from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1050, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,24,25]))
pari: [g,chi] = znchar(Mod(61,1050))
Basic properties
| Modulus: | \(1050\) | |
| Conductor: | \(175\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{175}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1050.bq
\(\chi_{1050}(31,\cdot)\) \(\chi_{1050}(61,\cdot)\) \(\chi_{1050}(241,\cdot)\) \(\chi_{1050}(271,\cdot)\) \(\chi_{1050}(481,\cdot)\) \(\chi_{1050}(661,\cdot)\) \(\chi_{1050}(691,\cdot)\) \(\chi_{1050}(871,\cdot)\)
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: | 30.0.4764432829290274543179606325793429277837276458740234375.1 |
Values on generators
\((701,127,451)\) → \((1,e\left(\frac{4}{5}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1050 }(61, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)