from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4000, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,15,14]))
pari: [g,chi] = znchar(Mod(2493,4000))
Basic properties
| Modulus: | \(4000\) | |
| Conductor: | \(800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{800}(253,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4000.bx
\(\chi_{4000}(93,\cdot)\) \(\chi_{4000}(357,\cdot)\) \(\chi_{4000}(493,\cdot)\) \(\chi_{4000}(757,\cdot)\) \(\chi_{4000}(893,\cdot)\) \(\chi_{4000}(1157,\cdot)\) \(\chi_{4000}(1293,\cdot)\) \(\chi_{4000}(1957,\cdot)\) \(\chi_{4000}(2093,\cdot)\) \(\chi_{4000}(2357,\cdot)\) \(\chi_{4000}(2493,\cdot)\) \(\chi_{4000}(2757,\cdot)\) \(\chi_{4000}(2893,\cdot)\) \(\chi_{4000}(3157,\cdot)\) \(\chi_{4000}(3293,\cdot)\) \(\chi_{4000}(3957,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.0.386856262276681335905976320000000000000000000000000000000000000000000000000000000000000000000000.1 |
Values on generators
\((2751,2501,1377)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{7}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 4000 }(2493, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(-1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{29}{40}\right)\) |
sage: chi.jacobi_sum(n)