from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4275, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([12,27,26]))
pari: [g,chi] = znchar(Mod(193,4275))
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 \(\chi_{855}(193,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4275.fi
\(\chi_{4275}(193,\cdot)\) \(\chi_{4275}(382,\cdot)\) \(\chi_{4275}(718,\cdot)\) \(\chi_{4275}(1732,\cdot)\) \(\chi_{4275}(2218,\cdot)\) \(\chi_{4275}(2257,\cdot)\) \(\chi_{4275}(2293,\cdot)\) \(\chi_{4275}(2518,\cdot)\) \(\chi_{4275}(3118,\cdot)\) \(\chi_{4275}(3757,\cdot)\) \(\chi_{4275}(3832,\cdot)\) \(\chi_{4275}(4057,\cdot)\)
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: | 36.36.17849776228866488737715206984999102954438314099226129130939288096733391284942626953125.1 |
Values on generators
\((1901,1027,1351)\) → \((e\left(\frac{1}{3}\right),-i,e\left(\frac{13}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
| \( \chi_{ 4275 }(193, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) |
sage: chi.jacobi_sum(n)