from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4508, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,8,0]))
pari: [g,chi] = znchar(Mod(93,4508))
Basic properties
| Modulus: | \(4508\) | |
| Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{49}(44,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4508.z
\(\chi_{4508}(93,\cdot)\) \(\chi_{4508}(277,\cdot)\) \(\chi_{4508}(737,\cdot)\) \(\chi_{4508}(921,\cdot)\) \(\chi_{4508}(1381,\cdot)\) \(\chi_{4508}(1565,\cdot)\) \(\chi_{4508}(2025,\cdot)\) \(\chi_{4508}(2209,\cdot)\) \(\chi_{4508}(2669,\cdot)\) \(\chi_{4508}(2853,\cdot)\) \(\chi_{4508}(3957,\cdot)\) \(\chi_{4508}(4141,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 21 polynomial |
Values on generators
\((2255,1473,1569)\) → \((1,e\left(\frac{4}{21}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 4508 }(93, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) |
sage: chi.jacobi_sum(n)