from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([4,3]))
pari: [g,chi] = znchar(Mod(1011,2009))
Basic properties
Modulus: | \(2009\) | |
Conductor: | \(287\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{287}(150,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2009.bb
\(\chi_{2009}(68,\cdot)\) \(\chi_{2009}(178,\cdot)\) \(\chi_{2009}(325,\cdot)\) \(\chi_{2009}(864,\cdot)\) \(\chi_{2009}(1011,\cdot)\) \(\chi_{2009}(1244,\cdot)\) \(\chi_{2009}(1391,\cdot)\) \(\chi_{2009}(1930,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
Fixed field: | 24.24.589415824352273084266952490343550409844469452348841.1 |
Values on generators
\((493,785)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{1}{8}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
\( \chi_{ 2009 }(1011, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) |
sage: chi.jacobi_sum(n)