from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(100014, base_ring=CyclotomicField(2730))
M = H._module
chi = DirichletCharacter(H, M([1365,385,2236]))
pari: [g,chi] = znchar(Mod(503,100014))
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 \(\chi_{50007}(503,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 100014.lt
\(\chi_{100014}(47,\cdot)\) \(\chi_{100014}(53,\cdot)\) \(\chi_{100014}(503,\cdot)\) \(\chi_{100014}(695,\cdot)\) \(\chi_{100014}(983,\cdot)\) \(\chi_{100014}(1007,\cdot)\) \(\chi_{100014}(1061,\cdot)\) \(\chi_{100014}(1181,\cdot)\) \(\chi_{100014}(1259,\cdot)\) \(\chi_{100014}(1475,\cdot)\) \(\chi_{100014}(1481,\cdot)\) \(\chi_{100014}(1529,\cdot)\)
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
\((66677,1267,32707)\) → \((-1,e\left(\frac{11}{78}\right),e\left(\frac{86}{105}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 100014 }(503, a) \) | \(1\) | \(1\) | \(e\left(\frac{977}{2730}\right)\) | \(e\left(\frac{293}{910}\right)\) | \(e\left(\frac{2117}{2730}\right)\) | \(e\left(\frac{1007}{1365}\right)\) | \(e\left(\frac{617}{1365}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{977}{1365}\right)\) | \(e\left(\frac{902}{1365}\right)\) | \(e\left(\frac{8}{91}\right)\) |
sage: chi.jacobi_sum(n)