from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2008, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,0,21]))
pari: [g,chi] = znchar(Mod(1657,2008))
Basic properties
Modulus: | \(2008\) | |
Conductor: | \(251\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{251}(151,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2008.s
\(\chi_{2008}(377,\cdot)\) \(\chi_{2008}(433,\cdot)\) \(\chi_{2008}(497,\cdot)\) \(\chi_{2008}(673,\cdot)\) \(\chi_{2008}(689,\cdot)\) \(\chi_{2008}(737,\cdot)\) \(\chi_{2008}(761,\cdot)\) \(\chi_{2008}(793,\cdot)\) \(\chi_{2008}(881,\cdot)\) \(\chi_{2008}(913,\cdot)\) \(\chi_{2008}(953,\cdot)\) \(\chi_{2008}(1161,\cdot)\) \(\chi_{2008}(1257,\cdot)\) \(\chi_{2008}(1265,\cdot)\) \(\chi_{2008}(1305,\cdot)\) \(\chi_{2008}(1481,\cdot)\) \(\chi_{2008}(1553,\cdot)\) \(\chi_{2008}(1657,\cdot)\) \(\chi_{2008}(1753,\cdot)\) \(\chi_{2008}(1945,\cdot)\)
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
\((503,1005,257)\) → \((1,1,e\left(\frac{21}{50}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 2008 }(1657, a) \) | \(-1\) | \(1\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) |
sage: chi.jacobi_sum(n)