from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2008, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,0,3]))
pari: [g,chi] = znchar(Mod(377,2008))
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 \(\chi_{251}(126,\cdot)\) | 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
\((503,1005,257)\) → \((1,1,e\left(\frac{3}{50}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2008 }(377, a) \) | \(-1\) | \(1\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{21}{25}\right)\) |
sage: chi.jacobi_sum(n)