from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2365, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,6,40]))
pari: [g,chi] = znchar(Mod(178,2365))
Basic properties
| Modulus: | \(2365\) | |
| Conductor: | \(2365\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2365.co
\(\chi_{2365}(178,\cdot)\) \(\chi_{2365}(337,\cdot)\) \(\chi_{2365}(393,\cdot)\) \(\chi_{2365}(552,\cdot)\) \(\chi_{2365}(767,\cdot)\) \(\chi_{2365}(853,\cdot)\) \(\chi_{2365}(952,\cdot)\) \(\chi_{2365}(1283,\cdot)\) \(\chi_{2365}(1382,\cdot)\) \(\chi_{2365}(1498,\cdot)\) \(\chi_{2365}(1597,\cdot)\) \(\chi_{2365}(1713,\cdot)\) \(\chi_{2365}(1812,\cdot)\) \(\chi_{2365}(1898,\cdot)\) \(\chi_{2365}(2272,\cdot)\) \(\chi_{2365}(2328,\cdot)\)
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
\((947,431,1981)\) → \((-i,e\left(\frac{1}{10}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 2365 }(178, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) |
sage: chi.jacobi_sum(n)