from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1215, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([16,27]))
pari: [g,chi] = znchar(Mod(829,1215))
Basic properties
Modulus: | \(1215\) | |
Conductor: | \(405\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{405}(169,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1215.t
\(\chi_{1215}(19,\cdot)\) \(\chi_{1215}(64,\cdot)\) \(\chi_{1215}(154,\cdot)\) \(\chi_{1215}(199,\cdot)\) \(\chi_{1215}(289,\cdot)\) \(\chi_{1215}(334,\cdot)\) \(\chi_{1215}(424,\cdot)\) \(\chi_{1215}(469,\cdot)\) \(\chi_{1215}(559,\cdot)\) \(\chi_{1215}(604,\cdot)\) \(\chi_{1215}(694,\cdot)\) \(\chi_{1215}(739,\cdot)\) \(\chi_{1215}(829,\cdot)\) \(\chi_{1215}(874,\cdot)\) \(\chi_{1215}(964,\cdot)\) \(\chi_{1215}(1009,\cdot)\) \(\chi_{1215}(1099,\cdot)\) \(\chi_{1215}(1144,\cdot)\)
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
\((731,487)\) → \((e\left(\frac{8}{27}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 1215 }(829, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) |
sage: chi.jacobi_sum(n)