from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3900, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,0,24,20]))
pari: [g,chi] = znchar(Mod(61,3900))
Basic properties
| Modulus: | \(3900\) | |
| Conductor: | \(325\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(15\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{325}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3900.ds
\(\chi_{3900}(61,\cdot)\) \(\chi_{3900}(841,\cdot)\) \(\chi_{3900}(1381,\cdot)\) \(\chi_{3900}(1621,\cdot)\) \(\chi_{3900}(2161,\cdot)\) \(\chi_{3900}(2941,\cdot)\) \(\chi_{3900}(3181,\cdot)\) \(\chi_{3900}(3721,\cdot)\)
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
\((1951,1301,3277,301)\) → \((1,1,e\left(\frac{4}{5}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3900 }(61, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)