from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1075, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,1]))
pari: [g,chi] = znchar(Mod(476,1075))
Basic properties
| Modulus: | \(1075\) | |
| Conductor: | \(43\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{43}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1075.bg
\(\chi_{1075}(26,\cdot)\) \(\chi_{1075}(76,\cdot)\) \(\chi_{1075}(201,\cdot)\) \(\chi_{1075}(276,\cdot)\) \(\chi_{1075}(476,\cdot)\) \(\chi_{1075}(501,\cdot)\) \(\chi_{1075}(751,\cdot)\) \(\chi_{1075}(851,\cdot)\) \(\chi_{1075}(951,\cdot)\) \(\chi_{1075}(976,\cdot)\) \(\chi_{1075}(1001,\cdot)\) \(\chi_{1075}(1051,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((302,476)\) → \((1,e\left(\frac{1}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 1075 }(476, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) |
sage: chi.jacobi_sum(n)