from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,0,11]))
pari: [g,chi] = znchar(Mod(664,2015))
Basic properties
| Modulus: | \(2015\) | |
| Conductor: | \(155\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{155}(44,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2015.fv
\(\chi_{2015}(79,\cdot)\) \(\chi_{2015}(664,\cdot)\) \(\chi_{2015}(859,\cdot)\) \(\chi_{2015}(1119,\cdot)\) \(\chi_{2015}(1314,\cdot)\) \(\chi_{2015}(1509,\cdot)\) \(\chi_{2015}(1574,\cdot)\) \(\chi_{2015}(1964,\cdot)\)
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: | 30.0.542049797152523742060051576582522667353340789794921875.1 |
Values on generators
\((807,1861,716)\) → \((-1,1,e\left(\frac{11}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 2015 }(664, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) |
sage: chi.jacobi_sum(n)