from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1024, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,15]))
pari: [g,chi] = znchar(Mod(33,1024))
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 \(\chi_{128}(13,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1024.k
\(\chi_{1024}(33,\cdot)\) \(\chi_{1024}(97,\cdot)\) \(\chi_{1024}(161,\cdot)\) \(\chi_{1024}(225,\cdot)\) \(\chi_{1024}(289,\cdot)\) \(\chi_{1024}(353,\cdot)\) \(\chi_{1024}(417,\cdot)\) \(\chi_{1024}(481,\cdot)\) \(\chi_{1024}(545,\cdot)\) \(\chi_{1024}(609,\cdot)\) \(\chi_{1024}(673,\cdot)\) \(\chi_{1024}(737,\cdot)\) \(\chi_{1024}(801,\cdot)\) \(\chi_{1024}(865,\cdot)\) \(\chi_{1024}(929,\cdot)\) \(\chi_{1024}(993,\cdot)\)
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: | \(\Q(\zeta_{128})^+\) |
Values on generators
\((1023,5)\) → \((1,e\left(\frac{15}{32}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1024 }(33, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) |
sage: chi.jacobi_sum(n)