from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3332, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,12,21]))
pari: [g,chi] = znchar(Mod(225,3332))
Basic properties
| Modulus: | \(3332\) | |
| Conductor: | \(833\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{833}(225,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3332.bu
\(\chi_{3332}(225,\cdot)\) \(\chi_{3332}(421,\cdot)\) \(\chi_{3332}(701,\cdot)\) \(\chi_{3332}(897,\cdot)\) \(\chi_{3332}(1653,\cdot)\) \(\chi_{3332}(1849,\cdot)\) \(\chi_{3332}(2129,\cdot)\) \(\chi_{3332}(2325,\cdot)\) \(\chi_{3332}(2605,\cdot)\) \(\chi_{3332}(2801,\cdot)\) \(\chi_{3332}(3081,\cdot)\) \(\chi_{3332}(3277,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((1667,885,785)\) → \((1,e\left(\frac{3}{7}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 3332 }(225, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(-1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) |
sage: chi.jacobi_sum(n)