from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1407, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,44,40]))
pari: [g,chi] = znchar(Mod(1145,1407))
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.cg
\(\chi_{1407}(116,\cdot)\) \(\chi_{1407}(170,\cdot)\) \(\chi_{1407}(284,\cdot)\) \(\chi_{1407}(389,\cdot)\) \(\chi_{1407}(569,\cdot)\) \(\chi_{1407}(620,\cdot)\) \(\chi_{1407}(674,\cdot)\) \(\chi_{1407}(758,\cdot)\) \(\chi_{1407}(926,\cdot)\) \(\chi_{1407}(977,\cdot)\) \(\chi_{1407}(1031,\cdot)\) \(\chi_{1407}(1052,\cdot)\) \(\chi_{1407}(1082,\cdot)\) \(\chi_{1407}(1145,\cdot)\) \(\chi_{1407}(1199,\cdot)\) \(\chi_{1407}(1229,\cdot)\) \(\chi_{1407}(1241,\cdot)\) \(\chi_{1407}(1262,\cdot)\) \(\chi_{1407}(1271,\cdot)\) \(\chi_{1407}(1292,\cdot)\)
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
\((470,1207,337)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{20}{33}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 1407 }(1145, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{13}{33}\right)\) |
sage: chi.jacobi_sum(n)