from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,54,45]))
pari: [g,chi] = znchar(Mod(1194,1309))
Basic properties
| Modulus: | \(1309\) | |
| Conductor: | \(1309\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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 1309.cp
\(\chi_{1309}(30,\cdot)\) \(\chi_{1309}(72,\cdot)\) \(\chi_{1309}(123,\cdot)\) \(\chi_{1309}(149,\cdot)\) \(\chi_{1309}(200,\cdot)\) \(\chi_{1309}(310,\cdot)\) \(\chi_{1309}(387,\cdot)\) \(\chi_{1309}(480,\cdot)\) \(\chi_{1309}(557,\cdot)\) \(\chi_{1309}(667,\cdot)\) \(\chi_{1309}(744,\cdot)\) \(\chi_{1309}(1075,\cdot)\) \(\chi_{1309}(1152,\cdot)\) \(\chi_{1309}(1194,\cdot)\) \(\chi_{1309}(1262,\cdot)\) \(\chi_{1309}(1271,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((1123,596,309)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{9}{10}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(1194, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{9}{10}\right)\) |
sage: chi.jacobi_sum(n)