from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9900, base_ring=CyclotomicField(10))
M = H._module
chi = DirichletCharacter(H, M([0,5,3,0]))
pari: [g,chi] = znchar(Mod(89,9900))
Basic properties
| Modulus: | \(9900\) | |
| Conductor: | \(75\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(10\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{75}(14,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9900.by
\(\chi_{9900}(89,\cdot)\) \(\chi_{9900}(2069,\cdot)\) \(\chi_{9900}(6029,\cdot)\) \(\chi_{9900}(8009,\cdot)\)
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: | \(\Q(\zeta_{5})\) |
| Fixed field: | 10.0.185394287109375.1 |
Values on generators
\((4951,5501,2377,4501)\) → \((1,-1,e\left(\frac{3}{10}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 9900 }(89, a) \) | \(-1\) | \(1\) | \(-1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)