from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8640, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,8,9]))
pari: [g,chi] = znchar(Mod(367,8640))
Basic properties
| Modulus: | \(8640\) | |
| Conductor: | \(2160\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2160}(1987,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8640.fw
\(\chi_{8640}(367,\cdot)\) \(\chi_{8640}(463,\cdot)\) \(\chi_{8640}(1327,\cdot)\) \(\chi_{8640}(2383,\cdot)\) \(\chi_{8640}(3247,\cdot)\) \(\chi_{8640}(3343,\cdot)\) \(\chi_{8640}(4207,\cdot)\) \(\chi_{8640}(5263,\cdot)\) \(\chi_{8640}(6127,\cdot)\) \(\chi_{8640}(6223,\cdot)\) \(\chi_{8640}(7087,\cdot)\) \(\chi_{8640}(8143,\cdot)\)
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
\((2431,3781,6401,3457)\) → \((-1,-i,e\left(\frac{2}{9}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8640 }(367, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{18}\right)\) |
sage: chi.jacobi_sum(n)