from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,22,11,8]))
pari: [g,chi] = znchar(Mod(487,1840))
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 \(\chi_{920}(27,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1840.co
\(\chi_{1840}(87,\cdot)\) \(\chi_{1840}(167,\cdot)\) \(\chi_{1840}(407,\cdot)\) \(\chi_{1840}(423,\cdot)\) \(\chi_{1840}(487,\cdot)\) \(\chi_{1840}(583,\cdot)\) \(\chi_{1840}(647,\cdot)\) \(\chi_{1840}(807,\cdot)\) \(\chi_{1840}(823,\cdot)\) \(\chi_{1840}(887,\cdot)\) \(\chi_{1840}(903,\cdot)\) \(\chi_{1840}(1047,\cdot)\) \(\chi_{1840}(1143,\cdot)\) \(\chi_{1840}(1223,\cdot)\) \(\chi_{1840}(1383,\cdot)\) \(\chi_{1840}(1527,\cdot)\) \(\chi_{1840}(1543,\cdot)\) \(\chi_{1840}(1623,\cdot)\) \(\chi_{1840}(1687,\cdot)\) \(\chi_{1840}(1783,\cdot)\)
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
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 1840 }(487, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) |
sage: chi.jacobi_sum(n)