from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1155, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,40,36]))
pari: [g,chi] = znchar(Mod(53,1155))
Basic properties
Modulus: | \(1155\) | |
Conductor: | \(1155\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | 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 1155.dl
\(\chi_{1155}(53,\cdot)\) \(\chi_{1155}(137,\cdot)\) \(\chi_{1155}(158,\cdot)\) \(\chi_{1155}(212,\cdot)\) \(\chi_{1155}(317,\cdot)\) \(\chi_{1155}(368,\cdot)\) \(\chi_{1155}(422,\cdot)\) \(\chi_{1155}(443,\cdot)\) \(\chi_{1155}(548,\cdot)\) \(\chi_{1155}(632,\cdot)\) \(\chi_{1155}(653,\cdot)\) \(\chi_{1155}(863,\cdot)\) \(\chi_{1155}(872,\cdot)\) \(\chi_{1155}(977,\cdot)\) \(\chi_{1155}(1082,\cdot)\) \(\chi_{1155}(1103,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((386,232,661,211)\) → \((-1,-i,e\left(\frac{2}{3}\right),e\left(\frac{3}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
\( \chi_{ 1155 }(53, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) |
sage: chi.jacobi_sum(n)