from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1312, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,35,31]))
pari: [g,chi] = znchar(Mod(13,1312))
Basic properties
| Modulus: | \(1312\) | |
| Conductor: | \(1312\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | 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 1312.cv
\(\chi_{1312}(13,\cdot)\) \(\chi_{1312}(93,\cdot)\) \(\chi_{1312}(101,\cdot)\) \(\chi_{1312}(117,\cdot)\) \(\chi_{1312}(157,\cdot)\) \(\chi_{1312}(293,\cdot)\) \(\chi_{1312}(309,\cdot)\) \(\chi_{1312}(381,\cdot)\) \(\chi_{1312}(685,\cdot)\) \(\chi_{1312}(805,\cdot)\) \(\chi_{1312}(837,\cdot)\) \(\chi_{1312}(885,\cdot)\) \(\chi_{1312}(909,\cdot)\) \(\chi_{1312}(917,\cdot)\) \(\chi_{1312}(973,\cdot)\) \(\chi_{1312}(1053,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((575,165,129)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{31}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1312 }(13, a) \) | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(-1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{40}\right)\) |
sage: chi.jacobi_sum(n)