from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3900, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,36,35]))
pari: [g,chi] = znchar(Mod(271,3900))
Basic properties
| Modulus: | \(3900\) | |
| Conductor: | \(1300\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1300}(271,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3900.fr
\(\chi_{3900}(271,\cdot)\) \(\chi_{3900}(331,\cdot)\) \(\chi_{3900}(631,\cdot)\) \(\chi_{3900}(691,\cdot)\) \(\chi_{3900}(1111,\cdot)\) \(\chi_{3900}(1411,\cdot)\) \(\chi_{3900}(1471,\cdot)\) \(\chi_{3900}(1831,\cdot)\) \(\chi_{3900}(1891,\cdot)\) \(\chi_{3900}(2191,\cdot)\) \(\chi_{3900}(2611,\cdot)\) \(\chi_{3900}(2671,\cdot)\) \(\chi_{3900}(2971,\cdot)\) \(\chi_{3900}(3031,\cdot)\) \(\chi_{3900}(3391,\cdot)\) \(\chi_{3900}(3811,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((1951,1301,3277,301)\) → \((-1,1,e\left(\frac{3}{5}\right),e\left(\frac{7}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3900 }(271, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)