from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1640, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,20,10,13]))
pari: [g,chi] = znchar(Mod(147,1640))
Basic properties
| Modulus: | \(1640\) | |
| Conductor: | \(1640\) | 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 1640.dw
\(\chi_{1640}(67,\cdot)\) \(\chi_{1640}(147,\cdot)\) \(\chi_{1640}(227,\cdot)\) \(\chi_{1640}(347,\cdot)\) \(\chi_{1640}(403,\cdot)\) \(\chi_{1640}(427,\cdot)\) \(\chi_{1640}(507,\cdot)\) \(\chi_{1640}(563,\cdot)\) \(\chi_{1640}(603,\cdot)\) \(\chi_{1640}(643,\cdot)\) \(\chi_{1640}(867,\cdot)\) \(\chi_{1640}(1243,\cdot)\) \(\chi_{1640}(1283,\cdot)\) \(\chi_{1640}(1323,\cdot)\) \(\chi_{1640}(1347,\cdot)\) \(\chi_{1640}(1483,\cdot)\)
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: | 40.0.850100520307178141323742388002497873911873722563896105590771728372465664000000000000000000000000000000.1 |
Values on generators
\((1231,821,657,1441)\) → \((-1,-1,i,e\left(\frac{13}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1640 }(147, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(i\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) |
sage: chi.jacobi_sum(n)