from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7942, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([21,5]))
pari: [g,chi] = znchar(Mod(293,7942))
Basic properties
| Modulus: | \(7942\) | |
| Conductor: | \(209\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{209}(84,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7942.t
\(\chi_{7942}(293,\cdot)\) \(\chi_{7942}(1513,\cdot)\) \(\chi_{7942}(2235,\cdot)\) \(\chi_{7942}(2459,\cdot)\) \(\chi_{7942}(3181,\cdot)\) \(\chi_{7942}(5123,\cdot)\) \(\chi_{7942}(6069,\cdot)\) \(\chi_{7942}(7289,\cdot)\)
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: | 30.30.1220232317838205647399552173000517992590495082743137882605129.1 |
Values on generators
\((5777,6139)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 7942 }(293, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{14}{15}\right)\) |
sage: chi.jacobi_sum(n)