from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14520, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,22,22,33,28]))
pari: [g,chi] = znchar(Mod(7283,14520))
Basic properties
| Modulus: | \(14520\) | |
| Conductor: | \(14520\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 14520.fn
\(\chi_{14520}(683,\cdot)\) \(\chi_{14520}(947,\cdot)\) \(\chi_{14520}(2003,\cdot)\) \(\chi_{14520}(2267,\cdot)\) \(\chi_{14520}(3323,\cdot)\) \(\chi_{14520}(3587,\cdot)\) \(\chi_{14520}(4643,\cdot)\) \(\chi_{14520}(4907,\cdot)\) \(\chi_{14520}(5963,\cdot)\) \(\chi_{14520}(6227,\cdot)\) \(\chi_{14520}(7283,\cdot)\) \(\chi_{14520}(7547,\cdot)\) \(\chi_{14520}(8603,\cdot)\) \(\chi_{14520}(8867,\cdot)\) \(\chi_{14520}(10187,\cdot)\) \(\chi_{14520}(11243,\cdot)\) \(\chi_{14520}(11507,\cdot)\) \(\chi_{14520}(12563,\cdot)\) \(\chi_{14520}(13883,\cdot)\) \(\chi_{14520}(14147,\cdot)\)
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
\((3631,7261,4841,11617,14401)\) → \((-1,-1,-1,-i,e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 14520 }(7283, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) |
sage: chi.jacobi_sum(n)