from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3100, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,21,14]))
pari: [g,chi] = znchar(Mod(203,3100))
Basic properties
| Modulus: | \(3100\) | |
| Conductor: | \(3100\) | 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 3100.gd
\(\chi_{3100}(127,\cdot)\) \(\chi_{3100}(203,\cdot)\) \(\chi_{3100}(323,\cdot)\) \(\chi_{3100}(383,\cdot)\) \(\chi_{3100}(623,\cdot)\) \(\chi_{3100}(663,\cdot)\) \(\chi_{3100}(787,\cdot)\) \(\chi_{3100}(947,\cdot)\) \(\chi_{3100}(1603,\cdot)\) \(\chi_{3100}(1667,\cdot)\) \(\chi_{3100}(2347,\cdot)\) \(\chi_{3100}(2367,\cdot)\) \(\chi_{3100}(2563,\cdot)\) \(\chi_{3100}(2687,\cdot)\) \(\chi_{3100}(2783,\cdot)\) \(\chi_{3100}(2927,\cdot)\)
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
\((1551,2977,1801)\) → \((-1,e\left(\frac{7}{20}\right),e\left(\frac{7}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 3100 }(203, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) |
sage: chi.jacobi_sum(n)