from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3311, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,8]))
pari: [g,chi] = znchar(Mod(111,3311))
Basic properties
| Modulus: | \(3311\) | |
| Conductor: | \(301\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{301}(111,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3311.dp
\(\chi_{3311}(111,\cdot)\) \(\chi_{3311}(496,\cdot)\) \(\chi_{3311}(573,\cdot)\) \(\chi_{3311}(1343,\cdot)\) \(\chi_{3311}(1651,\cdot)\) \(\chi_{3311}(1959,\cdot)\) \(\chi_{3311}(2036,\cdot)\) \(\chi_{3311}(2190,\cdot)\) \(\chi_{3311}(2267,\cdot)\) \(\chi_{3311}(2421,\cdot)\) \(\chi_{3311}(2575,\cdot)\) \(\chi_{3311}(3191,\cdot)\)
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: | 42.0.121842012423466724043945276342536999203112328184628981033554149680639161638328200007.1 |
Values on generators
\((1893,904,2927)\) → \((-1,1,e\left(\frac{4}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 3311 }(111, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) |
sage: chi.jacobi_sum(n)