from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([17,0]))
pari: [g,chi] = znchar(Mod(124,2009))
Basic properties
| Modulus: | \(2009\) | |
| Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{49}(26,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2009.bk
\(\chi_{2009}(124,\cdot)\) \(\chi_{2009}(206,\cdot)\) \(\chi_{2009}(493,\cdot)\) \(\chi_{2009}(698,\cdot)\) \(\chi_{2009}(780,\cdot)\) \(\chi_{2009}(985,\cdot)\) \(\chi_{2009}(1067,\cdot)\) \(\chi_{2009}(1272,\cdot)\) \(\chi_{2009}(1559,\cdot)\) \(\chi_{2009}(1641,\cdot)\) \(\chi_{2009}(1846,\cdot)\) \(\chi_{2009}(1928,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((493,785)\) → \((e\left(\frac{17}{42}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 2009 }(124, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) |
sage: chi.jacobi_sum(n)