from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4560, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,18,27,22]))
pari: [g,chi] = znchar(Mod(4043,4560))
Basic properties
Modulus: | \(4560\) | |
Conductor: | \(4560\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4560.ic
\(\chi_{4560}(203,\cdot)\) \(\chi_{4560}(1427,\cdot)\) \(\chi_{4560}(1667,\cdot)\) \(\chi_{4560}(1883,\cdot)\) \(\chi_{4560}(2123,\cdot)\) \(\chi_{4560}(3107,\cdot)\) \(\chi_{4560}(3347,\cdot)\) \(\chi_{4560}(3563,\cdot)\) \(\chi_{4560}(3587,\cdot)\) \(\chi_{4560}(3803,\cdot)\) \(\chi_{4560}(4043,\cdot)\) \(\chi_{4560}(4307,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((1711,1141,3041,2737,1921)\) → \((-1,i,-1,-i,e\left(\frac{11}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 4560 }(4043, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) |
sage: chi.jacobi_sum(n)