from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8640, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,28,18]))
pari: [g,chi] = znchar(Mod(49,8640))
Basic properties
Modulus: | \(8640\) | |
Conductor: | \(2160\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2160}(1669,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8640.gk
\(\chi_{8640}(49,\cdot)\) \(\chi_{8640}(529,\cdot)\) \(\chi_{8640}(1489,\cdot)\) \(\chi_{8640}(1969,\cdot)\) \(\chi_{8640}(2929,\cdot)\) \(\chi_{8640}(3409,\cdot)\) \(\chi_{8640}(4369,\cdot)\) \(\chi_{8640}(4849,\cdot)\) \(\chi_{8640}(5809,\cdot)\) \(\chi_{8640}(6289,\cdot)\) \(\chi_{8640}(7249,\cdot)\) \(\chi_{8640}(7729,\cdot)\)
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
\((2431,3781,6401,3457)\) → \((1,i,e\left(\frac{7}{9}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 8640 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{13}{18}\right)\) |
sage: chi.jacobi_sum(n)