from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4275, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([24,27,2]))
pari: [g,chi] = znchar(Mod(268,4275))
Basic properties
Modulus: | \(4275\) | |
Conductor: | \(855\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{855}(268,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4275.fc
\(\chi_{4275}(268,\cdot)\) \(\chi_{4275}(832,\cdot)\) \(\chi_{4275}(868,\cdot)\) \(\chi_{4275}(907,\cdot)\) \(\chi_{4275}(1093,\cdot)\) \(\chi_{4275}(1618,\cdot)\) \(\chi_{4275}(1807,\cdot)\) \(\chi_{4275}(2407,\cdot)\) \(\chi_{4275}(2632,\cdot)\) \(\chi_{4275}(3157,\cdot)\) \(\chi_{4275}(3568,\cdot)\) \(\chi_{4275}(3643,\cdot)\)
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: | 36.36.17849776228866488737715206984999102954438314099226129130939288096733391284942626953125.2 |
Values on generators
\((1901,1027,1351)\) → \((e\left(\frac{2}{3}\right),-i,e\left(\frac{1}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
\( \chi_{ 4275 }(268, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(-i\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) |
sage: chi.jacobi_sum(n)