from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,33,44,54]))
pari: [g,chi] = znchar(Mod(1271,7728))
Basic properties
| Modulus: | \(7728\) | |
| Conductor: | \(3864\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3864}(3203,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7728.gk
\(\chi_{7728}(1271,\cdot)\) \(\chi_{7728}(2279,\cdot)\) \(\chi_{7728}(2375,\cdot)\) \(\chi_{7728}(2615,\cdot)\) \(\chi_{7728}(3383,\cdot)\) \(\chi_{7728}(3623,\cdot)\) \(\chi_{7728}(3719,\cdot)\) \(\chi_{7728}(3959,\cdot)\) \(\chi_{7728}(4631,\cdot)\) \(\chi_{7728}(4727,\cdot)\) \(\chi_{7728}(5063,\cdot)\) \(\chi_{7728}(5303,\cdot)\) \(\chi_{7728}(5639,\cdot)\) \(\chi_{7728}(5735,\cdot)\) \(\chi_{7728}(5975,\cdot)\) \(\chi_{7728}(6311,\cdot)\) \(\chi_{7728}(6407,\cdot)\) \(\chi_{7728}(6743,\cdot)\) \(\chi_{7728}(7079,\cdot)\) \(\chi_{7728}(7415,\cdot)\)
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: | Number field defined by a degree 66 polynomial |
Values on generators
\((4831,5797,5153,6625,6721)\) → \((-1,-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{9}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7728 }(1271, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) |
sage: chi.jacobi_sum(n)