from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,42,10]))
pari: [g,chi] = znchar(Mod(59,2800))
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.ft
\(\chi_{2800}(19,\cdot)\) \(\chi_{2800}(59,\cdot)\) \(\chi_{2800}(339,\cdot)\) \(\chi_{2800}(579,\cdot)\) \(\chi_{2800}(619,\cdot)\) \(\chi_{2800}(859,\cdot)\) \(\chi_{2800}(1139,\cdot)\) \(\chi_{2800}(1179,\cdot)\) \(\chi_{2800}(1419,\cdot)\) \(\chi_{2800}(1459,\cdot)\) \(\chi_{2800}(1739,\cdot)\) \(\chi_{2800}(1979,\cdot)\) \(\chi_{2800}(2019,\cdot)\) \(\chi_{2800}(2259,\cdot)\) \(\chi_{2800}(2539,\cdot)\) \(\chi_{2800}(2579,\cdot)\)
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
\((351,2101,2577,801)\) → \((-1,i,e\left(\frac{7}{10}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 2800 }(59, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{15}\right)\) |
sage: chi.jacobi_sum(n)