from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1600, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,5,16]))
pari: [g,chi] = znchar(Mod(231,1600))
Basic properties
| Modulus: | \(1600\) | |
| 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 \(\chi_{800}(731,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1600.cj
\(\chi_{1600}(71,\cdot)\) \(\chi_{1600}(231,\cdot)\) \(\chi_{1600}(311,\cdot)\) \(\chi_{1600}(391,\cdot)\) \(\chi_{1600}(471,\cdot)\) \(\chi_{1600}(631,\cdot)\) \(\chi_{1600}(711,\cdot)\) \(\chi_{1600}(791,\cdot)\) \(\chi_{1600}(871,\cdot)\) \(\chi_{1600}(1031,\cdot)\) \(\chi_{1600}(1111,\cdot)\) \(\chi_{1600}(1191,\cdot)\) \(\chi_{1600}(1271,\cdot)\) \(\chi_{1600}(1431,\cdot)\) \(\chi_{1600}(1511,\cdot)\) \(\chi_{1600}(1591,\cdot)\)
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
\((1151,901,577)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{2}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1600 }(231, a) \) | \(-1\) | \(1\) | \(e\left(\frac{27}{40}\right)\) | \(-i\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) |
sage: chi.jacobi_sum(n)