from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(100014, base_ring=CyclotomicField(2730))
M = H._module
chi = DirichletCharacter(H, M([1365,35,13]))
pari: [g,chi] = znchar(Mod(635,100014))
Basic properties
| Modulus: | \(100014\) | |
| Conductor: | \(50007\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2730\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{50007}(635,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 100014.lq
\(\chi_{100014}(29,\cdot)\) \(\chi_{100014}(35,\cdot)\) \(\chi_{100014}(149,\cdot)\) \(\chi_{100014}(323,\cdot)\) \(\chi_{100014}(353,\cdot)\) \(\chi_{100014}(425,\cdot)\) \(\chi_{100014}(461,\cdot)\) \(\chi_{100014}(587,\cdot)\) \(\chi_{100014}(635,\cdot)\) \(\chi_{100014}(935,\cdot)\) \(\chi_{100014}(977,\cdot)\) \(\chi_{100014}(1301,\cdot)\) \(\chi_{100014}(1397,\cdot)\) \(\chi_{100014}(1499,\cdot)\)
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: | $\Q(\zeta_{1365})$ |
| Fixed field: | Number field defined by a degree 2730 polynomial (not computed) |
Values on generators
\((66677,1267,32707)\) → \((-1,e\left(\frac{1}{78}\right),e\left(\frac{1}{210}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 100014 }(635, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2521}{2730}\right)\) | \(e\left(\frac{466}{1365}\right)\) | \(e\left(\frac{391}{2730}\right)\) | \(e\left(\frac{166}{1365}\right)\) | \(e\left(\frac{1957}{2730}\right)\) | \(e\left(\frac{28}{195}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1156}{1365}\right)\) | \(e\left(\frac{449}{910}\right)\) | \(e\left(\frac{145}{546}\right)\) |
sage: chi.jacobi_sum(n)