from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3100, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,27,26]))
pari: [g,chi] = znchar(Mod(2287,3100))
Basic properties
Modulus: | \(3100\) | |
Conductor: | \(3100\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3100.gi
\(\chi_{3100}(327,\cdot)\) \(\chi_{3100}(703,\cdot)\) \(\chi_{3100}(823,\cdot)\) \(\chi_{3100}(1067,\cdot)\) \(\chi_{3100}(1367,\cdot)\) \(\chi_{3100}(1447,\cdot)\) \(\chi_{3100}(1727,\cdot)\) \(\chi_{3100}(1903,\cdot)\) \(\chi_{3100}(2163,\cdot)\) \(\chi_{3100}(2183,\cdot)\) \(\chi_{3100}(2223,\cdot)\) \(\chi_{3100}(2287,\cdot)\) \(\chi_{3100}(2483,\cdot)\) \(\chi_{3100}(2647,\cdot)\) \(\chi_{3100}(2863,\cdot)\) \(\chi_{3100}(2987,\cdot)\)
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
\((1551,2977,1801)\) → \((-1,e\left(\frac{9}{20}\right),e\left(\frac{13}{30}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 3100 }(2287, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)