from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4015, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,36,35]))
pari: [g,chi] = znchar(Mod(581,4015))
Basic properties
| Modulus: | \(4015\) | |
| Conductor: | \(803\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{803}(581,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4015.eu
\(\chi_{4015}(581,\cdot)\) \(\chi_{4015}(806,\cdot)\) \(\chi_{4015}(1071,\cdot)\) \(\chi_{4015}(1171,\cdot)\) \(\chi_{4015}(1411,\cdot)\) \(\chi_{4015}(1676,\cdot)\) \(\chi_{4015}(1776,\cdot)\) \(\chi_{4015}(1901,\cdot)\) \(\chi_{4015}(2506,\cdot)\) \(\chi_{4015}(2896,\cdot)\) \(\chi_{4015}(2996,\cdot)\) \(\chi_{4015}(3261,\cdot)\) \(\chi_{4015}(3501,\cdot)\) \(\chi_{4015}(3601,\cdot)\) \(\chi_{4015}(3866,\cdot)\) \(\chi_{4015}(3991,\cdot)\)
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
\((1607,2191,881)\) → \((1,e\left(\frac{3}{5}\right),e\left(\frac{7}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 4015 }(581, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) |
sage: chi.jacobi_sum(n)