from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4730, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,11]))
pari: [g,chi] = znchar(Mod(4201,4730))
Basic properties
| Modulus: | \(4730\) | |
| Conductor: | \(473\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{473}(417,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4730.cg
\(\chi_{4730}(241,\cdot)\) \(\chi_{4730}(571,\cdot)\) \(\chi_{4730}(1121,\cdot)\) \(\chi_{4730}(1781,\cdot)\) \(\chi_{4730}(2221,\cdot)\) \(\chi_{4730}(2441,\cdot)\) \(\chi_{4730}(2771,\cdot)\) \(\chi_{4730}(3101,\cdot)\) \(\chi_{4730}(3211,\cdot)\) \(\chi_{4730}(3431,\cdot)\) \(\chi_{4730}(3761,\cdot)\) \(\chi_{4730}(4201,\cdot)\)
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: | 42.42.69414958297778203866028207940161306079257307292058478902474460052228735844266189326476073.1 |
Values on generators
\((947,431,1981)\) → \((1,-1,e\left(\frac{11}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 4730 }(4201, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) |
sage: chi.jacobi_sum(n)