from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2013, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,36,59]))
pari: [g,chi] = znchar(Mod(31,2013))
Basic properties
| Modulus: | \(2013\) | |
| Conductor: | \(671\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{671}(31,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2013.gj
\(\chi_{2013}(31,\cdot)\) \(\chi_{2013}(148,\cdot)\) \(\chi_{2013}(181,\cdot)\) \(\chi_{2013}(322,\cdot)\) \(\chi_{2013}(421,\cdot)\) \(\chi_{2013}(823,\cdot)\) \(\chi_{2013}(922,\cdot)\) \(\chi_{2013}(982,\cdot)\) \(\chi_{2013}(994,\cdot)\) \(\chi_{2013}(1027,\cdot)\) \(\chi_{2013}(1039,\cdot)\) \(\chi_{2013}(1072,\cdot)\) \(\chi_{2013}(1081,\cdot)\) \(\chi_{2013}(1291,\cdot)\) \(\chi_{2013}(1324,\cdot)\) \(\chi_{2013}(1945,\cdot)\)
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
\((1343,1465,1222)\) → \((1,e\left(\frac{3}{5}\right),e\left(\frac{59}{60}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 2013 }(31, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(-i\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{37}{60}\right)\) |
sage: chi.jacobi_sum(n)