from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1664, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,29,24]))
pari: [g,chi] = znchar(Mod(213,1664))
Basic properties
Modulus: | \(1664\) | |
Conductor: | \(1664\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1664.ct
\(\chi_{1664}(5,\cdot)\) \(\chi_{1664}(125,\cdot)\) \(\chi_{1664}(213,\cdot)\) \(\chi_{1664}(333,\cdot)\) \(\chi_{1664}(421,\cdot)\) \(\chi_{1664}(541,\cdot)\) \(\chi_{1664}(629,\cdot)\) \(\chi_{1664}(749,\cdot)\) \(\chi_{1664}(837,\cdot)\) \(\chi_{1664}(957,\cdot)\) \(\chi_{1664}(1045,\cdot)\) \(\chi_{1664}(1165,\cdot)\) \(\chi_{1664}(1253,\cdot)\) \(\chi_{1664}(1373,\cdot)\) \(\chi_{1664}(1461,\cdot)\) \(\chi_{1664}(1581,\cdot)\)
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: | 32.0.1703607828843094180875218713781072594708784688655057584678292529110044142123224137728.2 |
Values on generators
\((1535,261,769)\) → \((1,e\left(\frac{29}{32}\right),-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
\( \chi_{ 1664 }(213, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) |
sage: chi.jacobi_sum(n)