from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2008, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,0,46]))
pari: [g,chi] = znchar(Mod(351,2008))
Basic properties
Modulus: | \(2008\) | |
Conductor: | \(1004\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1004}(351,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2008.t
\(\chi_{2008}(63,\cdot)\) \(\chi_{2008}(255,\cdot)\) \(\chi_{2008}(351,\cdot)\) \(\chi_{2008}(455,\cdot)\) \(\chi_{2008}(527,\cdot)\) \(\chi_{2008}(703,\cdot)\) \(\chi_{2008}(743,\cdot)\) \(\chi_{2008}(751,\cdot)\) \(\chi_{2008}(847,\cdot)\) \(\chi_{2008}(1055,\cdot)\) \(\chi_{2008}(1095,\cdot)\) \(\chi_{2008}(1127,\cdot)\) \(\chi_{2008}(1215,\cdot)\) \(\chi_{2008}(1247,\cdot)\) \(\chi_{2008}(1271,\cdot)\) \(\chi_{2008}(1319,\cdot)\) \(\chi_{2008}(1335,\cdot)\) \(\chi_{2008}(1511,\cdot)\) \(\chi_{2008}(1575,\cdot)\) \(\chi_{2008}(1631,\cdot)\)
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
\((503,1005,257)\) → \((-1,1,e\left(\frac{23}{25}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 2008 }(351, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) |
sage: chi.jacobi_sum(n)