from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1805, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([9,14]))
pari: [g,chi] = znchar(Mod(1382,1805))
Basic properties
| Modulus: | \(1805\) | |
| Conductor: | \(95\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{95}(52,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1805.s
\(\chi_{1805}(127,\cdot)\) \(\chi_{1805}(262,\cdot)\) \(\chi_{1805}(307,\cdot)\) \(\chi_{1805}(333,\cdot)\) \(\chi_{1805}(477,\cdot)\) \(\chi_{1805}(488,\cdot)\) \(\chi_{1805}(623,\cdot)\) \(\chi_{1805}(668,\cdot)\) \(\chi_{1805}(838,\cdot)\) \(\chi_{1805}(1382,\cdot)\) \(\chi_{1805}(1743,\cdot)\) \(\chi_{1805}(1777,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | \(\Q(\zeta_{95})^+\) |
Values on generators
\((362,1446)\) → \((i,e\left(\frac{7}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 1805 }(1382, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{25}{36}\right)\) |
sage: chi.jacobi_sum(n)