from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,15,10,9]))
pari: [g,chi] = znchar(Mod(2839,3600))
Basic properties
Modulus: | \(3600\) | |
Conductor: | \(1800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1800}(139,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3600.ex
\(\chi_{3600}(439,\cdot)\) \(\chi_{3600}(679,\cdot)\) \(\chi_{3600}(1159,\cdot)\) \(\chi_{3600}(1879,\cdot)\) \(\chi_{3600}(2119,\cdot)\) \(\chi_{3600}(2839,\cdot)\) \(\chi_{3600}(3319,\cdot)\) \(\chi_{3600}(3559,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((3151,901,2801,577)\) → \((-1,-1,e\left(\frac{1}{3}\right),e\left(\frac{3}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 3600 }(2839, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) |
sage: chi.jacobi_sum(n)