from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3484, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,22,18]))
pari: [g,chi] = znchar(Mod(107,3484))
Basic properties
| Modulus: | \(3484\) | |
| Conductor: | \(3484\) | 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 3484.dc
\(\chi_{3484}(107,\cdot)\) \(\chi_{3484}(159,\cdot)\) \(\chi_{3484}(263,\cdot)\) \(\chi_{3484}(399,\cdot)\) \(\chi_{3484}(627,\cdot)\) \(\chi_{3484}(679,\cdot)\) \(\chi_{3484}(1179,\cdot)\) \(\chi_{3484}(1231,\cdot)\) \(\chi_{3484}(1335,\cdot)\) \(\chi_{3484}(1355,\cdot)\) \(\chi_{3484}(1563,\cdot)\) \(\chi_{3484}(1667,\cdot)\) \(\chi_{3484}(1699,\cdot)\) \(\chi_{3484}(1751,\cdot)\) \(\chi_{3484}(1823,\cdot)\) \(\chi_{3484}(2427,\cdot)\) \(\chi_{3484}(2635,\cdot)\) \(\chi_{3484}(2739,\cdot)\) \(\chi_{3484}(2811,\cdot)\) \(\chi_{3484}(2895,\cdot)\)
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
\((1743,1341,3017)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{3}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 3484 }(107, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{31}{66}\right)\) |
sage: chi.jacobi_sum(n)