from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1215, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([32,27]))
pari: [g,chi] = znchar(Mod(298,1215))
Basic properties
Modulus: | \(1215\) | |
Conductor: | \(135\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{135}(88,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1215.s
\(\chi_{1215}(28,\cdot)\) \(\chi_{1215}(217,\cdot)\) \(\chi_{1215}(298,\cdot)\) \(\chi_{1215}(352,\cdot)\) \(\chi_{1215}(433,\cdot)\) \(\chi_{1215}(622,\cdot)\) \(\chi_{1215}(703,\cdot)\) \(\chi_{1215}(757,\cdot)\) \(\chi_{1215}(838,\cdot)\) \(\chi_{1215}(1027,\cdot)\) \(\chi_{1215}(1108,\cdot)\) \(\chi_{1215}(1162,\cdot)\)
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: | 36.0.7225377334561374804949923918873673793376691639423370361328125.1 |
Values on generators
\((731,487)\) → \((e\left(\frac{8}{9}\right),-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 1215 }(298, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)