from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(586971, base_ring=CyclotomicField(25410))
M = H._module
chi = DirichletCharacter(H, M([8470,19965,25221]))
pari: [g,chi] = znchar(Mod(13,586971))
Basic properties
| Modulus: | \(586971\) | |
| Conductor: | \(586971\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(25410\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 586971.sg
\(\chi_{586971}(13,\cdot)\) \(\chi_{586971}(139,\cdot)\) \(\chi_{586971}(160,\cdot)\) \(\chi_{586971}(349,\cdot)\) \(\chi_{586971}(580,\cdot)\) \(\chi_{586971}(601,\cdot)\) \(\chi_{586971}(706,\cdot)\) \(\chi_{586971}(853,\cdot)\) \(\chi_{586971}(1042,\cdot)\) \(\chi_{586971}(1084,\cdot)\) \(\chi_{586971}(1168,\cdot)\) \(\chi_{586971}(1294,\cdot)\) \(\chi_{586971}(1399,\cdot)\) \(\chi_{586971}(1525,\cdot)\)
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: | $\Q(\zeta_{12705})$ |
| Fixed field: | Number field defined by a degree 25410 polynomial (not computed) |
Values on generators
\((130439,179686,73207)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{11}{14}\right),e\left(\frac{1201}{1210}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 586971 }(13, a) \) | \(1\) | \(1\) | \(e\left(\frac{19171}{25410}\right)\) | \(e\left(\frac{6466}{12705}\right)\) | \(e\left(\frac{2129}{25410}\right)\) | \(e\left(\frac{2231}{8470}\right)\) | \(e\left(\frac{710}{847}\right)\) | \(e\left(\frac{8413}{12705}\right)\) | \(e\left(\frac{227}{12705}\right)\) | \(e\left(\frac{1949}{4235}\right)\) | \(e\left(\frac{314}{605}\right)\) | \(e\left(\frac{15061}{25410}\right)\) |
sage: chi.jacobi_sum(n)