from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1407, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,13]))
pari: [g,chi] = znchar(Mod(85,1407))
Basic properties
| Modulus: | \(1407\) | |
| Conductor: | \(67\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{67}(18,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1407.cn
\(\chi_{1407}(85,\cdot)\) \(\chi_{1407}(232,\cdot)\) \(\chi_{1407}(316,\cdot)\) \(\chi_{1407}(337,\cdot)\) \(\chi_{1407}(379,\cdot)\) \(\chi_{1407}(463,\cdot)\) \(\chi_{1407}(526,\cdot)\) \(\chi_{1407}(547,\cdot)\) \(\chi_{1407}(568,\cdot)\) \(\chi_{1407}(610,\cdot)\) \(\chi_{1407}(631,\cdot)\) \(\chi_{1407}(757,\cdot)\) \(\chi_{1407}(778,\cdot)\) \(\chi_{1407}(883,\cdot)\) \(\chi_{1407}(988,\cdot)\) \(\chi_{1407}(1051,\cdot)\) \(\chi_{1407}(1135,\cdot)\) \(\chi_{1407}(1219,\cdot)\) \(\chi_{1407}(1240,\cdot)\) \(\chi_{1407}(1324,\cdot)\)
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
\((470,1207,337)\) → \((1,1,e\left(\frac{13}{66}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1407 }(85, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) |
sage: chi.jacobi_sum(n)