from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3840, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,19,16,8]))
pari: [g,chi] = znchar(Mod(137,3840))
Basic properties
Modulus: | \(3840\) | |
Conductor: | \(1920\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1920}(317,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3840.dj
\(\chi_{3840}(137,\cdot)\) \(\chi_{3840}(473,\cdot)\) \(\chi_{3840}(617,\cdot)\) \(\chi_{3840}(953,\cdot)\) \(\chi_{3840}(1097,\cdot)\) \(\chi_{3840}(1433,\cdot)\) \(\chi_{3840}(1577,\cdot)\) \(\chi_{3840}(1913,\cdot)\) \(\chi_{3840}(2057,\cdot)\) \(\chi_{3840}(2393,\cdot)\) \(\chi_{3840}(2537,\cdot)\) \(\chi_{3840}(2873,\cdot)\) \(\chi_{3840}(3017,\cdot)\) \(\chi_{3840}(3353,\cdot)\) \(\chi_{3840}(3497,\cdot)\) \(\chi_{3840}(3833,\cdot)\)
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: | 32.32.8052845212573000012543979797231296934933304854055472857088000000000000000000000000.2 |
Values on generators
\((511,2821,2561,1537)\) → \((1,e\left(\frac{19}{32}\right),-1,i)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 3840 }(137, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(-i\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) |
sage: chi.jacobi_sum(n)