from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4034, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([4]))
pari: [g,chi] = znchar(Mod(899,4034))
Basic properties
Modulus: | \(4034\) | |
Conductor: | \(2017\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2017}(899,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4034.m
\(\chi_{4034}(899,\cdot)\) \(\chi_{4034}(1039,\cdot)\) \(\chi_{4034}(1401,\cdot)\) \(\chi_{4034}(1761,\cdot)\) \(\chi_{4034}(1785,\cdot)\) \(\chi_{4034}(1963,\cdot)\) \(\chi_{4034}(2277,\cdot)\) \(\chi_{4034}(2387,\cdot)\) \(\chi_{4034}(2443,\cdot)\) \(\chi_{4034}(3009,\cdot)\) \(\chi_{4034}(3399,\cdot)\) \(\chi_{4034}(3859,\cdot)\)
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
\(5\) → \(e\left(\frac{2}{21}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 4034 }(899, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) |
sage: chi.jacobi_sum(n)