from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5550, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,11]))
pari: [g,chi] = znchar(Mod(457,5550))
Basic properties
| Modulus: | \(5550\) | |
| Conductor: | \(185\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{185}(87,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5550.de
\(\chi_{5550}(457,\cdot)\) \(\chi_{5550}(757,\cdot)\) \(\chi_{5550}(907,\cdot)\) \(\chi_{5550}(943,\cdot)\) \(\chi_{5550}(1093,\cdot)\) \(\chi_{5550}(1393,\cdot)\) \(\chi_{5550}(1993,\cdot)\) \(\chi_{5550}(2107,\cdot)\) \(\chi_{5550}(3493,\cdot)\) \(\chi_{5550}(3907,\cdot)\) \(\chi_{5550}(5293,\cdot)\) \(\chi_{5550}(5407,\cdot)\)
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: | 36.36.57444765302724909954814307473256133361395843470561362005770206451416015625.2 |
Values on generators
\((3701,1777,2851)\) → \((1,i,e\left(\frac{11}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
| \( \chi_{ 5550 }(457, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(-i\) | \(e\left(\frac{11}{18}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)