from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1716, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,0,12,25]))
pari: [g,chi] = znchar(Mod(49,1716))
Basic properties
| Modulus: | \(1716\) | |
| Conductor: | \(143\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{143}(49,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1716.de
\(\chi_{1716}(49,\cdot)\) \(\chi_{1716}(361,\cdot)\) \(\chi_{1716}(433,\cdot)\) \(\chi_{1716}(829,\cdot)\) \(\chi_{1716}(1213,\cdot)\) \(\chi_{1716}(1369,\cdot)\) \(\chi_{1716}(1609,\cdot)\) \(\chi_{1716}(1681,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.30.69503752297329754905479727341904896738456941915804813.1 |
Values on generators
\((859,1145,937,925)\) → \((1,1,e\left(\frac{2}{5}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 1716 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) |
sage: chi.jacobi_sum(n)