from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2013, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,4,13]))
pari: [g,chi] = znchar(Mod(37,2013))
Basic properties
Modulus: | \(2013\) | |
Conductor: | \(671\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{671}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2013.dp
\(\chi_{2013}(37,\cdot)\) \(\chi_{2013}(268,\cdot)\) \(\chi_{2013}(313,\cdot)\) \(\chi_{2013}(328,\cdot)\) \(\chi_{2013}(526,\cdot)\) \(\chi_{2013}(907,\cdot)\) \(\chi_{2013}(1675,\cdot)\) \(\chi_{2013}(1741,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | 20.0.383309715998561417281698106667019879208291343451301.3 |
Values on generators
\((1343,1465,1222)\) → \((1,e\left(\frac{1}{5}\right),e\left(\frac{13}{20}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
\( \chi_{ 2013 }(37, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(i\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) |
sage: chi.jacobi_sum(n)