from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1280, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,3,16]))
pari: [g,chi] = znchar(Mod(9,1280))
Basic properties
| Modulus: | \(1280\) | |
| Conductor: | \(640\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{640}(509,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1280.bo
\(\chi_{1280}(9,\cdot)\) \(\chi_{1280}(89,\cdot)\) \(\chi_{1280}(169,\cdot)\) \(\chi_{1280}(249,\cdot)\) \(\chi_{1280}(329,\cdot)\) \(\chi_{1280}(409,\cdot)\) \(\chi_{1280}(489,\cdot)\) \(\chi_{1280}(569,\cdot)\) \(\chi_{1280}(649,\cdot)\) \(\chi_{1280}(729,\cdot)\) \(\chi_{1280}(809,\cdot)\) \(\chi_{1280}(889,\cdot)\) \(\chi_{1280}(969,\cdot)\) \(\chi_{1280}(1049,\cdot)\) \(\chi_{1280}(1129,\cdot)\) \(\chi_{1280}(1209,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | 32.32.478904856520590268236983445984471619880855975682375680000000000000000.1 |
Values on generators
\((511,261,257)\) → \((1,e\left(\frac{3}{32}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1280 }(9, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{32}\right)\) |
sage: chi.jacobi_sum(n)