from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3484, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,44,7]))
pari: [g,chi] = znchar(Mod(61,3484))
Basic properties
Modulus: | \(3484\) | |
Conductor: | \(871\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{871}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3484.dm
\(\chi_{3484}(61,\cdot)\) \(\chi_{3484}(757,\cdot)\) \(\chi_{3484}(1017,\cdot)\) \(\chi_{3484}(1049,\cdot)\) \(\chi_{3484}(1173,\cdot)\) \(\chi_{3484}(1257,\cdot)\) \(\chi_{3484}(1381,\cdot)\) \(\chi_{3484}(1485,\cdot)\) \(\chi_{3484}(1537,\cdot)\) \(\chi_{3484}(1569,\cdot)\) \(\chi_{3484}(1621,\cdot)\) \(\chi_{3484}(1725,\cdot)\) \(\chi_{3484}(1933,\cdot)\) \(\chi_{3484}(2109,\cdot)\) \(\chi_{3484}(2525,\cdot)\) \(\chi_{3484}(2577,\cdot)\) \(\chi_{3484}(2661,\cdot)\) \(\chi_{3484}(2765,\cdot)\) \(\chi_{3484}(3285,\cdot)\) \(\chi_{3484}(3357,\cdot)\)
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
\((1743,1341,3017)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{7}{66}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
\( \chi_{ 3484 }(61, a) \) | \(-1\) | \(1\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) |
sage: chi.jacobi_sum(n)