from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1312, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,35,37]))
pari: [g,chi] = znchar(Mod(179,1312))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1312.di
\(\chi_{1312}(67,\cdot)\) \(\chi_{1312}(99,\cdot)\) \(\chi_{1312}(147,\cdot)\) \(\chi_{1312}(171,\cdot)\) \(\chi_{1312}(179,\cdot)\) \(\chi_{1312}(235,\cdot)\) \(\chi_{1312}(315,\cdot)\) \(\chi_{1312}(587,\cdot)\) \(\chi_{1312}(667,\cdot)\) \(\chi_{1312}(675,\cdot)\) \(\chi_{1312}(691,\cdot)\) \(\chi_{1312}(731,\cdot)\) \(\chi_{1312}(867,\cdot)\) \(\chi_{1312}(883,\cdot)\) \(\chi_{1312}(955,\cdot)\) \(\chi_{1312}(1259,\cdot)\)
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
\((575,165,129)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{37}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1312 }(179, a) \) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{40}\right)\) |
sage: chi.jacobi_sum(n)