from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,22,33,8]))
pari: [g,chi] = znchar(Mod(73,1840))
Basic properties
| Modulus: | \(1840\) | |
| 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 \(\chi_{920}(533,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1840.cs
\(\chi_{1840}(73,\cdot)\) \(\chi_{1840}(233,\cdot)\) \(\chi_{1840}(377,\cdot)\) \(\chi_{1840}(393,\cdot)\) \(\chi_{1840}(473,\cdot)\) \(\chi_{1840}(537,\cdot)\) \(\chi_{1840}(633,\cdot)\) \(\chi_{1840}(777,\cdot)\) \(\chi_{1840}(857,\cdot)\) \(\chi_{1840}(1097,\cdot)\) \(\chi_{1840}(1113,\cdot)\) \(\chi_{1840}(1177,\cdot)\) \(\chi_{1840}(1273,\cdot)\) \(\chi_{1840}(1337,\cdot)\) \(\chi_{1840}(1497,\cdot)\) \(\chi_{1840}(1513,\cdot)\) \(\chi_{1840}(1577,\cdot)\) \(\chi_{1840}(1593,\cdot)\) \(\chi_{1840}(1737,\cdot)\) \(\chi_{1840}(1833,\cdot)\)
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: | Number field defined by a degree 44 polynomial |
Values on generators
\((1151,1381,737,1201)\) → \((1,-1,-i,e\left(\frac{2}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 1840 }(73, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) |
sage: chi.jacobi_sum(n)