from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3332, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,41,21]))
pari: [g,chi] = znchar(Mod(33,3332))
Basic properties
| Modulus: | \(3332\) | |
| Conductor: | \(833\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{833}(33,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3332.cd
\(\chi_{3332}(33,\cdot)\) \(\chi_{3332}(101,\cdot)\) \(\chi_{3332}(577,\cdot)\) \(\chi_{3332}(985,\cdot)\) \(\chi_{3332}(1053,\cdot)\) \(\chi_{3332}(1461,\cdot)\) \(\chi_{3332}(1529,\cdot)\) \(\chi_{3332}(1937,\cdot)\) \(\chi_{3332}(2005,\cdot)\) \(\chi_{3332}(2413,\cdot)\) \(\chi_{3332}(2889,\cdot)\) \(\chi_{3332}(2957,\cdot)\)
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: | 42.0.8165380551374267651486876056553316266087570018721628631541634242656760638483343855871696919.1 |
Values on generators
\((1667,885,785)\) → \((1,e\left(\frac{41}{42}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 3332 }(33, a) \) | \(-1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) |
sage: chi.jacobi_sum(n)