from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3969, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,9]))
pari: [g,chi] = znchar(Mod(3268,3969))
Basic properties
| Modulus: | \(3969\) | |
| Conductor: | \(441\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{441}(328,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3969.bz
\(\chi_{3969}(55,\cdot)\) \(\chi_{3969}(433,\cdot)\) \(\chi_{3969}(622,\cdot)\) \(\chi_{3969}(1000,\cdot)\) \(\chi_{3969}(1189,\cdot)\) \(\chi_{3969}(1756,\cdot)\) \(\chi_{3969}(2134,\cdot)\) \(\chi_{3969}(2323,\cdot)\) \(\chi_{3969}(2701,\cdot)\) \(\chi_{3969}(3268,\cdot)\) \(\chi_{3969}(3457,\cdot)\) \(\chi_{3969}(3835,\cdot)\)
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: | Number field defined by a degree 42 polynomial |
Values on generators
\((2108,3727)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{3}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3969 }(3268, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)