from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2365, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,32]))
pari: [g,chi] = znchar(Mod(56,2365))
Basic properties
| Modulus: | \(2365\) | |
| 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 \(\chi_{43}(13,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2365.bt
\(\chi_{2365}(56,\cdot)\) \(\chi_{2365}(111,\cdot)\) \(\chi_{2365}(496,\cdot)\) \(\chi_{2365}(771,\cdot)\) \(\chi_{2365}(826,\cdot)\) \(\chi_{2365}(1046,\cdot)\) \(\chi_{2365}(1156,\cdot)\) \(\chi_{2365}(1321,\cdot)\) \(\chi_{2365}(1486,\cdot)\) \(\chi_{2365}(1651,\cdot)\) \(\chi_{2365}(1816,\cdot)\) \(\chi_{2365}(2036,\cdot)\)
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
\((947,431,1981)\) → \((1,1,e\left(\frac{16}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 2365 }(56, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) |
sage: chi.jacobi_sum(n)