from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1920, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,27,16,16]))
pari: [g,chi] = znchar(Mod(29,1920))
Basic properties
| Modulus: | \(1920\) | |
| Conductor: | \(1920\) | 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 1920.db
\(\chi_{1920}(29,\cdot)\) \(\chi_{1920}(149,\cdot)\) \(\chi_{1920}(269,\cdot)\) \(\chi_{1920}(389,\cdot)\) \(\chi_{1920}(509,\cdot)\) \(\chi_{1920}(629,\cdot)\) \(\chi_{1920}(749,\cdot)\) \(\chi_{1920}(869,\cdot)\) \(\chi_{1920}(989,\cdot)\) \(\chi_{1920}(1109,\cdot)\) \(\chi_{1920}(1229,\cdot)\) \(\chi_{1920}(1349,\cdot)\) \(\chi_{1920}(1469,\cdot)\) \(\chi_{1920}(1589,\cdot)\) \(\chi_{1920}(1709,\cdot)\) \(\chi_{1920}(1829,\cdot)\)
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: | 32.0.20615283744186880032112588280912120153429260426382010514145280000000000000000.1 |
Values on generators
\((511,901,641,1537)\) → \((1,e\left(\frac{27}{32}\right),-1,-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1920 }(29, a) \) | \(-1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(-i\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) |
sage: chi.jacobi_sum(n)