from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1540, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,40,24]))
pari: [g,chi] = znchar(Mod(907,1540))
Basic properties
Modulus: | \(1540\) | |
Conductor: | \(1540\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1540.dr
\(\chi_{1540}(163,\cdot)\) \(\chi_{1540}(207,\cdot)\) \(\chi_{1540}(247,\cdot)\) \(\chi_{1540}(443,\cdot)\) \(\chi_{1540}(487,\cdot)\) \(\chi_{1540}(543,\cdot)\) \(\chi_{1540}(807,\cdot)\) \(\chi_{1540}(823,\cdot)\) \(\chi_{1540}(863,\cdot)\) \(\chi_{1540}(907,\cdot)\) \(\chi_{1540}(1087,\cdot)\) \(\chi_{1540}(1103,\cdot)\) \(\chi_{1540}(1367,\cdot)\) \(\chi_{1540}(1423,\cdot)\) \(\chi_{1540}(1467,\cdot)\) \(\chi_{1540}(1523,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((771,617,661,981)\) → \((-1,i,e\left(\frac{2}{3}\right),e\left(\frac{2}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 1540 }(907, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) |
sage: chi.jacobi_sum(n)