from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,3,20]))
pari: [g,chi] = znchar(Mod(2627,2800))
Basic properties
| Modulus: | \(2800\) | |
| Conductor: | \(2800\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2800.fg
\(\chi_{2800}(67,\cdot)\) \(\chi_{2800}(123,\cdot)\) \(\chi_{2800}(387,\cdot)\) \(\chi_{2800}(627,\cdot)\) \(\chi_{2800}(683,\cdot)\) \(\chi_{2800}(947,\cdot)\) \(\chi_{2800}(1003,\cdot)\) \(\chi_{2800}(1187,\cdot)\) \(\chi_{2800}(1563,\cdot)\) \(\chi_{2800}(1747,\cdot)\) \(\chi_{2800}(1803,\cdot)\) \(\chi_{2800}(2067,\cdot)\) \(\chi_{2800}(2123,\cdot)\) \(\chi_{2800}(2363,\cdot)\) \(\chi_{2800}(2627,\cdot)\) \(\chi_{2800}(2683,\cdot)\)
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
\((351,2101,2577,801)\) → \((-1,-i,e\left(\frac{1}{20}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 2800 }(2627, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) |
sage: chi.jacobi_sum(n)