from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4016, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,0,48]))
pari: [g,chi] = znchar(Mod(241,4016))
Basic properties
| Modulus: | \(4016\) | |
| Conductor: | \(251\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(25\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{251}(241,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4016.y
\(\chi_{4016}(241,\cdot)\) \(\chi_{4016}(593,\cdot)\) \(\chi_{4016}(625,\cdot)\) \(\chi_{4016}(769,\cdot)\) \(\chi_{4016}(817,\cdot)\) \(\chi_{4016}(833,\cdot)\) \(\chi_{4016}(1009,\cdot)\) \(\chi_{4016}(1073,\cdot)\) \(\chi_{4016}(1569,\cdot)\) \(\chi_{4016}(1761,\cdot)\) \(\chi_{4016}(1857,\cdot)\) \(\chi_{4016}(2033,\cdot)\) \(\chi_{4016}(2209,\cdot)\) \(\chi_{4016}(2257,\cdot)\) \(\chi_{4016}(2353,\cdot)\) \(\chi_{4016}(2561,\cdot)\) \(\chi_{4016}(2721,\cdot)\) \(\chi_{4016}(2753,\cdot)\) \(\chi_{4016}(3137,\cdot)\) \(\chi_{4016}(3969,\cdot)\)
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 25 polynomial |
Values on generators
\((2511,3013,257)\) → \((1,1,e\left(\frac{24}{25}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 4016 }(241, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) |
sage: chi.jacobi_sum(n)