from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4100, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,32,3]))
pari: [g,chi] = znchar(Mod(11,4100))
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.hn
\(\chi_{4100}(11,\cdot)\) \(\chi_{4100}(311,\cdot)\) \(\chi_{4100}(731,\cdot)\) \(\chi_{4100}(791,\cdot)\) \(\chi_{4100}(971,\cdot)\) \(\chi_{4100}(1331,\cdot)\) \(\chi_{4100}(1411,\cdot)\) \(\chi_{4100}(1491,\cdot)\) \(\chi_{4100}(1571,\cdot)\) \(\chi_{4100}(1711,\cdot)\) \(\chi_{4100}(2031,\cdot)\) \(\chi_{4100}(2631,\cdot)\) \(\chi_{4100}(2671,\cdot)\) \(\chi_{4100}(2691,\cdot)\) \(\chi_{4100}(3391,\cdot)\) \(\chi_{4100}(3971,\cdot)\)
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
\((2051,1477,3901)\) → \((-1,e\left(\frac{4}{5}\right),e\left(\frac{3}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 4100 }(11, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(1\) | \(e\left(\frac{27}{40}\right)\) |
sage: chi.jacobi_sum(n)