from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2805, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,0,28,25]))
pari: [g,chi] = znchar(Mod(161,2805))
Basic properties
Modulus: | \(2805\) | |
Conductor: | \(561\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{561}(161,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2805.en
\(\chi_{2805}(161,\cdot)\) \(\chi_{2805}(206,\cdot)\) \(\chi_{2805}(281,\cdot)\) \(\chi_{2805}(491,\cdot)\) \(\chi_{2805}(536,\cdot)\) \(\chi_{2805}(926,\cdot)\) \(\chi_{2805}(1256,\cdot)\) \(\chi_{2805}(1436,\cdot)\) \(\chi_{2805}(1481,\cdot)\) \(\chi_{2805}(1691,\cdot)\) \(\chi_{2805}(1766,\cdot)\) \(\chi_{2805}(1811,\cdot)\) \(\chi_{2805}(2021,\cdot)\) \(\chi_{2805}(2246,\cdot)\) \(\chi_{2805}(2576,\cdot)\) \(\chi_{2805}(2756,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((1871,562,1531,496)\) → \((-1,1,e\left(\frac{7}{10}\right),e\left(\frac{5}{8}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(19\) | \(23\) | \(26\) |
\( \chi_{ 2805 }(161, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) |
sage: chi.jacobi_sum(n)