from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1849, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([2]))
pari: [g,chi] = znchar(Mod(1557,1849))
Basic properties
Modulus: | \(1849\) | |
Conductor: | \(43\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{43}(9,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1849.g
\(\chi_{1849}(210,\cdot)\) \(\chi_{1849}(361,\cdot)\) \(\chi_{1849}(367,\cdot)\) \(\chi_{1849}(660,\cdot)\) \(\chi_{1849}(891,\cdot)\) \(\chi_{1849}(1085,\cdot)\) \(\chi_{1849}(1261,\cdot)\) \(\chi_{1849}(1557,\cdot)\) \(\chi_{1849}(1561,\cdot)\) \(\chi_{1849}(1573,\cdot)\) \(\chi_{1849}(1588,\cdot)\) \(\chi_{1849}(1830,\cdot)\)
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 21 polynomial |
Values on generators
\(3\) → \(e\left(\frac{1}{21}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 1849 }(1557, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) |
sage: chi.jacobi_sum(n)