from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,39,10]))
pari: [g,chi] = znchar(Mod(217,3800))
Basic properties
| Modulus: | \(3800\) | |
| Conductor: | \(475\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{475}(217,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3800.ek
\(\chi_{3800}(217,\cdot)\) \(\chi_{3800}(297,\cdot)\) \(\chi_{3800}(673,\cdot)\) \(\chi_{3800}(753,\cdot)\) \(\chi_{3800}(977,\cdot)\) \(\chi_{3800}(1433,\cdot)\) \(\chi_{3800}(1513,\cdot)\) \(\chi_{3800}(1737,\cdot)\) \(\chi_{3800}(1817,\cdot)\) \(\chi_{3800}(2273,\cdot)\) \(\chi_{3800}(2497,\cdot)\) \(\chi_{3800}(2577,\cdot)\) \(\chi_{3800}(2953,\cdot)\) \(\chi_{3800}(3033,\cdot)\) \(\chi_{3800}(3337,\cdot)\) \(\chi_{3800}(3713,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((951,1901,1977,401)\) → \((1,1,e\left(\frac{13}{20}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 3800 }(217, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(i\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) |
sage: chi.jacobi_sum(n)