from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4030, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,25,8]))
pari: [g,chi] = znchar(Mod(19,4030))
Basic properties
Modulus: | \(4030\) | |
Conductor: | \(2015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2015}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4030.hr
\(\chi_{4030}(19,\cdot)\) \(\chi_{4030}(59,\cdot)\) \(\chi_{4030}(609,\cdot)\) \(\chi_{4030}(1099,\cdot)\) \(\chi_{4030}(1289,\cdot)\) \(\chi_{4030}(1619,\cdot)\) \(\chi_{4030}(1879,\cdot)\) \(\chi_{4030}(1909,\cdot)\) \(\chi_{4030}(2229,\cdot)\) \(\chi_{4030}(2459,\cdot)\) \(\chi_{4030}(3079,\cdot)\) \(\chi_{4030}(3109,\cdot)\) \(\chi_{4030}(3269,\cdot)\) \(\chi_{4030}(3729,\cdot)\) \(\chi_{4030}(3789,\cdot)\) \(\chi_{4030}(4019,\cdot)\)
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
\((807,1861,2731)\) → \((-1,e\left(\frac{5}{12}\right),e\left(\frac{2}{15}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 4030 }(19, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) |
sage: chi.jacobi_sum(n)