from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8640, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,0,32,9]))
pari: [g,chi] = znchar(Mod(1087,8640))
Basic properties
| Modulus: | \(8640\) | |
| Conductor: | \(540\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{540}(7,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8640.fv
\(\chi_{8640}(1087,\cdot)\) \(\chi_{8640}(1663,\cdot)\) \(\chi_{8640}(2047,\cdot)\) \(\chi_{8640}(2623,\cdot)\) \(\chi_{8640}(3967,\cdot)\) \(\chi_{8640}(4543,\cdot)\) \(\chi_{8640}(4927,\cdot)\) \(\chi_{8640}(5503,\cdot)\) \(\chi_{8640}(6847,\cdot)\) \(\chi_{8640}(7423,\cdot)\) \(\chi_{8640}(7807,\cdot)\) \(\chi_{8640}(8383,\cdot)\)
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: | Number field defined by a degree 36 polynomial |
Values on generators
\((2431,3781,6401,3457)\) → \((-1,1,e\left(\frac{8}{9}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8640 }(1087, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{9}\right)\) |
sage: chi.jacobi_sum(n)