from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4034, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([27]))
pari: [g,chi] = znchar(Mod(3251,4034))
Basic properties
Modulus: | \(4034\) | |
Conductor: | \(2017\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2017}(1234,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4034.o
\(\chi_{4034}(783,\cdot)\) \(\chi_{4034}(1153,\cdot)\) \(\chi_{4034}(1347,\cdot)\) \(\chi_{4034}(1529,\cdot)\) \(\chi_{4034}(1695,\cdot)\) \(\chi_{4034}(1955,\cdot)\) \(\chi_{4034}(2079,\cdot)\) \(\chi_{4034}(2339,\cdot)\) \(\chi_{4034}(2505,\cdot)\) \(\chi_{4034}(2687,\cdot)\) \(\chi_{4034}(2881,\cdot)\) \(\chi_{4034}(3251,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\(5\) → \(e\left(\frac{27}{28}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 4034 }(3251, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(-i\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)