from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,25,11]))
pari: [g,chi] = znchar(Mod(881,2015))
Basic properties
| Modulus: | \(2015\) | |
| Conductor: | \(403\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{403}(75,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2015.fh
\(\chi_{2015}(166,\cdot)\) \(\chi_{2015}(251,\cdot)\) \(\chi_{2015}(446,\cdot)\) \(\chi_{2015}(706,\cdot)\) \(\chi_{2015}(881,\cdot)\) \(\chi_{2015}(1076,\cdot)\) \(\chi_{2015}(1226,\cdot)\) \(\chi_{2015}(1726,\cdot)\)
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: | 30.0.125335162623872041511671899825337780820428323909681753896697202993418803.2 |
Values on generators
\((807,1861,716)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{11}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 2015 }(881, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) |
sage: chi.jacobi_sum(n)