from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,35,14]))
pari: [g,chi] = znchar(Mod(141,2015))
Basic properties
| Modulus: | \(2015\) | |
| Conductor: | \(403\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{403}(141,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2015.gi
\(\chi_{2015}(141,\cdot)\) \(\chi_{2015}(176,\cdot)\) \(\chi_{2015}(331,\cdot)\) \(\chi_{2015}(561,\cdot)\) \(\chi_{2015}(631,\cdot)\) \(\chi_{2015}(756,\cdot)\) \(\chi_{2015}(786,\cdot)\) \(\chi_{2015}(1016,\cdot)\) \(\chi_{2015}(1181,\cdot)\) \(\chi_{2015}(1346,\cdot)\) \(\chi_{2015}(1376,\cdot)\) \(\chi_{2015}(1501,\cdot)\) \(\chi_{2015}(1536,\cdot)\) \(\chi_{2015}(1541,\cdot)\) \(\chi_{2015}(1636,\cdot)\) \(\chi_{2015}(1696,\cdot)\)
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
\((807,1861,716)\) → \((1,e\left(\frac{7}{12}\right),e\left(\frac{7}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 2015 }(141, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(-i\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) |
sage: chi.jacobi_sum(n)