from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2736, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,6,22]))
pari: [g,chi] = znchar(Mod(1307,2736))
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.gx
\(\chi_{2736}(59,\cdot)\) \(\chi_{2736}(371,\cdot)\) \(\chi_{2736}(659,\cdot)\) \(\chi_{2736}(1067,\cdot)\) \(\chi_{2736}(1211,\cdot)\) \(\chi_{2736}(1307,\cdot)\) \(\chi_{2736}(1427,\cdot)\) \(\chi_{2736}(1739,\cdot)\) \(\chi_{2736}(2027,\cdot)\) \(\chi_{2736}(2435,\cdot)\) \(\chi_{2736}(2579,\cdot)\) \(\chi_{2736}(2675,\cdot)\)
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: | 36.0.1106979957984580051751811127579726572670357439523131838827096361582468940533158389988438920881242112.1 |
Values on generators
\((1711,2053,1217,1009)\) → \((-1,i,e\left(\frac{1}{6}\right),e\left(\frac{11}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2736 }(1307, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(1\) | \(e\left(\frac{7}{36}\right)\) |
sage: chi.jacobi_sum(n)