from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3328, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,11,8]))
pari: [g,chi] = znchar(Mod(1737,3328))
Basic properties
| Modulus: | \(3328\) | |
| Conductor: | \(1664\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1664}(1373,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3328.ct
\(\chi_{3328}(57,\cdot)\) \(\chi_{3328}(73,\cdot)\) \(\chi_{3328}(473,\cdot)\) \(\chi_{3328}(489,\cdot)\) \(\chi_{3328}(889,\cdot)\) \(\chi_{3328}(905,\cdot)\) \(\chi_{3328}(1305,\cdot)\) \(\chi_{3328}(1321,\cdot)\) \(\chi_{3328}(1721,\cdot)\) \(\chi_{3328}(1737,\cdot)\) \(\chi_{3328}(2137,\cdot)\) \(\chi_{3328}(2153,\cdot)\) \(\chi_{3328}(2553,\cdot)\) \(\chi_{3328}(2569,\cdot)\) \(\chi_{3328}(2969,\cdot)\) \(\chi_{3328}(2985,\cdot)\)
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
\((1535,261,769)\) → \((1,e\left(\frac{11}{32}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 3328 }(1737, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) |
sage: chi.jacobi_sum(n)