from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2151, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,24]))
pari: [g,chi] = znchar(Mod(283,2151))
Basic properties
Modulus: | \(2151\) | |
Conductor: | \(2151\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | 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 2151.n
\(\chi_{2151}(283,\cdot)\) \(\chi_{2151}(337,\cdot)\) \(\chi_{2151}(502,\cdot)\) \(\chi_{2151}(679,\cdot)\) \(\chi_{2151}(727,\cdot)\) \(\chi_{2151}(817,\cdot)\) \(\chi_{2151}(1219,\cdot)\) \(\chi_{2151}(1444,\cdot)\) \(\chi_{2151}(1534,\cdot)\) \(\chi_{2151}(1717,\cdot)\) \(\chi_{2151}(1771,\cdot)\) \(\chi_{2151}(2113,\cdot)\)
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 21 polynomial |
Values on generators
\((479,1441)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{4}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 2151 }(283, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) |
sage: chi.jacobi_sum(n)