from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1216, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,16]))
pari: [g,chi] = znchar(Mod(47,1216))
Basic properties
| Modulus: | \(1216\) | |
| Conductor: | \(304\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{304}(275,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1216.bt
\(\chi_{1216}(47,\cdot)\) \(\chi_{1216}(111,\cdot)\) \(\chi_{1216}(175,\cdot)\) \(\chi_{1216}(207,\cdot)\) \(\chi_{1216}(271,\cdot)\) \(\chi_{1216}(367,\cdot)\) \(\chi_{1216}(655,\cdot)\) \(\chi_{1216}(719,\cdot)\) \(\chi_{1216}(783,\cdot)\) \(\chi_{1216}(815,\cdot)\) \(\chi_{1216}(879,\cdot)\) \(\chi_{1216}(975,\cdot)\)
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: | 36.0.52733281945045886724167383478270850720626086921526306402773390818541568.1 |
Values on generators
\((191,837,705)\) → \((-1,-i,e\left(\frac{4}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 1216 }(47, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) |
sage: chi.jacobi_sum(n)