from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4620, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,15,0,5,21]))
pari: [g,chi] = znchar(Mod(1151,4620))
Basic properties
Modulus: | \(4620\) | |
Conductor: | \(924\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{924}(227,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4620.fq
\(\chi_{4620}(1151,\cdot)\) \(\chi_{4620}(1811,\cdot)\) \(\chi_{4620}(2411,\cdot)\) \(\chi_{4620}(3071,\cdot)\) \(\chi_{4620}(3251,\cdot)\) \(\chi_{4620}(3671,\cdot)\) \(\chi_{4620}(3911,\cdot)\) \(\chi_{4620}(4331,\cdot)\)
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: | 30.30.270877037062811393670853967610983706070206882960369770568766980096.1 |
Values on generators
\((2311,1541,3697,661,2521)\) → \((-1,-1,1,e\left(\frac{1}{6}\right),e\left(\frac{7}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
\( \chi_{ 4620 }(1151, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(1\) | \(e\left(\frac{13}{30}\right)\) |
sage: chi.jacobi_sum(n)