from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2013, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,54,43]))
pari: [g,chi] = znchar(Mod(1568,2013))
Basic properties
Modulus: | \(2013\) | |
Conductor: | \(2013\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2013.fz
\(\chi_{2013}(200,\cdot)\) \(\chi_{2013}(299,\cdot)\) \(\chi_{2013}(359,\cdot)\) \(\chi_{2013}(458,\cdot)\) \(\chi_{2013}(734,\cdot)\) \(\chi_{2013}(767,\cdot)\) \(\chi_{2013}(860,\cdot)\) \(\chi_{2013}(959,\cdot)\) \(\chi_{2013}(1238,\cdot)\) \(\chi_{2013}(1250,\cdot)\) \(\chi_{2013}(1271,\cdot)\) \(\chi_{2013}(1349,\cdot)\) \(\chi_{2013}(1535,\cdot)\) \(\chi_{2013}(1568,\cdot)\) \(\chi_{2013}(1856,\cdot)\) \(\chi_{2013}(1889,\cdot)\)
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
\((1343,1465,1222)\) → \((-1,e\left(\frac{9}{10}\right),e\left(\frac{43}{60}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
\( \chi_{ 2013 }(1568, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) |
sage: chi.jacobi_sum(n)