from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1920, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,13,8,12]))
pari: [g,chi] = znchar(Mod(953,1920))
Basic properties
| Modulus: | \(1920\) | |
| Conductor: | \(960\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{960}(533,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1920.cr
\(\chi_{1920}(137,\cdot)\) \(\chi_{1920}(473,\cdot)\) \(\chi_{1920}(617,\cdot)\) \(\chi_{1920}(953,\cdot)\) \(\chi_{1920}(1097,\cdot)\) \(\chi_{1920}(1433,\cdot)\) \(\chi_{1920}(1577,\cdot)\) \(\chi_{1920}(1913,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
| Fixed field: | 16.16.968232702940866945220608000000000000.1 |
Values on generators
\((511,901,641,1537)\) → \((1,e\left(\frac{13}{16}\right),-1,-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1920 }(953, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(-1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) |
sage: chi.jacobi_sum(n)