from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3330, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,0,11]))
pari: [g,chi] = znchar(Mod(161,3330))
Basic properties
| Modulus: | \(3330\) | |
| Conductor: | \(111\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{111}(50,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3330.fg
\(\chi_{3330}(161,\cdot)\) \(\chi_{3330}(431,\cdot)\) \(\chi_{3330}(611,\cdot)\) \(\chi_{3330}(701,\cdot)\) \(\chi_{3330}(1241,\cdot)\) \(\chi_{3330}(1421,\cdot)\) \(\chi_{3330}(1781,\cdot)\) \(\chi_{3330}(2141,\cdot)\) \(\chi_{3330}(2501,\cdot)\) \(\chi_{3330}(2681,\cdot)\) \(\chi_{3330}(3221,\cdot)\) \(\chi_{3330}(3311,\cdot)\)
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: | \(\Q(\zeta_{111})^+\) |
Values on generators
\((371,667,631)\) → \((-1,1,e\left(\frac{11}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
| \( \chi_{ 3330 }(161, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(-i\) | \(e\left(\frac{1}{9}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)