from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8040, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,33,0,14]))
pari: [g,chi] = znchar(Mod(371,8040))
Basic properties
| Modulus: | \(8040\) | |
| Conductor: | \(1608\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1608}(371,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8040.fp
\(\chi_{8040}(371,\cdot)\) \(\chi_{8040}(851,\cdot)\) \(\chi_{8040}(971,\cdot)\) \(\chi_{8040}(1091,\cdot)\) \(\chi_{8040}(1691,\cdot)\) \(\chi_{8040}(1931,\cdot)\) \(\chi_{8040}(2891,\cdot)\) \(\chi_{8040}(3131,\cdot)\) \(\chi_{8040}(3251,\cdot)\) \(\chi_{8040}(3371,\cdot)\) \(\chi_{8040}(3611,\cdot)\) \(\chi_{8040}(4091,\cdot)\) \(\chi_{8040}(4811,\cdot)\) \(\chi_{8040}(5051,\cdot)\) \(\chi_{8040}(5291,\cdot)\) \(\chi_{8040}(5651,\cdot)\) \(\chi_{8040}(6371,\cdot)\) \(\chi_{8040}(6851,\cdot)\) \(\chi_{8040}(7091,\cdot)\) \(\chi_{8040}(7811,\cdot)\)
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
\((6031,4021,2681,3217,5161)\) → \((-1,-1,-1,1,e\left(\frac{7}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8040 }(371, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{49}{66}\right)\) |
sage: chi.jacobi_sum(n)