from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3528, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,0,38]))
pari: [g,chi] = znchar(Mod(415,3528))
Basic properties
Modulus: | \(3528\) | |
Conductor: | \(196\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{196}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3528.fc
\(\chi_{3528}(415,\cdot)\) \(\chi_{3528}(487,\cdot)\) \(\chi_{3528}(919,\cdot)\) \(\chi_{3528}(991,\cdot)\) \(\chi_{3528}(1423,\cdot)\) \(\chi_{3528}(1495,\cdot)\) \(\chi_{3528}(1927,\cdot)\) \(\chi_{3528}(1999,\cdot)\) \(\chi_{3528}(2503,\cdot)\) \(\chi_{3528}(2935,\cdot)\) \(\chi_{3528}(3439,\cdot)\) \(\chi_{3528}(3511,\cdot)\)
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: | 42.0.74252462132603256348231837398371002884673933378885582779211491265789772693504.1 |
Values on generators
\((2647,1765,785,1081)\) → \((-1,1,1,e\left(\frac{19}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 3528 }(415, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{20}{21}\right)\) |
sage: chi.jacobi_sum(n)