from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8624, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,0,5,9]))
pari: [g,chi] = znchar(Mod(129,8624))
Basic properties
Modulus: | \(8624\) | |
Conductor: | \(77\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{77}(52,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8624.dv
\(\chi_{8624}(129,\cdot)\) \(\chi_{8624}(2273,\cdot)\) \(\chi_{8624}(2481,\cdot)\) \(\chi_{8624}(3841,\cdot)\) \(\chi_{8624}(5409,\cdot)\) \(\chi_{8624}(5617,\cdot)\) \(\chi_{8624}(7185,\cdot)\) \(\chi_{8624}(7761,\cdot)\)
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: | \(\Q(\zeta_{77})^+\) |
Values on generators
\((5391,6469,7745,3137)\) → \((1,1,e\left(\frac{1}{6}\right),e\left(\frac{3}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 8624 }(129, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) |
sage: chi.jacobi_sum(n)