from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4016, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,25,21]))
pari: [g,chi] = znchar(Mod(151,4016))
Basic properties
| Modulus: | \(4016\) | |
| Conductor: | \(2008\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2008}(1155,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4016.bc
\(\chi_{4016}(151,\cdot)\) \(\chi_{4016}(247,\cdot)\) \(\chi_{4016}(439,\cdot)\) \(\chi_{4016}(935,\cdot)\) \(\chi_{4016}(999,\cdot)\) \(\chi_{4016}(1175,\cdot)\) \(\chi_{4016}(1191,\cdot)\) \(\chi_{4016}(1239,\cdot)\) \(\chi_{4016}(1383,\cdot)\) \(\chi_{4016}(1415,\cdot)\) \(\chi_{4016}(1767,\cdot)\) \(\chi_{4016}(2055,\cdot)\) \(\chi_{4016}(2887,\cdot)\) \(\chi_{4016}(3271,\cdot)\) \(\chi_{4016}(3303,\cdot)\) \(\chi_{4016}(3463,\cdot)\) \(\chi_{4016}(3671,\cdot)\) \(\chi_{4016}(3767,\cdot)\) \(\chi_{4016}(3815,\cdot)\) \(\chi_{4016}(3991,\cdot)\)
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
\((2511,3013,257)\) → \((-1,-1,e\left(\frac{21}{50}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 4016 }(151, a) \) | \(1\) | \(1\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{19}{50}\right)\) |
sage: chi.jacobi_sum(n)