from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4536, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,27,37,0]))
pari: [g,chi] = znchar(Mod(1163,4536))
Basic properties
Modulus: | \(4536\) | |
Conductor: | \(648\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{648}(515,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4536.fw
\(\chi_{4536}(155,\cdot)\) \(\chi_{4536}(491,\cdot)\) \(\chi_{4536}(659,\cdot)\) \(\chi_{4536}(995,\cdot)\) \(\chi_{4536}(1163,\cdot)\) \(\chi_{4536}(1499,\cdot)\) \(\chi_{4536}(1667,\cdot)\) \(\chi_{4536}(2003,\cdot)\) \(\chi_{4536}(2171,\cdot)\) \(\chi_{4536}(2507,\cdot)\) \(\chi_{4536}(2675,\cdot)\) \(\chi_{4536}(3011,\cdot)\) \(\chi_{4536}(3179,\cdot)\) \(\chi_{4536}(3515,\cdot)\) \(\chi_{4536}(3683,\cdot)\) \(\chi_{4536}(4019,\cdot)\) \(\chi_{4536}(4187,\cdot)\) \(\chi_{4536}(4523,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((1135,2269,3809,2593)\) → \((-1,-1,e\left(\frac{37}{54}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 4536 }(1163, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{5}{18}\right)\) |
sage: chi.jacobi_sum(n)