from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1053, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([2,9]))
pari: [g,chi] = znchar(Mod(8,1053))
Basic properties
| Modulus: | \(1053\) | |
| Conductor: | \(351\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{351}(164,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1053.bw
\(\chi_{1053}(8,\cdot)\) \(\chi_{1053}(44,\cdot)\) \(\chi_{1053}(125,\cdot)\) \(\chi_{1053}(278,\cdot)\) \(\chi_{1053}(359,\cdot)\) \(\chi_{1053}(395,\cdot)\) \(\chi_{1053}(476,\cdot)\) \(\chi_{1053}(629,\cdot)\) \(\chi_{1053}(710,\cdot)\) \(\chi_{1053}(746,\cdot)\) \(\chi_{1053}(827,\cdot)\) \(\chi_{1053}(980,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((326,730)\) → \((e\left(\frac{1}{18}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 1053 }(8, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)