from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1900, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,3,10]))
pari: [g,chi] = znchar(Mod(27,1900))
Basic properties
| Modulus: | \(1900\) | |
| Conductor: | \(1900\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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 1900.ch
\(\chi_{1900}(27,\cdot)\) \(\chi_{1900}(103,\cdot)\) \(\chi_{1900}(183,\cdot)\) \(\chi_{1900}(483,\cdot)\) \(\chi_{1900}(487,\cdot)\) \(\chi_{1900}(563,\cdot)\) \(\chi_{1900}(787,\cdot)\) \(\chi_{1900}(863,\cdot)\) \(\chi_{1900}(867,\cdot)\) \(\chi_{1900}(1167,\cdot)\) \(\chi_{1900}(1247,\cdot)\) \(\chi_{1900}(1323,\cdot)\) \(\chi_{1900}(1547,\cdot)\) \(\chi_{1900}(1623,\cdot)\) \(\chi_{1900}(1627,\cdot)\) \(\chi_{1900}(1703,\cdot)\)
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
\((951,77,401)\) → \((-1,e\left(\frac{1}{20}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1900 }(27, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(-i\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{14}{15}\right)\) |
sage: chi.jacobi_sum(n)