from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3328, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,3,24]))
pari: [g,chi] = znchar(Mod(265,3328))
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 \(\chi_{1664}(1149,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3328.cm
\(\chi_{3328}(265,\cdot)\) \(\chi_{3328}(281,\cdot)\) \(\chi_{3328}(681,\cdot)\) \(\chi_{3328}(697,\cdot)\) \(\chi_{3328}(1097,\cdot)\) \(\chi_{3328}(1113,\cdot)\) \(\chi_{3328}(1513,\cdot)\) \(\chi_{3328}(1529,\cdot)\) \(\chi_{3328}(1929,\cdot)\) \(\chi_{3328}(1945,\cdot)\) \(\chi_{3328}(2345,\cdot)\) \(\chi_{3328}(2361,\cdot)\) \(\chi_{3328}(2761,\cdot)\) \(\chi_{3328}(2777,\cdot)\) \(\chi_{3328}(3177,\cdot)\) \(\chi_{3328}(3193,\cdot)\)
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: | 32.0.1703607828843094180875218713781072594708784688655057584678292529110044142123224137728.1 |
Values on generators
\((1535,261,769)\) → \((1,e\left(\frac{3}{32}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 3328 }(265, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) |
sage: chi.jacobi_sum(n)