from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3969, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([35,4]))
pari: [g,chi] = znchar(Mod(620,3969))
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 \(\chi_{441}(32,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3969.bx
\(\chi_{3969}(53,\cdot)\) \(\chi_{3969}(107,\cdot)\) \(\chi_{3969}(620,\cdot)\) \(\chi_{3969}(674,\cdot)\) \(\chi_{3969}(1187,\cdot)\) \(\chi_{3969}(1241,\cdot)\) \(\chi_{3969}(1754,\cdot)\) \(\chi_{3969}(1808,\cdot)\) \(\chi_{3969}(2375,\cdot)\) \(\chi_{3969}(2888,\cdot)\) \(\chi_{3969}(2942,\cdot)\) \(\chi_{3969}(3455,\cdot)\)
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
\((2108,3727)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{2}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 3969 }(620, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)