from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,35,38]))
pari: [g,chi] = znchar(Mod(973,2015))
Basic properties
Modulus: | \(2015\) | |
Conductor: | \(2015\) | 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 2015.hp
\(\chi_{2015}(228,\cdot)\) \(\chi_{2015}(292,\cdot)\) \(\chi_{2015}(327,\cdot)\) \(\chi_{2015}(487,\cdot)\) \(\chi_{2015}(778,\cdot)\) \(\chi_{2015}(943,\cdot)\) \(\chi_{2015}(973,\cdot)\) \(\chi_{2015}(1137,\cdot)\) \(\chi_{2015}(1138,\cdot)\) \(\chi_{2015}(1233,\cdot)\) \(\chi_{2015}(1367,\cdot)\) \(\chi_{2015}(1562,\cdot)\) \(\chi_{2015}(1592,\cdot)\) \(\chi_{2015}(1753,\cdot)\) \(\chi_{2015}(1788,\cdot)\) \(\chi_{2015}(1822,\cdot)\)
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
\((807,1861,716)\) → \((-i,e\left(\frac{7}{12}\right),e\left(\frac{19}{30}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
\( \chi_{ 2015 }(973, a) \) | \(-1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(-i\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) |
sage: chi.jacobi_sum(n)