from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9900, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,40,3,48]))
pari: [g,chi] = znchar(Mod(4777,9900))
Basic properties
| Modulus: | \(9900\) | |
| Conductor: | \(2475\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2475}(2302,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9900.mf
\(\chi_{9900}(97,\cdot)\) \(\chi_{9900}(1633,\cdot)\) \(\chi_{9900}(3217,\cdot)\) \(\chi_{9900}(3337,\cdot)\) \(\chi_{9900}(3397,\cdot)\) \(\chi_{9900}(4453,\cdot)\) \(\chi_{9900}(4777,\cdot)\) \(\chi_{9900}(5173,\cdot)\) \(\chi_{9900}(5713,\cdot)\) \(\chi_{9900}(6637,\cdot)\) \(\chi_{9900}(7753,\cdot)\) \(\chi_{9900}(8077,\cdot)\) \(\chi_{9900}(8233,\cdot)\) \(\chi_{9900}(8473,\cdot)\) \(\chi_{9900}(9013,\cdot)\) \(\chi_{9900}(9817,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((4951,5501,2377,4501)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{1}{20}\right),e\left(\frac{4}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 9900 }(4777, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)