from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2057, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([24,33]))
pari: [g,chi] = znchar(Mod(89,2057))
Basic properties
Modulus: | \(2057\) | |
Conductor: | \(2057\) | 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 2057.x
\(\chi_{2057}(89,\cdot)\) \(\chi_{2057}(166,\cdot)\) \(\chi_{2057}(276,\cdot)\) \(\chi_{2057}(353,\cdot)\) \(\chi_{2057}(463,\cdot)\) \(\chi_{2057}(540,\cdot)\) \(\chi_{2057}(650,\cdot)\) \(\chi_{2057}(837,\cdot)\) \(\chi_{2057}(914,\cdot)\) \(\chi_{2057}(1024,\cdot)\) \(\chi_{2057}(1101,\cdot)\) \(\chi_{2057}(1288,\cdot)\) \(\chi_{2057}(1398,\cdot)\) \(\chi_{2057}(1475,\cdot)\) \(\chi_{2057}(1585,\cdot)\) \(\chi_{2057}(1662,\cdot)\) \(\chi_{2057}(1772,\cdot)\) \(\chi_{2057}(1849,\cdot)\) \(\chi_{2057}(1959,\cdot)\) \(\chi_{2057}(2036,\cdot)\)
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
\((970,122)\) → \((e\left(\frac{6}{11}\right),-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 2057 }(89, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(-i\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(-1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) |
sage: chi.jacobi_sum(n)