from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1009, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([31]))
pari: [g,chi] = znchar(Mod(24,1009))
Basic properties
| Modulus: | \(1009\) | |
| Conductor: | \(1009\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1009.q
\(\chi_{1009}(24,\cdot)\) \(\chi_{1009}(60,\cdot)\) \(\chi_{1009}(81,\cdot)\) \(\chi_{1009}(185,\cdot)\) \(\chi_{1009}(242,\cdot)\) \(\chi_{1009}(246,\cdot)\) \(\chi_{1009}(254,\cdot)\) \(\chi_{1009}(433,\cdot)\) \(\chi_{1009}(436,\cdot)\) \(\chi_{1009}(502,\cdot)\) \(\chi_{1009}(605,\cdot)\) \(\chi_{1009}(967,\cdot)\)
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
\(11\) → \(e\left(\frac{31}{42}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1009 }(24, a) \) | \(1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) |
sage: chi.jacobi_sum(n)