from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2960, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,27,11]))
pari: [g,chi] = znchar(Mod(13,2960))
Basic properties
| Modulus: | \(2960\) | |
| Conductor: | \(2960\) | 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 2960.gq
\(\chi_{2960}(13,\cdot)\) \(\chi_{2960}(93,\cdot)\) \(\chi_{2960}(277,\cdot)\) \(\chi_{2960}(357,\cdot)\) \(\chi_{2960}(893,\cdot)\) \(\chi_{2960}(1053,\cdot)\) \(\chi_{2960}(1293,\cdot)\) \(\chi_{2960}(1613,\cdot)\) \(\chi_{2960}(1717,\cdot)\) \(\chi_{2960}(2037,\cdot)\) \(\chi_{2960}(2277,\cdot)\) \(\chi_{2960}(2437,\cdot)\)
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
\((2591,741,1777,2481)\) → \((1,-i,-i,e\left(\frac{11}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2960 }(13, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)