from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4600, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,22,11,6]))
pari: [g,chi] = znchar(Mod(4357,4600))
Basic properties
| Modulus: | \(4600\) | |
| Conductor: | \(920\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{920}(677,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4600.cm
\(\chi_{4600}(157,\cdot)\) \(\chi_{4600}(293,\cdot)\) \(\chi_{4600}(493,\cdot)\) \(\chi_{4600}(557,\cdot)\) \(\chi_{4600}(757,\cdot)\) \(\chi_{4600}(893,\cdot)\) \(\chi_{4600}(957,\cdot)\) \(\chi_{4600}(1157,\cdot)\) \(\chi_{4600}(1293,\cdot)\) \(\chi_{4600}(1493,\cdot)\) \(\chi_{4600}(1693,\cdot)\) \(\chi_{4600}(1893,\cdot)\) \(\chi_{4600}(2357,\cdot)\) \(\chi_{4600}(2757,\cdot)\) \(\chi_{4600}(3093,\cdot)\) \(\chi_{4600}(3493,\cdot)\) \(\chi_{4600}(3557,\cdot)\) \(\chi_{4600}(4157,\cdot)\) \(\chi_{4600}(4293,\cdot)\) \(\chi_{4600}(4357,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | 44.44.13383169230192059253459701104387771124501004765020501667165784506368000000000000000000000000000000000.1 |
Values on generators
\((1151,2301,2577,1201)\) → \((1,-1,i,e\left(\frac{3}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 4600 }(4357, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) |
sage: chi.jacobi_sum(n)