from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1122, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,8,25]))
pari: [g,chi] = znchar(Mod(59,1122))
Basic properties
Modulus: | \(1122\) | |
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 \(\chi_{561}(59,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1122.bj
\(\chi_{1122}(53,\cdot)\) \(\chi_{1122}(59,\cdot)\) \(\chi_{1122}(179,\cdot)\) \(\chi_{1122}(185,\cdot)\) \(\chi_{1122}(257,\cdot)\) \(\chi_{1122}(383,\cdot)\) \(\chi_{1122}(389,\cdot)\) \(\chi_{1122}(467,\cdot)\) \(\chi_{1122}(587,\cdot)\) \(\chi_{1122}(665,\cdot)\) \(\chi_{1122}(773,\cdot)\) \(\chi_{1122}(797,\cdot)\) \(\chi_{1122}(971,\cdot)\) \(\chi_{1122}(977,\cdot)\) \(\chi_{1122}(995,\cdot)\) \(\chi_{1122}(1103,\cdot)\)
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
\((749,409,1057)\) → \((-1,e\left(\frac{1}{5}\right),e\left(\frac{5}{8}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
\( \chi_{ 1122 }(59, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{40}\right)\) |
sage: chi.jacobi_sum(n)