from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9072, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,22,24]))
pari: [g,chi] = znchar(Mod(179,9072))
Basic properties
Modulus: | \(9072\) | |
Conductor: | \(3024\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3024}(1859,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9072.hf
\(\chi_{9072}(179,\cdot)\) \(\chi_{9072}(1115,\cdot)\) \(\chi_{9072}(1691,\cdot)\) \(\chi_{9072}(2627,\cdot)\) \(\chi_{9072}(3203,\cdot)\) \(\chi_{9072}(4139,\cdot)\) \(\chi_{9072}(4715,\cdot)\) \(\chi_{9072}(5651,\cdot)\) \(\chi_{9072}(6227,\cdot)\) \(\chi_{9072}(7163,\cdot)\) \(\chi_{9072}(7739,\cdot)\) \(\chi_{9072}(8675,\cdot)\)
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
\((1135,6805,3809,2593)\) → \((-1,-i,e\left(\frac{11}{18}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 9072 }(179, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)