from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,11,8]))
pari: [g,chi] = znchar(Mod(729,1840))
Basic properties
Modulus: | \(1840\) | |
Conductor: | \(920\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{920}(269,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1840.bv
\(\chi_{1840}(9,\cdot)\) \(\chi_{1840}(169,\cdot)\) \(\chi_{1840}(409,\cdot)\) \(\chi_{1840}(489,\cdot)\) \(\chi_{1840}(729,\cdot)\) \(\chi_{1840}(809,\cdot)\) \(\chi_{1840}(969,\cdot)\) \(\chi_{1840}(1129,\cdot)\) \(\chi_{1840}(1209,\cdot)\) \(\chi_{1840}(1369,\cdot)\)
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
\((1151,1381,737,1201)\) → \((1,-1,-1,e\left(\frac{4}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 1840 }(729, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) |
sage: chi.jacobi_sum(n)