from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1920, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,3,0,16]))
pari: [g,chi] = znchar(Mod(1789,1920))
Basic properties
| Modulus: | \(1920\) | |
| Conductor: | \(640\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{640}(509,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1920.cz
\(\chi_{1920}(109,\cdot)\) \(\chi_{1920}(229,\cdot)\) \(\chi_{1920}(349,\cdot)\) \(\chi_{1920}(469,\cdot)\) \(\chi_{1920}(589,\cdot)\) \(\chi_{1920}(709,\cdot)\) \(\chi_{1920}(829,\cdot)\) \(\chi_{1920}(949,\cdot)\) \(\chi_{1920}(1069,\cdot)\) \(\chi_{1920}(1189,\cdot)\) \(\chi_{1920}(1309,\cdot)\) \(\chi_{1920}(1429,\cdot)\) \(\chi_{1920}(1549,\cdot)\) \(\chi_{1920}(1669,\cdot)\) \(\chi_{1920}(1789,\cdot)\) \(\chi_{1920}(1909,\cdot)\)
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: | 32.32.478904856520590268236983445984471619880855975682375680000000000000000.1 |
Values on generators
\((511,901,641,1537)\) → \((1,e\left(\frac{3}{32}\right),1,-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1920 }(1789, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(-i\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) |
sage: chi.jacobi_sum(n)