![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3744, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,9,20,2]))
![Copy content]()
pari:[g,chi] = znchar(Mod(509,3744))
| Modulus: | \(3744\) | |
| Conductor: | \(3744\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(24\) |
![Copy content]()
sage:chi.multiplicative_order()
![Copy content]()
pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
pari:#znconreyconductor(g,chi)==1
|
| Minimal: | yes |
| Parity: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
pari:zncharisodd(g,chi)
|
\(\chi_{3744}(461,\cdot)\)
\(\chi_{3744}(509,\cdot)\)
\(\chi_{3744}(869,\cdot)\)
\(\chi_{3744}(1541,\cdot)\)
\(\chi_{3744}(2333,\cdot)\)
\(\chi_{3744}(2381,\cdot)\)
\(\chi_{3744}(2741,\cdot)\)
\(\chi_{3744}(3413,\cdot)\)
![Copy content]()
sage:chi.galois_orbit()
![Copy content]()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((703,2341,2081,2017)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{5}{6}\right),e\left(\frac{1}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3744 }(509, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{24}\right)\) | \(1\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
