from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5290, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,14]))
pari: [g,chi] = znchar(Mod(63,5290))
Basic properties
| Modulus: | \(5290\) | |
| Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{115}(63,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5290.m
\(\chi_{5290}(63,\cdot)\) \(\chi_{5290}(263,\cdot)\) \(\chi_{5290}(557,\cdot)\) \(\chi_{5290}(803,\cdot)\) \(\chi_{5290}(1253,\cdot)\) \(\chi_{5290}(1417,\cdot)\) \(\chi_{5290}(1717,\cdot)\) \(\chi_{5290}(2527,\cdot)\) \(\chi_{5290}(2673,\cdot)\) \(\chi_{5290}(2687,\cdot)\) \(\chi_{5290}(2997,\cdot)\) \(\chi_{5290}(3237,\cdot)\) \(\chi_{5290}(3437,\cdot)\) \(\chi_{5290}(3533,\cdot)\) \(\chi_{5290}(3833,\cdot)\) \(\chi_{5290}(3977,\cdot)\) \(\chi_{5290}(4427,\cdot)\) \(\chi_{5290}(4643,\cdot)\) \(\chi_{5290}(4803,\cdot)\) \(\chi_{5290}(5113,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | \(\Q(\zeta_{115})^+\) |
Values on generators
\((2117,2121)\) → \((-i,e\left(\frac{7}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 5290 }(63, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) |
sage: chi.jacobi_sum(n)