from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3312, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,22,48]))
pari: [g,chi] = znchar(Mod(49,3312))
Basic properties
| Modulus: | \(3312\) | |
| Conductor: | \(207\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{207}(49,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3312.cm
\(\chi_{3312}(49,\cdot)\) \(\chi_{3312}(193,\cdot)\) \(\chi_{3312}(625,\cdot)\) \(\chi_{3312}(673,\cdot)\) \(\chi_{3312}(817,\cdot)\) \(\chi_{3312}(913,\cdot)\) \(\chi_{3312}(961,\cdot)\) \(\chi_{3312}(1393,\cdot)\) \(\chi_{3312}(1681,\cdot)\) \(\chi_{3312}(1777,\cdot)\) \(\chi_{3312}(1825,\cdot)\) \(\chi_{3312}(1921,\cdot)\) \(\chi_{3312}(2065,\cdot)\) \(\chi_{3312}(2257,\cdot)\) \(\chi_{3312}(2401,\cdot)\) \(\chi_{3312}(2497,\cdot)\) \(\chi_{3312}(2785,\cdot)\) \(\chi_{3312}(2833,\cdot)\) \(\chi_{3312}(2929,\cdot)\) \(\chi_{3312}(3121,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | 33.33.70011645999218458416472683122408534303895571350166174758601569.1 |
Values on generators
\((415,2485,2945,2305)\) → \((1,1,e\left(\frac{1}{3}\right),e\left(\frac{8}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3312 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)