from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1805, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,22]))
pari: [g,chi] = znchar(Mod(1711,1805))
Basic properties
Modulus: | \(1805\) | |
Conductor: | \(361\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(19\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{361}(267,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1805.q
\(\chi_{1805}(96,\cdot)\) \(\chi_{1805}(191,\cdot)\) \(\chi_{1805}(286,\cdot)\) \(\chi_{1805}(381,\cdot)\) \(\chi_{1805}(476,\cdot)\) \(\chi_{1805}(571,\cdot)\) \(\chi_{1805}(666,\cdot)\) \(\chi_{1805}(761,\cdot)\) \(\chi_{1805}(856,\cdot)\) \(\chi_{1805}(951,\cdot)\) \(\chi_{1805}(1046,\cdot)\) \(\chi_{1805}(1141,\cdot)\) \(\chi_{1805}(1236,\cdot)\) \(\chi_{1805}(1331,\cdot)\) \(\chi_{1805}(1426,\cdot)\) \(\chi_{1805}(1521,\cdot)\) \(\chi_{1805}(1616,\cdot)\) \(\chi_{1805}(1711,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
Fixed field: | 19.19.10842505080063916320800450434338728415281531281.1 |
Values on generators
\((362,1446)\) → \((1,e\left(\frac{11}{19}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 1805 }(1711, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) |
sage: chi.jacobi_sum(n)