from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2365, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([7,0,10]))
pari: [g,chi] = znchar(Mod(452,2365))
Basic properties
| Modulus: | \(2365\) | |
| Conductor: | \(215\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{215}(22,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2365.bw
\(\chi_{2365}(452,\cdot)\) \(\chi_{2365}(518,\cdot)\) \(\chi_{2365}(672,\cdot)\) \(\chi_{2365}(727,\cdot)\) \(\chi_{2365}(782,\cdot)\) \(\chi_{2365}(892,\cdot)\) \(\chi_{2365}(1398,\cdot)\) \(\chi_{2365}(1618,\cdot)\) \(\chi_{2365}(1673,\cdot)\) \(\chi_{2365}(1728,\cdot)\) \(\chi_{2365}(1838,\cdot)\) \(\chi_{2365}(1937,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | 28.28.1407829094312471215113334241722636006626629352569580078125.1 |
Values on generators
\((947,431,1981)\) → \((i,1,e\left(\frac{5}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 2365 }(452, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) |
sage: chi.jacobi_sum(n)