from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2668, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,40,33]))
pari: [g,chi] = znchar(Mod(771,2668))
Basic properties
Modulus: | \(2668\) | |
Conductor: | \(2668\) | 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 2668.bj
\(\chi_{2668}(75,\cdot)\) \(\chi_{2668}(215,\cdot)\) \(\chi_{2668}(307,\cdot)\) \(\chi_{2668}(331,\cdot)\) \(\chi_{2668}(423,\cdot)\) \(\chi_{2668}(679,\cdot)\) \(\chi_{2668}(771,\cdot)\) \(\chi_{2668}(795,\cdot)\) \(\chi_{2668}(887,\cdot)\) \(\chi_{2668}(1143,\cdot)\) \(\chi_{2668}(1235,\cdot)\) \(\chi_{2668}(1375,\cdot)\) \(\chi_{2668}(1467,\cdot)\) \(\chi_{2668}(2187,\cdot)\) \(\chi_{2668}(2279,\cdot)\) \(\chi_{2668}(2303,\cdot)\) \(\chi_{2668}(2395,\cdot)\) \(\chi_{2668}(2419,\cdot)\) \(\chi_{2668}(2511,\cdot)\) \(\chi_{2668}(2651,\cdot)\)
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
\((1335,465,553)\) → \((-1,e\left(\frac{10}{11}\right),-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 2668 }(771, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) |
sage: chi.jacobi_sum(n)