from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,40,8]))
pari: [g,chi] = znchar(Mod(763,2015))
Basic properties
| Modulus: | \(2015\) | |
| Conductor: | \(2015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2015.gq
\(\chi_{2015}(133,\cdot)\) \(\chi_{2015}(152,\cdot)\) \(\chi_{2015}(237,\cdot)\) \(\chi_{2015}(328,\cdot)\) \(\chi_{2015}(503,\cdot)\) \(\chi_{2015}(692,\cdot)\) \(\chi_{2015}(763,\cdot)\) \(\chi_{2015}(958,\cdot)\) \(\chi_{2015}(1043,\cdot)\) \(\chi_{2015}(1192,\cdot)\) \(\chi_{2015}(1342,\cdot)\) \(\chi_{2015}(1498,\cdot)\) \(\chi_{2015}(1537,\cdot)\) \(\chi_{2015}(1712,\cdot)\) \(\chi_{2015}(1972,\cdot)\) \(\chi_{2015}(1998,\cdot)\)
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
\((807,1861,716)\) → \((-i,e\left(\frac{2}{3}\right),e\left(\frac{2}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 2015 }(763, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) |
sage: chi.jacobi_sum(n)