from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1728, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,27,16]))
pari: [g,chi] = znchar(Mod(91,1728))
Basic properties
| Modulus: | \(1728\) | |
| Conductor: | \(576\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{576}(283,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1728.bw
\(\chi_{1728}(19,\cdot)\) \(\chi_{1728}(91,\cdot)\) \(\chi_{1728}(235,\cdot)\) \(\chi_{1728}(307,\cdot)\) \(\chi_{1728}(451,\cdot)\) \(\chi_{1728}(523,\cdot)\) \(\chi_{1728}(667,\cdot)\) \(\chi_{1728}(739,\cdot)\) \(\chi_{1728}(883,\cdot)\) \(\chi_{1728}(955,\cdot)\) \(\chi_{1728}(1099,\cdot)\) \(\chi_{1728}(1171,\cdot)\) \(\chi_{1728}(1315,\cdot)\) \(\chi_{1728}(1387,\cdot)\) \(\chi_{1728}(1531,\cdot)\) \(\chi_{1728}(1603,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((703,325,1217)\) → \((-1,e\left(\frac{9}{16}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1728 }(91, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(-i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)