from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,25,6]))
pari: [g,chi] = znchar(Mod(58,2015))
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.gb
\(\chi_{2015}(58,\cdot)\) \(\chi_{2015}(232,\cdot)\) \(\chi_{2015}(418,\cdot)\) \(\chi_{2015}(492,\cdot)\) \(\chi_{2015}(643,\cdot)\) \(\chi_{2015}(678,\cdot)\) \(\chi_{2015}(773,\cdot)\) \(\chi_{2015}(852,\cdot)\) \(\chi_{2015}(1077,\cdot)\) \(\chi_{2015}(1112,\cdot)\) \(\chi_{2015}(1207,\cdot)\) \(\chi_{2015}(1263,\cdot)\) \(\chi_{2015}(1393,\cdot)\) \(\chi_{2015}(1697,\cdot)\) \(\chi_{2015}(1813,\cdot)\) \(\chi_{2015}(1827,\cdot)\)
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
\((807,1861,716)\) → \((-i,e\left(\frac{5}{12}\right),e\left(\frac{1}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
\( \chi_{ 2015 }(58, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) |
sage: chi.jacobi_sum(n)