![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1860, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,15,15,1]))
![Copy content]()
pari:[g,chi] = znchar(Mod(1739,1860))
| Modulus: | \(1860\) | |
| Conductor: | \(1860\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(30\) |
![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)
|
\(\chi_{1860}(179,\cdot)\)
\(\chi_{1860}(239,\cdot)\)
\(\chi_{1860}(539,\cdot)\)
\(\chi_{1860}(1199,\cdot)\)
\(\chi_{1860}(1319,\cdot)\)
\(\chi_{1860}(1439,\cdot)\)
\(\chi_{1860}(1499,\cdot)\)
\(\chi_{1860}(1739,\cdot)\)
![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 ]
\((931,1241,1117,1801)\) → \((-1,-1,-1,e\left(\frac{1}{30}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1860 }(1739, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
