from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4000, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,35,32]))
pari: [g,chi] = znchar(Mod(51,4000))
Basic properties
| Modulus: | \(4000\) | |
| Conductor: | \(800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{800}(211,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4000.cc
\(\chi_{4000}(51,\cdot)\) \(\chi_{4000}(451,\cdot)\) \(\chi_{4000}(651,\cdot)\) \(\chi_{4000}(851,\cdot)\) \(\chi_{4000}(1051,\cdot)\) \(\chi_{4000}(1451,\cdot)\) \(\chi_{4000}(1651,\cdot)\) \(\chi_{4000}(1851,\cdot)\) \(\chi_{4000}(2051,\cdot)\) \(\chi_{4000}(2451,\cdot)\) \(\chi_{4000}(2651,\cdot)\) \(\chi_{4000}(2851,\cdot)\) \(\chi_{4000}(3051,\cdot)\) \(\chi_{4000}(3451,\cdot)\) \(\chi_{4000}(3651,\cdot)\) \(\chi_{4000}(3851,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((2751,2501,1377)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{4}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 4000 }(51, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{40}\right)\) | \(i\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) |
sage: chi.jacobi_sum(n)