from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2013, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,3,20]))
pari: [g,chi] = znchar(Mod(13,2013))
Basic properties
| Modulus: | \(2013\) | |
| Conductor: | \(671\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{671}(13,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2013.ep
\(\chi_{2013}(13,\cdot)\) \(\chi_{2013}(535,\cdot)\) \(\chi_{2013}(745,\cdot)\) \(\chi_{2013}(1084,\cdot)\) \(\chi_{2013}(1267,\cdot)\) \(\chi_{2013}(1294,\cdot)\) \(\chi_{2013}(1843,\cdot)\) \(\chi_{2013}(1999,\cdot)\)
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: | 30.0.6671126861411628234277079565695510248853566661816916032716309571.1 |
Values on generators
\((1343,1465,1222)\) → \((1,e\left(\frac{1}{10}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 2013 }(13, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) |
sage: chi.jacobi_sum(n)