from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4730, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,34]))
pari: [g,chi] = znchar(Mod(1321,4730))
Basic properties
| Modulus: | \(4730\) | |
| Conductor: | \(43\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{43}(31,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4730.bt
\(\chi_{4730}(111,\cdot)\) \(\chi_{4730}(771,\cdot)\) \(\chi_{4730}(1321,\cdot)\) \(\chi_{4730}(1651,\cdot)\) \(\chi_{4730}(2421,\cdot)\) \(\chi_{4730}(2861,\cdot)\) \(\chi_{4730}(3191,\cdot)\) \(\chi_{4730}(3411,\cdot)\) \(\chi_{4730}(3521,\cdot)\) \(\chi_{4730}(3851,\cdot)\) \(\chi_{4730}(4181,\cdot)\) \(\chi_{4730}(4401,\cdot)\)
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
\((947,431,1981)\) → \((1,1,e\left(\frac{17}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 4730 }(1321, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{21}\right)\) |
sage: chi.jacobi_sum(n)