from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1407, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,8]))
pari: [g,chi] = znchar(Mod(55,1407))
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 \(\chi_{469}(55,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1407.cw
\(\chi_{1407}(55,\cdot)\) \(\chi_{1407}(160,\cdot)\) \(\chi_{1407}(181,\cdot)\) \(\chi_{1407}(307,\cdot)\) \(\chi_{1407}(328,\cdot)\) \(\chi_{1407}(370,\cdot)\) \(\chi_{1407}(391,\cdot)\) \(\chi_{1407}(412,\cdot)\) \(\chi_{1407}(475,\cdot)\) \(\chi_{1407}(559,\cdot)\) \(\chi_{1407}(601,\cdot)\) \(\chi_{1407}(622,\cdot)\) \(\chi_{1407}(706,\cdot)\) \(\chi_{1407}(853,\cdot)\) \(\chi_{1407}(1021,\cdot)\) \(\chi_{1407}(1105,\cdot)\) \(\chi_{1407}(1126,\cdot)\) \(\chi_{1407}(1210,\cdot)\) \(\chi_{1407}(1294,\cdot)\) \(\chi_{1407}(1357,\cdot)\)
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
\((470,1207,337)\) → \((1,-1,e\left(\frac{4}{33}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 1407 }(55, a) \) | \(-1\) | \(1\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{47}{66}\right)\) |
sage: chi.jacobi_sum(n)