from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3200, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,15,14]))
pari: [g,chi] = znchar(Mod(303,3200))
Basic properties
| Modulus: | \(3200\) | |
| Conductor: | \(800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{800}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3200.ct
\(\chi_{3200}(303,\cdot)\) \(\chi_{3200}(527,\cdot)\) \(\chi_{3200}(623,\cdot)\) \(\chi_{3200}(847,\cdot)\) \(\chi_{3200}(1167,\cdot)\) \(\chi_{3200}(1263,\cdot)\) \(\chi_{3200}(1487,\cdot)\) \(\chi_{3200}(1583,\cdot)\) \(\chi_{3200}(1903,\cdot)\) \(\chi_{3200}(2127,\cdot)\) \(\chi_{3200}(2223,\cdot)\) \(\chi_{3200}(2447,\cdot)\) \(\chi_{3200}(2767,\cdot)\) \(\chi_{3200}(2863,\cdot)\) \(\chi_{3200}(3087,\cdot)\) \(\chi_{3200}(3183,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.40.386856262276681335905976320000000000000000000000000000000000000000000000000000000000000000000000.1 |
Values on generators
\((1151,901,2177)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{7}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 3200 }(303, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{40}\right)\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{9}{40}\right)\) |
sage: chi.jacobi_sum(n)