from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1407, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,17]))
pari: [g,chi] = znchar(Mod(20,1407))
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.cx
\(\chi_{1407}(20,\cdot)\) \(\chi_{1407}(41,\cdot)\) \(\chi_{1407}(146,\cdot)\) \(\chi_{1407}(251,\cdot)\) \(\chi_{1407}(314,\cdot)\) \(\chi_{1407}(398,\cdot)\) \(\chi_{1407}(482,\cdot)\) \(\chi_{1407}(503,\cdot)\) \(\chi_{1407}(587,\cdot)\) \(\chi_{1407}(755,\cdot)\) \(\chi_{1407}(902,\cdot)\) \(\chi_{1407}(986,\cdot)\) \(\chi_{1407}(1007,\cdot)\) \(\chi_{1407}(1049,\cdot)\) \(\chi_{1407}(1133,\cdot)\) \(\chi_{1407}(1196,\cdot)\) \(\chi_{1407}(1217,\cdot)\) \(\chi_{1407}(1238,\cdot)\) \(\chi_{1407}(1280,\cdot)\) \(\chi_{1407}(1301,\cdot)\)
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
\((470,1207,337)\) → \((-1,-1,e\left(\frac{17}{66}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 1407 }(20, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) |
sage: chi.jacobi_sum(n)