from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1024, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,17]))
pari: [g,chi] = znchar(Mod(31,1024))
Basic properties
Modulus: | \(1024\) | |
Conductor: | \(128\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{128}(59,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1024.l
\(\chi_{1024}(31,\cdot)\) \(\chi_{1024}(95,\cdot)\) \(\chi_{1024}(159,\cdot)\) \(\chi_{1024}(223,\cdot)\) \(\chi_{1024}(287,\cdot)\) \(\chi_{1024}(351,\cdot)\) \(\chi_{1024}(415,\cdot)\) \(\chi_{1024}(479,\cdot)\) \(\chi_{1024}(543,\cdot)\) \(\chi_{1024}(607,\cdot)\) \(\chi_{1024}(671,\cdot)\) \(\chi_{1024}(735,\cdot)\) \(\chi_{1024}(799,\cdot)\) \(\chi_{1024}(863,\cdot)\) \(\chi_{1024}(927,\cdot)\) \(\chi_{1024}(991,\cdot)\)
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: | 32.0.3138550867693340381917894711603833208051177722232017256448.1 |
Values on generators
\((1023,5)\) → \((-1,e\left(\frac{17}{32}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 1024 }(31, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) |
sage: chi.jacobi_sum(n)