from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1920, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,21,16,8]))
pari: [g,chi] = znchar(Mod(1547,1920))
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.cv
\(\chi_{1920}(83,\cdot)\) \(\chi_{1920}(107,\cdot)\) \(\chi_{1920}(323,\cdot)\) \(\chi_{1920}(347,\cdot)\) \(\chi_{1920}(563,\cdot)\) \(\chi_{1920}(587,\cdot)\) \(\chi_{1920}(803,\cdot)\) \(\chi_{1920}(827,\cdot)\) \(\chi_{1920}(1043,\cdot)\) \(\chi_{1920}(1067,\cdot)\) \(\chi_{1920}(1283,\cdot)\) \(\chi_{1920}(1307,\cdot)\) \(\chi_{1920}(1523,\cdot)\) \(\chi_{1920}(1547,\cdot)\) \(\chi_{1920}(1763,\cdot)\) \(\chi_{1920}(1787,\cdot)\)
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: | 32.0.8052845212573000012543979797231296934933304854055472857088000000000000000000000000.1 |
Values on generators
\((511,901,641,1537)\) → \((-1,e\left(\frac{21}{32}\right),-1,i)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 1920 }(1547, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(-i\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) |
sage: chi.jacobi_sum(n)