from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3528, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,7,8]))
pari: [g,chi] = znchar(Mod(1073,3528))
Basic properties
| Modulus: | \(3528\) | |
| Conductor: | \(441\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{441}(191,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3528.gc
\(\chi_{3528}(65,\cdot)\) \(\chi_{3528}(473,\cdot)\) \(\chi_{3528}(977,\cdot)\) \(\chi_{3528}(1073,\cdot)\) \(\chi_{3528}(1481,\cdot)\) \(\chi_{3528}(1577,\cdot)\) \(\chi_{3528}(1985,\cdot)\) \(\chi_{3528}(2081,\cdot)\) \(\chi_{3528}(2489,\cdot)\) \(\chi_{3528}(2585,\cdot)\) \(\chi_{3528}(2993,\cdot)\) \(\chi_{3528}(3089,\cdot)\)
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: | Number field defined by a degree 42 polynomial |
Values on generators
\((2647,1765,785,1081)\) → \((1,1,e\left(\frac{1}{6}\right),e\left(\frac{4}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3528 }(1073, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{21}\right)\) |
sage: chi.jacobi_sum(n)