from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8550, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,21,40]))
pari: [g,chi] = znchar(Mod(353,8550))
Basic properties
| Modulus: | \(8550\) | |
| Conductor: | \(4275\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{4275}(353,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8550.fp
\(\chi_{8550}(353,\cdot)\) \(\chi_{8550}(653,\cdot)\) \(\chi_{8550}(1037,\cdot)\) \(\chi_{8550}(1337,\cdot)\) \(\chi_{8550}(2063,\cdot)\) \(\chi_{8550}(2363,\cdot)\) \(\chi_{8550}(2747,\cdot)\) \(\chi_{8550}(3047,\cdot)\) \(\chi_{8550}(3773,\cdot)\) \(\chi_{8550}(4073,\cdot)\) \(\chi_{8550}(5483,\cdot)\) \(\chi_{8550}(5783,\cdot)\) \(\chi_{8550}(6167,\cdot)\) \(\chi_{8550}(6467,\cdot)\) \(\chi_{8550}(7877,\cdot)\) \(\chi_{8550}(8177,\cdot)\)
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
\((1901,1027,1351)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{7}{20}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8550 }(353, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)