from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1664, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,13,16]))
pari: [g,chi] = znchar(Mod(363,1664))
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.co
\(\chi_{1664}(51,\cdot)\) \(\chi_{1664}(155,\cdot)\) \(\chi_{1664}(259,\cdot)\) \(\chi_{1664}(363,\cdot)\) \(\chi_{1664}(467,\cdot)\) \(\chi_{1664}(571,\cdot)\) \(\chi_{1664}(675,\cdot)\) \(\chi_{1664}(779,\cdot)\) \(\chi_{1664}(883,\cdot)\) \(\chi_{1664}(987,\cdot)\) \(\chi_{1664}(1091,\cdot)\) \(\chi_{1664}(1195,\cdot)\) \(\chi_{1664}(1299,\cdot)\) \(\chi_{1664}(1403,\cdot)\) \(\chi_{1664}(1507,\cdot)\) \(\chi_{1664}(1611,\cdot)\)
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: | 32.0.2088443876129429457733048543333054873029337200425307489036314630977400864768.1 |
Values on generators
\((1535,261,769)\) → \((-1,e\left(\frac{13}{32}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
\( \chi_{ 1664 }(363, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) |
sage: chi.jacobi_sum(n)