from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3724, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,19,7]))
pari: [g,chi] = znchar(Mod(1851,3724))
Basic properties
Modulus: | \(3724\) | |
Conductor: | \(3724\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3724.dl
\(\chi_{3724}(255,\cdot)\) \(\chi_{3724}(787,\cdot)\) \(\chi_{3724}(943,\cdot)\) \(\chi_{3724}(1319,\cdot)\) \(\chi_{3724}(1475,\cdot)\) \(\chi_{3724}(1851,\cdot)\) \(\chi_{3724}(2007,\cdot)\) \(\chi_{3724}(2539,\cdot)\) \(\chi_{3724}(2915,\cdot)\) \(\chi_{3724}(3071,\cdot)\) \(\chi_{3724}(3447,\cdot)\) \(\chi_{3724}(3603,\cdot)\)
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
\((1863,3041,3137)\) → \((-1,e\left(\frac{19}{42}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 3724 }(1851, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) |
sage: chi.jacobi_sum(n)