from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3969, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([25,36]))
pari: [g,chi] = znchar(Mod(263,3969))
Basic properties
| Modulus: | \(3969\) | |
| Conductor: | \(567\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{567}(263,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3969.cd
\(\chi_{3969}(263,\cdot)\) \(\chi_{3969}(275,\cdot)\) \(\chi_{3969}(704,\cdot)\) \(\chi_{3969}(716,\cdot)\) \(\chi_{3969}(1145,\cdot)\) \(\chi_{3969}(1157,\cdot)\) \(\chi_{3969}(1586,\cdot)\) \(\chi_{3969}(1598,\cdot)\) \(\chi_{3969}(2027,\cdot)\) \(\chi_{3969}(2039,\cdot)\) \(\chi_{3969}(2468,\cdot)\) \(\chi_{3969}(2480,\cdot)\) \(\chi_{3969}(2909,\cdot)\) \(\chi_{3969}(2921,\cdot)\) \(\chi_{3969}(3350,\cdot)\) \(\chi_{3969}(3362,\cdot)\) \(\chi_{3969}(3791,\cdot)\) \(\chi_{3969}(3803,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((2108,3727)\) → \((e\left(\frac{25}{54}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3969 }(263, a) \) | \(-1\) | \(1\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) |
sage: chi.jacobi_sum(n)