from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1407, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,44,20]))
pari: [g,chi] = znchar(Mod(830,1407))
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.cz
\(\chi_{1407}(23,\cdot)\) \(\chi_{1407}(65,\cdot)\) \(\chi_{1407}(86,\cdot)\) \(\chi_{1407}(317,\cdot)\) \(\chi_{1407}(368,\cdot)\) \(\chi_{1407}(473,\cdot)\) \(\chi_{1407}(485,\cdot)\) \(\chi_{1407}(557,\cdot)\) \(\chi_{1407}(590,\cdot)\) \(\chi_{1407}(725,\cdot)\) \(\chi_{1407}(821,\cdot)\) \(\chi_{1407}(830,\cdot)\) \(\chi_{1407}(851,\cdot)\) \(\chi_{1407}(998,\cdot)\) \(\chi_{1407}(1040,\cdot)\) \(\chi_{1407}(1061,\cdot)\) \(\chi_{1407}(1178,\cdot)\) \(\chi_{1407}(1283,\cdot)\) \(\chi_{1407}(1346,\cdot)\) \(\chi_{1407}(1376,\cdot)\)
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
\((470,1207,337)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{10}{33}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 1407 }(830, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) |
sage: chi.jacobi_sum(n)