from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4031, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([9,28]))
pari: [g,chi] = znchar(Mod(876,4031))
Basic properties
Modulus: | \(4031\) | |
Conductor: | \(4031\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | 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 4031.u
\(\chi_{4031}(42,\cdot)\) \(\chi_{4031}(96,\cdot)\) \(\chi_{4031}(876,\cdot)\) \(\chi_{4031}(1347,\cdot)\) \(\chi_{4031}(1571,\cdot)\) \(\chi_{4031}(1849,\cdot)\) \(\chi_{4031}(2181,\cdot)\) \(\chi_{4031}(2266,\cdot)\) \(\chi_{4031}(2822,\cdot)\) \(\chi_{4031}(2876,\cdot)\) \(\chi_{4031}(3154,\cdot)\) \(\chi_{4031}(3571,\cdot)\)
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
\((3337,697)\) → \((e\left(\frac{3}{14}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 4031 }(876, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{42}\right)\) |
sage: chi.jacobi_sum(n)