from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3969, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,11]))
pari: [g,chi] = znchar(Mod(404,3969))
Basic properties
| Modulus: | \(3969\) | |
| Conductor: | \(147\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{147}(110,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3969.bv
\(\chi_{3969}(404,\cdot)\) \(\chi_{3969}(647,\cdot)\) \(\chi_{3969}(971,\cdot)\) \(\chi_{3969}(1214,\cdot)\) \(\chi_{3969}(1781,\cdot)\) \(\chi_{3969}(2105,\cdot)\) \(\chi_{3969}(2348,\cdot)\) \(\chi_{3969}(2672,\cdot)\) \(\chi_{3969}(2915,\cdot)\) \(\chi_{3969}(3239,\cdot)\) \(\chi_{3969}(3482,\cdot)\) \(\chi_{3969}(3806,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | \(\Q(\zeta_{147})^+\) |
Values on generators
\((2108,3727)\) → \((-1,e\left(\frac{11}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3969 }(404, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)