from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,11,18]))
pari: [g,chi] = znchar(Mod(719,1840))
Basic properties
Modulus: | \(1840\) | |
Conductor: | \(460\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{460}(259,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1840.bq
\(\chi_{1840}(239,\cdot)\) \(\chi_{1840}(399,\cdot)\) \(\chi_{1840}(639,\cdot)\) \(\chi_{1840}(719,\cdot)\) \(\chi_{1840}(959,\cdot)\) \(\chi_{1840}(1039,\cdot)\) \(\chi_{1840}(1199,\cdot)\) \(\chi_{1840}(1359,\cdot)\) \(\chi_{1840}(1439,\cdot)\) \(\chi_{1840}(1599,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
Fixed field: | Number field defined by a degree 22 polynomial |
Values on generators
\((1151,1381,737,1201)\) → \((-1,1,-1,e\left(\frac{9}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 1840 }(719, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) |
sage: chi.jacobi_sum(n)