from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4100, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,4,37]))
pari: [g,chi] = znchar(Mod(179,4100))
Basic properties
| Modulus: | \(4100\) | |
| Conductor: | \(4100\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4100.hc
\(\chi_{4100}(179,\cdot)\) \(\chi_{4100}(539,\cdot)\) \(\chi_{4100}(919,\cdot)\) \(\chi_{4100}(1059,\cdot)\) \(\chi_{4100}(1219,\cdot)\) \(\chi_{4100}(1379,\cdot)\) \(\chi_{4100}(1659,\cdot)\) \(\chi_{4100}(1839,\cdot)\) \(\chi_{4100}(2079,\cdot)\) \(\chi_{4100}(2359,\cdot)\) \(\chi_{4100}(2939,\cdot)\) \(\chi_{4100}(2959,\cdot)\) \(\chi_{4100}(3539,\cdot)\) \(\chi_{4100}(3579,\cdot)\) \(\chi_{4100}(3619,\cdot)\) \(\chi_{4100}(3919,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((2051,1477,3901)\) → \((-1,e\left(\frac{1}{10}\right),e\left(\frac{37}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 4100 }(179, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{40}\right)\) |
sage: chi.jacobi_sum(n)