from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1407, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,11,37]))
pari: [g,chi] = znchar(Mod(185,1407))
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.ct
\(\chi_{1407}(80,\cdot)\) \(\chi_{1407}(185,\cdot)\) \(\chi_{1407}(299,\cdot)\) \(\chi_{1407}(353,\cdot)\) \(\chi_{1407}(584,\cdot)\) \(\chi_{1407}(605,\cdot)\) \(\chi_{1407}(614,\cdot)\) \(\chi_{1407}(635,\cdot)\) \(\chi_{1407}(647,\cdot)\) \(\chi_{1407}(677,\cdot)\) \(\chi_{1407}(731,\cdot)\) \(\chi_{1407}(794,\cdot)\) \(\chi_{1407}(824,\cdot)\) \(\chi_{1407}(845,\cdot)\) \(\chi_{1407}(899,\cdot)\) \(\chi_{1407}(950,\cdot)\) \(\chi_{1407}(1118,\cdot)\) \(\chi_{1407}(1202,\cdot)\) \(\chi_{1407}(1256,\cdot)\) \(\chi_{1407}(1307,\cdot)\)
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
\((470,1207,337)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{37}{66}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1407 }(185, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{29}{66}\right)\) |
sage: chi.jacobi_sum(n)