from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3840, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,5,0,8]))
pari: [g,chi] = znchar(Mod(7,3840))
Basic properties
| Modulus: | \(3840\) | |
| 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 \(\chi_{640}(587,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3840.cx
\(\chi_{3840}(7,\cdot)\) \(\chi_{3840}(343,\cdot)\) \(\chi_{3840}(487,\cdot)\) \(\chi_{3840}(823,\cdot)\) \(\chi_{3840}(967,\cdot)\) \(\chi_{3840}(1303,\cdot)\) \(\chi_{3840}(1447,\cdot)\) \(\chi_{3840}(1783,\cdot)\) \(\chi_{3840}(1927,\cdot)\) \(\chi_{3840}(2263,\cdot)\) \(\chi_{3840}(2407,\cdot)\) \(\chi_{3840}(2743,\cdot)\) \(\chi_{3840}(2887,\cdot)\) \(\chi_{3840}(3223,\cdot)\) \(\chi_{3840}(3367,\cdot)\) \(\chi_{3840}(3703,\cdot)\)
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: | 32.32.187072209578355573530071658587684226515959365500928000000000000000000000000.2 |
Values on generators
\((511,2821,2561,1537)\) → \((-1,e\left(\frac{5}{32}\right),1,i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3840 }(7, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(-i\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) |
sage: chi.jacobi_sum(n)