from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7744, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,15,24]))
pari: [g,chi] = znchar(Mod(9,7744))
Basic properties
| Modulus: | \(7744\) | |
| Conductor: | \(352\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{352}(317,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7744.br
\(\chi_{7744}(9,\cdot)\) \(\chi_{7744}(729,\cdot)\) \(\chi_{7744}(1049,\cdot)\) \(\chi_{7744}(1721,\cdot)\) \(\chi_{7744}(1945,\cdot)\) \(\chi_{7744}(2665,\cdot)\) \(\chi_{7744}(2985,\cdot)\) \(\chi_{7744}(3657,\cdot)\) \(\chi_{7744}(3881,\cdot)\) \(\chi_{7744}(4601,\cdot)\) \(\chi_{7744}(4921,\cdot)\) \(\chi_{7744}(5593,\cdot)\) \(\chi_{7744}(5817,\cdot)\) \(\chi_{7744}(6537,\cdot)\) \(\chi_{7744}(6857,\cdot)\) \(\chi_{7744}(7529,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.40.96430685261162182749113906515642066253992366248338958954046471967872161601814528.1 |
Values on generators
\((5567,4357,6657)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{3}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 7744 }(9, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)