from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5850, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([8,3,11]))
pari: [g,chi] = znchar(Mod(7,5850))
Basic properties
| Modulus: | \(5850\) | |
| Conductor: | \(585\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{585}(7,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5850.ed
\(\chi_{5850}(7,\cdot)\) \(\chi_{5850}(3343,\cdot)\) \(\chi_{5850}(4543,\cdot)\) \(\chi_{5850}(5107,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
| Fixed field: | 12.12.150677008729220317728515625.2 |
Values on generators
\((3251,3277,2251)\) → \((e\left(\frac{2}{3}\right),i,e\left(\frac{11}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5850 }(7, a) \) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(i\) | \(i\) |
sage: chi.jacobi_sum(n)