from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3528, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,14,29]))
pari: [g,chi] = znchar(Mod(103,3528))
Basic properties
| Modulus: | \(3528\) | |
| Conductor: | \(1764\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1764}(103,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3528.eg
\(\chi_{3528}(103,\cdot)\) \(\chi_{3528}(367,\cdot)\) \(\chi_{3528}(871,\cdot)\) \(\chi_{3528}(1111,\cdot)\) \(\chi_{3528}(1375,\cdot)\) \(\chi_{3528}(1615,\cdot)\) \(\chi_{3528}(1879,\cdot)\) \(\chi_{3528}(2119,\cdot)\) \(\chi_{3528}(2623,\cdot)\) \(\chi_{3528}(2887,\cdot)\) \(\chi_{3528}(3127,\cdot)\) \(\chi_{3528}(3391,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.42.272018952124861435139193115870691546077971916868404958924735840700688722215619164268098368399776141017088.1 |
Values on generators
\((2647,1765,785,1081)\) → \((-1,1,e\left(\frac{1}{3}\right),e\left(\frac{29}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3528 }(103, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(1\) | \(e\left(\frac{2}{21}\right)\) |
sage: chi.jacobi_sum(n)