from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,21]))
pari: [g,chi] = znchar(Mod(472,2009))
Basic properties
| Modulus: | \(2009\) | |
| Conductor: | \(287\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{287}(185,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2009.bq
\(\chi_{2009}(80,\cdot)\) \(\chi_{2009}(166,\cdot)\) \(\chi_{2009}(374,\cdot)\) \(\chi_{2009}(472,\cdot)\) \(\chi_{2009}(607,\cdot)\) \(\chi_{2009}(705,\cdot)\) \(\chi_{2009}(717,\cdot)\) \(\chi_{2009}(815,\cdot)\) \(\chi_{2009}(1109,\cdot)\) \(\chi_{2009}(1146,\cdot)\) \(\chi_{2009}(1440,\cdot)\) \(\chi_{2009}(1538,\cdot)\) \(\chi_{2009}(1550,\cdot)\) \(\chi_{2009}(1648,\cdot)\) \(\chi_{2009}(1783,\cdot)\) \(\chi_{2009}(1881,\cdot)\)
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
\((493,785)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{7}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 2009 }(472, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) |
sage: chi.jacobi_sum(n)