from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2100, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,15,12,25]))
pari: [g,chi] = znchar(Mod(131,2100))
Basic properties
| Modulus: | \(2100\) | |
| Conductor: | \(2100\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2100.dj
\(\chi_{2100}(131,\cdot)\) \(\chi_{2100}(311,\cdot)\) \(\chi_{2100}(731,\cdot)\) \(\chi_{2100}(971,\cdot)\) \(\chi_{2100}(1391,\cdot)\) \(\chi_{2100}(1571,\cdot)\) \(\chi_{2100}(1811,\cdot)\) \(\chi_{2100}(1991,\cdot)\)
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: | Number field defined by a degree 30 polynomial |
Values on generators
\((1051,701,1177,1501)\) → \((-1,-1,e\left(\frac{2}{5}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 2100 }(131, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)