from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2550, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,38,25]))
pari: [g,chi] = znchar(Mod(263,2550))
Basic properties
| Modulus: | \(2550\) | |
| Conductor: | \(1275\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1275}(263,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2550.ch
\(\chi_{2550}(263,\cdot)\) \(\chi_{2550}(287,\cdot)\) \(\chi_{2550}(383,\cdot)\) \(\chi_{2550}(767,\cdot)\) \(\chi_{2550}(773,\cdot)\) \(\chi_{2550}(797,\cdot)\) \(\chi_{2550}(1277,\cdot)\) \(\chi_{2550}(1283,\cdot)\) \(\chi_{2550}(1403,\cdot)\) \(\chi_{2550}(1787,\cdot)\) \(\chi_{2550}(1817,\cdot)\) \(\chi_{2550}(1913,\cdot)\) \(\chi_{2550}(2297,\cdot)\) \(\chi_{2550}(2303,\cdot)\) \(\chi_{2550}(2327,\cdot)\) \(\chi_{2550}(2423,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((851,1327,751)\) → \((-1,e\left(\frac{19}{20}\right),e\left(\frac{5}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 2550 }(263, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)