from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1407, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,55,15]))
pari: [g,chi] = znchar(Mod(5,1407))
Basic properties
| Modulus: | \(1407\) | |
| Conductor: | \(1407\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1407.cq
\(\chi_{1407}(5,\cdot)\) \(\chi_{1407}(110,\cdot)\) \(\chi_{1407}(206,\cdot)\) \(\chi_{1407}(311,\cdot)\) \(\chi_{1407}(320,\cdot)\) \(\chi_{1407}(362,\cdot)\) \(\chi_{1407}(521,\cdot)\) \(\chi_{1407}(563,\cdot)\) \(\chi_{1407}(656,\cdot)\) \(\chi_{1407}(740,\cdot)\) \(\chi_{1407}(782,\cdot)\) \(\chi_{1407}(857,\cdot)\) \(\chi_{1407}(929,\cdot)\) \(\chi_{1407}(941,\cdot)\) \(\chi_{1407}(983,\cdot)\) \(\chi_{1407}(1013,\cdot)\) \(\chi_{1407}(1130,\cdot)\) \(\chi_{1407}(1181,\cdot)\) \(\chi_{1407}(1214,\cdot)\) \(\chi_{1407}(1382,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((470,1207,337)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{5}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1407 }(5, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{29}{66}\right)\) |
sage: chi.jacobi_sum(n)