from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9680, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,10,5,16]))
pari: [g,chi] = znchar(Mod(487,9680))
Basic properties
| Modulus: | \(9680\) | |
| Conductor: | \(440\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{440}(267,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9680.cr
\(\chi_{9680}(487,\cdot)\) \(\chi_{9680}(807,\cdot)\) \(\chi_{9680}(1703,\cdot)\) \(\chi_{9680}(2423,\cdot)\) \(\chi_{9680}(2743,\cdot)\) \(\chi_{9680}(7287,\cdot)\) \(\chi_{9680}(9223,\cdot)\) \(\chi_{9680}(9447,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((3631,2421,1937,4721)\) → \((-1,-1,i,e\left(\frac{4}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 9680 }(487, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(-1\) | \(i\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) |
sage: chi.jacobi_sum(n)