from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([25,6]))
pari: [g,chi] = znchar(Mod(215,2009))
Basic properties
| Modulus: | \(2009\) | |
| Conductor: | \(287\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{287}(215,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2009.bf
\(\chi_{2009}(215,\cdot)\) \(\chi_{2009}(264,\cdot)\) \(\chi_{2009}(570,\cdot)\) \(\chi_{2009}(754,\cdot)\) \(\chi_{2009}(1158,\cdot)\) \(\chi_{2009}(1207,\cdot)\) \(\chi_{2009}(1636,\cdot)\) \(\chi_{2009}(1697,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.0.682752912803285903537497565889168449852341644059496446718727.1 |
Values on generators
\((493,785)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 2009 }(215, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) |
sage: chi.jacobi_sum(n)