from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3528, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,28,10]))
pari: [g,chi] = znchar(Mod(151,3528))
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 \(\chi_{1764}(151,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3528.fj
\(\chi_{3528}(151,\cdot)\) \(\chi_{3528}(247,\cdot)\) \(\chi_{3528}(751,\cdot)\) \(\chi_{3528}(1159,\cdot)\) \(\chi_{3528}(1663,\cdot)\) \(\chi_{3528}(1759,\cdot)\) \(\chi_{3528}(2167,\cdot)\) \(\chi_{3528}(2263,\cdot)\) \(\chi_{3528}(2671,\cdot)\) \(\chi_{3528}(2767,\cdot)\) \(\chi_{3528}(3175,\cdot)\) \(\chi_{3528}(3271,\cdot)\)
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: | 42.0.38859850303551633591313302267241649439710273838343565560676548671526960316517023466871195485682305859584.1 |
Values on generators
\((2647,1765,785,1081)\) → \((-1,1,e\left(\frac{2}{3}\right),e\left(\frac{5}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3528 }(151, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(-1\) | \(e\left(\frac{13}{21}\right)\) |
sage: chi.jacobi_sum(n)