from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4730, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,21,5]))
pari: [g,chi] = znchar(Mod(381,4730))
Basic properties
| Modulus: | \(4730\) | |
| Conductor: | \(473\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{473}(381,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4730.ca
\(\chi_{4730}(381,\cdot)\) \(\chi_{4730}(811,\cdot)\) \(\chi_{4730}(1641,\cdot)\) \(\chi_{4730}(2961,\cdot)\) \(\chi_{4730}(3361,\cdot)\) \(\chi_{4730}(3791,\cdot)\) \(\chi_{4730}(4221,\cdot)\) \(\chi_{4730}(4681,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.30.900147933423976629543618770038416552630751039074980685343506012294353.1 |
Values on generators
\((947,431,1981)\) → \((1,e\left(\frac{7}{10}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 4730 }(381, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) |
sage: chi.jacobi_sum(n)