from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4675, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([16,8,35]))
pari: [g,chi] = znchar(Mod(631,4675))
Basic properties
| Modulus: | \(4675\) | |
| Conductor: | \(4675\) | 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 4675.he
\(\chi_{4675}(236,\cdot)\) \(\chi_{4675}(416,\cdot)\) \(\chi_{4675}(631,\cdot)\) \(\chi_{4675}(971,\cdot)\) \(\chi_{4675}(1181,\cdot)\) \(\chi_{4675}(1521,\cdot)\) \(\chi_{4675}(2066,\cdot)\) \(\chi_{4675}(2161,\cdot)\) \(\chi_{4675}(2616,\cdot)\) \(\chi_{4675}(2711,\cdot)\) \(\chi_{4675}(2831,\cdot)\) \(\chi_{4675}(3171,\cdot)\) \(\chi_{4675}(3381,\cdot)\) \(\chi_{4675}(3721,\cdot)\) \(\chi_{4675}(4361,\cdot)\) \(\chi_{4675}(4541,\cdot)\)
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
\((4302,3401,3301)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{1}{5}\right),e\left(\frac{7}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 4675 }(631, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{8}\right)\) |
sage: chi.jacobi_sum(n)