from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1700, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,12,35]))
pari: [g,chi] = znchar(Mod(1039,1700))
Basic properties
| Modulus: | \(1700\) | |
| Conductor: | \(1700\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | 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 1700.cf
\(\chi_{1700}(19,\cdot)\) \(\chi_{1700}(59,\cdot)\) \(\chi_{1700}(179,\cdot)\) \(\chi_{1700}(219,\cdot)\) \(\chi_{1700}(359,\cdot)\) \(\chi_{1700}(519,\cdot)\) \(\chi_{1700}(559,\cdot)\) \(\chi_{1700}(739,\cdot)\) \(\chi_{1700}(859,\cdot)\) \(\chi_{1700}(1039,\cdot)\) \(\chi_{1700}(1079,\cdot)\) \(\chi_{1700}(1239,\cdot)\) \(\chi_{1700}(1379,\cdot)\) \(\chi_{1700}(1419,\cdot)\) \(\chi_{1700}(1539,\cdot)\) \(\chi_{1700}(1579,\cdot)\)
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: | 40.0.4333834970391607127648627101904836847739142552018165588378906250000000000000000000000000000000000000000.1 |
Values on generators
\((851,477,1601)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{7}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1700 }(1039, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) |
sage: chi.jacobi_sum(n)