from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8880, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,18,0,13]))
pari: [g,chi] = znchar(Mod(1421,8880))
Basic properties
| Modulus: | \(8880\) | |
| Conductor: | \(1776\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1776}(1421,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8880.ny
\(\chi_{8880}(1421,\cdot)\) \(\chi_{8880}(1781,\cdot)\) \(\chi_{8880}(2141,\cdot)\) \(\chi_{8880}(2501,\cdot)\) \(\chi_{8880}(3941,\cdot)\) \(\chi_{8880}(4901,\cdot)\) \(\chi_{8880}(5141,\cdot)\) \(\chi_{8880}(5981,\cdot)\) \(\chi_{8880}(6821,\cdot)\) \(\chi_{8880}(7661,\cdot)\) \(\chi_{8880}(7901,\cdot)\) \(\chi_{8880}(8861,\cdot)\)
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
\((5551,6661,5921,1777,8401)\) → \((1,-i,-1,1,e\left(\frac{13}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
| \( \chi_{ 8880 }(1421, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(e\left(\frac{13}{18}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)