from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1407, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,55,18]))
pari: [g,chi] = znchar(Mod(40,1407))
Basic properties
| Modulus: | \(1407\) | |
| Conductor: | \(469\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{469}(40,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1407.cr
\(\chi_{1407}(40,\cdot)\) \(\chi_{1407}(82,\cdot)\) \(\chi_{1407}(241,\cdot)\) \(\chi_{1407}(283,\cdot)\) \(\chi_{1407}(292,\cdot)\) \(\chi_{1407}(397,\cdot)\) \(\chi_{1407}(493,\cdot)\) \(\chi_{1407}(598,\cdot)\) \(\chi_{1407}(628,\cdot)\) \(\chi_{1407}(796,\cdot)\) \(\chi_{1407}(829,\cdot)\) \(\chi_{1407}(880,\cdot)\) \(\chi_{1407}(997,\cdot)\) \(\chi_{1407}(1027,\cdot)\) \(\chi_{1407}(1069,\cdot)\) \(\chi_{1407}(1081,\cdot)\) \(\chi_{1407}(1153,\cdot)\) \(\chi_{1407}(1228,\cdot)\) \(\chi_{1407}(1270,\cdot)\) \(\chi_{1407}(1354,\cdot)\)
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
\((470,1207,337)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{3}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1407 }(40, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{59}{66}\right)\) |
sage: chi.jacobi_sum(n)