from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3969, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([10,36]))
pari: [g,chi] = znchar(Mod(214,3969))
Basic properties
Modulus: | \(3969\) | |
Conductor: | \(567\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(27\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{567}(214,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3969.bq
\(\chi_{3969}(214,\cdot)\) \(\chi_{3969}(373,\cdot)\) \(\chi_{3969}(655,\cdot)\) \(\chi_{3969}(814,\cdot)\) \(\chi_{3969}(1096,\cdot)\) \(\chi_{3969}(1255,\cdot)\) \(\chi_{3969}(1537,\cdot)\) \(\chi_{3969}(1696,\cdot)\) \(\chi_{3969}(1978,\cdot)\) \(\chi_{3969}(2137,\cdot)\) \(\chi_{3969}(2419,\cdot)\) \(\chi_{3969}(2578,\cdot)\) \(\chi_{3969}(2860,\cdot)\) \(\chi_{3969}(3019,\cdot)\) \(\chi_{3969}(3301,\cdot)\) \(\chi_{3969}(3460,\cdot)\) \(\chi_{3969}(3742,\cdot)\) \(\chi_{3969}(3901,\cdot)\)
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 27 polynomial |
Values on generators
\((2108,3727)\) → \((e\left(\frac{5}{27}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 3969 }(214, a) \) | \(1\) | \(1\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) |
sage: chi.jacobi_sum(n)