.dd
\(\chi_{6027}(226,\cdot)\) \(\chi_{6027}(361,\cdot)\) \(\chi_{6027}(1402,\cdot)\) \(\chi_{6027}(1537,\cdot)\) \(\chi_{6027}(1843,\cdot)\) \(\chi_{6027}(2137,\cdot)\) \(\chi_{6027}(2578,\cdot)\) \(\chi_{6027}(2872,\cdot)\) \(\chi_{6027}(3301,\cdot)\) \(\chi_{6027}(3313,\cdot)\) \(\chi_{6027}(4477,\cdot)\) \(\chi_{6027}(4489,\cdot)\) \(\chi_{6027}(4918,\cdot)\) \(\chi_{6027}(5212,\cdot)\) \(\chi_{6027}(5653,\cdot)\) \(\chi_{6027}(5947,\cdot)\)
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
\((4019,493,2794)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{17}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 6027 }(4489, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) |
sage: chi.jacobi_sum(n)