from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9900, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,20,15,6]))
pari: [g,chi] = znchar(Mod(607,9900))
Basic properties
Modulus: | \(9900\) | |
Conductor: | \(1980\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1980}(607,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9900.mm
\(\chi_{9900}(7,\cdot)\) \(\chi_{9900}(607,\cdot)\) \(\chi_{9900}(943,\cdot)\) \(\chi_{9900}(1843,\cdot)\) \(\chi_{9900}(3307,\cdot)\) \(\chi_{9900}(3643,\cdot)\) \(\chi_{9900}(4243,\cdot)\) \(\chi_{9900}(4507,\cdot)\) \(\chi_{9900}(5143,\cdot)\) \(\chi_{9900}(5407,\cdot)\) \(\chi_{9900}(6343,\cdot)\) \(\chi_{9900}(6943,\cdot)\) \(\chi_{9900}(7207,\cdot)\) \(\chi_{9900}(7807,\cdot)\) \(\chi_{9900}(8707,\cdot)\) \(\chi_{9900}(9643,\cdot)\)
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
\((4951,5501,2377,4501)\) → \((-1,e\left(\frac{1}{3}\right),i,e\left(\frac{1}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 9900 }(607, a) \) | \(-1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)