from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3822, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,2,14]))
pari: [g,chi] = znchar(Mod(107,3822))
Basic properties
| Modulus: | \(3822\) | |
| Conductor: | \(1911\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1911}(107,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3822.dg
\(\chi_{3822}(107,\cdot)\) \(\chi_{3822}(347,\cdot)\) \(\chi_{3822}(653,\cdot)\) \(\chi_{3822}(893,\cdot)\) \(\chi_{3822}(1199,\cdot)\) \(\chi_{3822}(1985,\cdot)\) \(\chi_{3822}(2291,\cdot)\) \(\chi_{3822}(2531,\cdot)\) \(\chi_{3822}(2837,\cdot)\) \(\chi_{3822}(3077,\cdot)\) \(\chi_{3822}(3383,\cdot)\) \(\chi_{3822}(3623,\cdot)\)
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
\((2549,3433,1471)\) → \((-1,e\left(\frac{1}{21}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3822 }(107, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{23}{42}\right)\) |
sage: chi.jacobi_sum(n)