from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6400, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,35,12]))
pari: [g,chi] = znchar(Mod(289,6400))
Basic properties
| Modulus: | \(6400\) | |
| 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 \(\chi_{800}(589,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6400.co
\(\chi_{6400}(289,\cdot)\) \(\chi_{6400}(609,\cdot)\) \(\chi_{6400}(929,\cdot)\) \(\chi_{6400}(1569,\cdot)\) \(\chi_{6400}(1889,\cdot)\) \(\chi_{6400}(2209,\cdot)\) \(\chi_{6400}(2529,\cdot)\) \(\chi_{6400}(3169,\cdot)\) \(\chi_{6400}(3489,\cdot)\) \(\chi_{6400}(3809,\cdot)\) \(\chi_{6400}(4129,\cdot)\) \(\chi_{6400}(4769,\cdot)\) \(\chi_{6400}(5089,\cdot)\) \(\chi_{6400}(5409,\cdot)\) \(\chi_{6400}(5729,\cdot)\) \(\chi_{6400}(6369,\cdot)\)
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: | 40.40.15474250491067253436239052800000000000000000000000000000000000000000000000000000000000000000000.1 |
Values on generators
\((4351,4101,5377)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{3}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 6400 }(289, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{40}\right)\) | \(i\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) |
sage: chi.jacobi_sum(n)