from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8042, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([16]))
pari: [g,chi] = znchar(Mod(37,8042))
Basic properties
Modulus: | \(8042\) | |
Conductor: | \(4021\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(15\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{4021}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8042.i
\(\chi_{8042}(37,\cdot)\) \(\chi_{8042}(375,\cdot)\) \(\chi_{8042}(1369,\cdot)\) \(\chi_{8042}(3695,\cdot)\) \(\chi_{8042}(3911,\cdot)\) \(\chi_{8042}(5297,\cdot)\) \(\chi_{8042}(5751,\cdot)\) \(\chi_{8042}(7713,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | Number field defined by a degree 15 polynomial |
Values on generators
\(4023\) → \(e\left(\frac{8}{15}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 8042 }(37, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) |
sage: chi.jacobi_sum(n)