from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9200, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,22,38]))
pari: [g,chi] = znchar(Mod(99,9200))
Basic properties
| Modulus: | \(9200\) | |
| Conductor: | \(1840\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1840}(99,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9200.dx
\(\chi_{9200}(99,\cdot)\) \(\chi_{9200}(1299,\cdot)\) \(\chi_{9200}(1699,\cdot)\) \(\chi_{9200}(2499,\cdot)\) \(\chi_{9200}(3099,\cdot)\) \(\chi_{9200}(3299,\cdot)\) \(\chi_{9200}(3699,\cdot)\) \(\chi_{9200}(4099,\cdot)\) \(\chi_{9200}(4299,\cdot)\) \(\chi_{9200}(4499,\cdot)\) \(\chi_{9200}(4699,\cdot)\) \(\chi_{9200}(5899,\cdot)\) \(\chi_{9200}(6299,\cdot)\) \(\chi_{9200}(7099,\cdot)\) \(\chi_{9200}(7699,\cdot)\) \(\chi_{9200}(7899,\cdot)\) \(\chi_{9200}(8299,\cdot)\) \(\chi_{9200}(8699,\cdot)\) \(\chi_{9200}(8899,\cdot)\) \(\chi_{9200}(9099,\cdot)\)
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
\((1151,6901,2577,1201)\) → \((-1,-i,-1,e\left(\frac{19}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 9200 }(99, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) |
sage: chi.jacobi_sum(n)