from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3840, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,21,0,0]))
pari: [g,chi] = znchar(Mod(121,3840))
Basic properties
| Modulus: | \(3840\) | |
| Conductor: | \(128\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{128}(117,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3840.dc
\(\chi_{3840}(121,\cdot)\) \(\chi_{3840}(361,\cdot)\) \(\chi_{3840}(601,\cdot)\) \(\chi_{3840}(841,\cdot)\) \(\chi_{3840}(1081,\cdot)\) \(\chi_{3840}(1321,\cdot)\) \(\chi_{3840}(1561,\cdot)\) \(\chi_{3840}(1801,\cdot)\) \(\chi_{3840}(2041,\cdot)\) \(\chi_{3840}(2281,\cdot)\) \(\chi_{3840}(2521,\cdot)\) \(\chi_{3840}(2761,\cdot)\) \(\chi_{3840}(3001,\cdot)\) \(\chi_{3840}(3241,\cdot)\) \(\chi_{3840}(3481,\cdot)\) \(\chi_{3840}(3721,\cdot)\)
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: | \(\Q(\zeta_{128})^+\) |
Values on generators
\((511,2821,2561,1537)\) → \((1,e\left(\frac{21}{32}\right),1,1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3840 }(121, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(i\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) |
sage: chi.jacobi_sum(n)