from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3267, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([22,45]))
pari: [g,chi] = znchar(Mod(10,3267))
Basic properties
| Modulus: | \(3267\) | |
| Conductor: | \(1089\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1089}(373,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3267.bc
\(\chi_{3267}(10,\cdot)\) \(\chi_{3267}(208,\cdot)\) \(\chi_{3267}(307,\cdot)\) \(\chi_{3267}(505,\cdot)\) \(\chi_{3267}(802,\cdot)\) \(\chi_{3267}(901,\cdot)\) \(\chi_{3267}(1099,\cdot)\) \(\chi_{3267}(1198,\cdot)\) \(\chi_{3267}(1396,\cdot)\) \(\chi_{3267}(1495,\cdot)\) \(\chi_{3267}(1792,\cdot)\) \(\chi_{3267}(1990,\cdot)\) \(\chi_{3267}(2089,\cdot)\) \(\chi_{3267}(2287,\cdot)\) \(\chi_{3267}(2386,\cdot)\) \(\chi_{3267}(2584,\cdot)\) \(\chi_{3267}(2683,\cdot)\) \(\chi_{3267}(2881,\cdot)\) \(\chi_{3267}(2980,\cdot)\) \(\chi_{3267}(3178,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((3026,244)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{15}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 3267 }(10, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) |
sage: chi.jacobi_sum(n)