from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1755, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([4,9,9]))
pari: [g,chi] = znchar(Mod(112,1755))
Basic properties
| Modulus: | \(1755\) | |
| Conductor: | \(1755\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | 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 1755.gd
\(\chi_{1755}(112,\cdot)\) \(\chi_{1755}(148,\cdot)\) \(\chi_{1755}(502,\cdot)\) \(\chi_{1755}(538,\cdot)\) \(\chi_{1755}(697,\cdot)\) \(\chi_{1755}(733,\cdot)\) \(\chi_{1755}(1087,\cdot)\) \(\chi_{1755}(1123,\cdot)\) \(\chi_{1755}(1282,\cdot)\) \(\chi_{1755}(1318,\cdot)\) \(\chi_{1755}(1672,\cdot)\) \(\chi_{1755}(1708,\cdot)\)
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
\((326,352,1081)\) → \((e\left(\frac{1}{9}\right),i,i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 1755 }(112, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{29}{36}\right)\) |
sage: chi.jacobi_sum(n)