from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8880, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,18,0,34]))
pari: [g,chi] = znchar(Mod(731,8880))
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 \(\chi_{1776}(731,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8880.no
\(\chi_{8880}(731,\cdot)\) \(\chi_{8880}(2171,\cdot)\) \(\chi_{8880}(3371,\cdot)\) \(\chi_{8880}(3851,\cdot)\) \(\chi_{8880}(4091,\cdot)\) \(\chi_{8880}(4211,\cdot)\) \(\chi_{8880}(5171,\cdot)\) \(\chi_{8880}(6611,\cdot)\) \(\chi_{8880}(7811,\cdot)\) \(\chi_{8880}(8291,\cdot)\) \(\chi_{8880}(8531,\cdot)\) \(\chi_{8880}(8651,\cdot)\)
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
\((5551,6661,5921,1777,8401)\) → \((-1,i,-1,1,e\left(\frac{17}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
| \( \chi_{ 8880 }(731, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)