from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8035, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,38]))
pari: [g,chi] = znchar(Mod(1163,8035))
Basic properties
Modulus: | \(8035\) | |
Conductor: | \(8035\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8035.l
\(\chi_{8035}(1163,\cdot)\) \(\chi_{8035}(1692,\cdot)\) \(\chi_{8035}(1858,\cdot)\) \(\chi_{8035}(1978,\cdot)\) \(\chi_{8035}(2132,\cdot)\) \(\chi_{8035}(2168,\cdot)\) \(\chi_{8035}(2212,\cdot)\) \(\chi_{8035}(2417,\cdot)\) \(\chi_{8035}(3993,\cdot)\) \(\chi_{8035}(4377,\cdot)\) \(\chi_{8035}(4493,\cdot)\) \(\chi_{8035}(5072,\cdot)\) \(\chi_{8035}(5192,\cdot)\) \(\chi_{8035}(5382,\cdot)\) \(\chi_{8035}(6513,\cdot)\) \(\chi_{8035}(6953,\cdot)\) \(\chi_{8035}(7033,\cdot)\) \(\chi_{8035}(7207,\cdot)\) \(\chi_{8035}(7238,\cdot)\) \(\chi_{8035}(7707,\cdot)\)
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
\((4822,4826)\) → \((-i,e\left(\frac{19}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 8035 }(1163, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) |
sage: chi.jacobi_sum(n)