from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2736, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,12,20]))
pari: [g,chi] = znchar(Mod(283,2736))
Basic properties
| Modulus: | \(2736\) | |
| Conductor: | \(2736\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2736.gr
\(\chi_{2736}(283,\cdot)\) \(\chi_{2736}(403,\cdot)\) \(\chi_{2736}(499,\cdot)\) \(\chi_{2736}(643,\cdot)\) \(\chi_{2736}(1051,\cdot)\) \(\chi_{2736}(1339,\cdot)\) \(\chi_{2736}(1651,\cdot)\) \(\chi_{2736}(1771,\cdot)\) \(\chi_{2736}(1867,\cdot)\) \(\chi_{2736}(2011,\cdot)\) \(\chi_{2736}(2419,\cdot)\) \(\chi_{2736}(2707,\cdot)\)
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
\((1711,2053,1217,1009)\) → \((-1,i,e\left(\frac{1}{3}\right),e\left(\frac{5}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2736 }(283, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-i\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(-1\) | \(e\left(\frac{17}{36}\right)\) |
sage: chi.jacobi_sum(n)