from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3311, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,0,11]))
pari: [g,chi] = znchar(Mod(331,3311))
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 \(\chi_{301}(30,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3311.dx
\(\chi_{3311}(331,\cdot)\) \(\chi_{3311}(716,\cdot)\) \(\chi_{3311}(837,\cdot)\) \(\chi_{3311}(1101,\cdot)\) \(\chi_{3311}(2069,\cdot)\) \(\chi_{3311}(2179,\cdot)\) \(\chi_{3311}(2377,\cdot)\) \(\chi_{3311}(2454,\cdot)\) \(\chi_{3311}(2641,\cdot)\) \(\chi_{3311}(2685,\cdot)\) \(\chi_{3311}(2872,\cdot)\) \(\chi_{3311}(3301,\cdot)\)
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: | 42.0.4314711866802139421730881462743363761193335651876617911874600258889032535003889995189686443.1 |
Values on generators
\((1893,904,2927)\) → \((e\left(\frac{1}{3}\right),1,e\left(\frac{11}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
\( \chi_{ 3311 }(331, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{8}{21}\right)\) |
sage: chi.jacobi_sum(n)