from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3969, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,37]))
pari: [g,chi] = znchar(Mod(514,3969))
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 \(\chi_{441}(220,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3969.ca
\(\chi_{3969}(514,\cdot)\) \(\chi_{3969}(1027,\cdot)\) \(\chi_{3969}(1081,\cdot)\) \(\chi_{3969}(1594,\cdot)\) \(\chi_{3969}(2161,\cdot)\) \(\chi_{3969}(2215,\cdot)\) \(\chi_{3969}(2728,\cdot)\) \(\chi_{3969}(2782,\cdot)\) \(\chi_{3969}(3295,\cdot)\) \(\chi_{3969}(3349,\cdot)\) \(\chi_{3969}(3862,\cdot)\) \(\chi_{3969}(3916,\cdot)\)
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
\((2108,3727)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{37}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 3969 }(514, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)