from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5635, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,22,4]))
pari: [g,chi] = znchar(Mod(48,5635))
Basic properties
| Modulus: | \(5635\) | |
| Conductor: | \(805\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{805}(48,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5635.cd
\(\chi_{5635}(48,\cdot)\) \(\chi_{5635}(538,\cdot)\) \(\chi_{5635}(587,\cdot)\) \(\chi_{5635}(832,\cdot)\) \(\chi_{5635}(1028,\cdot)\) \(\chi_{5635}(1273,\cdot)\) \(\chi_{5635}(1567,\cdot)\) \(\chi_{5635}(1812,\cdot)\) \(\chi_{5635}(2302,\cdot)\) \(\chi_{5635}(2743,\cdot)\) \(\chi_{5635}(2792,\cdot)\) \(\chi_{5635}(3233,\cdot)\) \(\chi_{5635}(3282,\cdot)\) \(\chi_{5635}(3527,\cdot)\) \(\chi_{5635}(3968,\cdot)\) \(\chi_{5635}(4213,\cdot)\) \(\chi_{5635}(4948,\cdot)\) \(\chi_{5635}(4997,\cdot)\) \(\chi_{5635}(5193,\cdot)\) \(\chi_{5635}(5487,\cdot)\)
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: | Number field defined by a degree 44 polynomial |
Values on generators
\((3382,346,2696)\) → \((-i,-1,e\left(\frac{1}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 5635 }(48, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) |
sage: chi.jacobi_sum(n)