![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1740, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,14,14,9]))
![Copy content]()
pari:[g,chi] = znchar(Mod(599,1740))
| Modulus: | \(1740\) | |
| Conductor: | \(1740\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(28\) |
![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: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
pari:zncharisodd(g,chi)
|
\(\chi_{1740}(119,\cdot)\)
\(\chi_{1740}(359,\cdot)\)
\(\chi_{1740}(479,\cdot)\)
\(\chi_{1740}(599,\cdot)\)
\(\chi_{1740}(659,\cdot)\)
\(\chi_{1740}(839,\cdot)\)
\(\chi_{1740}(959,\cdot)\)
\(\chi_{1740}(1139,\cdot)\)
\(\chi_{1740}(1199,\cdot)\)
\(\chi_{1740}(1319,\cdot)\)
\(\chi_{1740}(1439,\cdot)\)
\(\chi_{1740}(1679,\cdot)\)
![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 ]
\((871,581,697,901)\) → \((-1,-1,-1,e\left(\frac{9}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1740 }(599, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(-i\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(-i\) | \(e\left(\frac{5}{28}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
