from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2005, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,13]))
pari: [g,chi] = znchar(Mod(16,2005))
Basic properties
| Modulus: | \(2005\) | |
| Conductor: | \(401\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{401}(16,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2005.bh
\(\chi_{2005}(16,\cdot)\) \(\chi_{2005}(41,\cdot)\) \(\chi_{2005}(86,\cdot)\) \(\chi_{2005}(146,\cdot)\) \(\chi_{2005}(206,\cdot)\) \(\chi_{2005}(276,\cdot)\) \(\chi_{2005}(376,\cdot)\) \(\chi_{2005}(396,\cdot)\) \(\chi_{2005}(471,\cdot)\) \(\chi_{2005}(481,\cdot)\) \(\chi_{2005}(546,\cdot)\) \(\chi_{2005}(606,\cdot)\) \(\chi_{2005}(751,\cdot)\) \(\chi_{2005}(816,\cdot)\) \(\chi_{2005}(1126,\cdot)\) \(\chi_{2005}(1426,\cdot)\) \(\chi_{2005}(1431,\cdot)\) \(\chi_{2005}(1516,\cdot)\) \(\chi_{2005}(1541,\cdot)\) \(\chi_{2005}(1781,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
| Fixed field: | Number field defined by a degree 50 polynomial |
Values on generators
\((402,1206)\) → \((1,e\left(\frac{13}{50}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 2005 }(16, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{47}{50}\right)\) |
sage: chi.jacobi_sum(n)