from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8040, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,0,52]))
pari: [g,chi] = znchar(Mod(121,8040))
Basic properties
| Modulus: | \(8040\) | |
| Conductor: | \(67\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{67}(54,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8040.ey
\(\chi_{8040}(121,\cdot)\) \(\chi_{8040}(361,\cdot)\) \(\chi_{8040}(601,\cdot)\) \(\chi_{8040}(961,\cdot)\) \(\chi_{8040}(1681,\cdot)\) \(\chi_{8040}(2161,\cdot)\) \(\chi_{8040}(2401,\cdot)\) \(\chi_{8040}(3121,\cdot)\) \(\chi_{8040}(3721,\cdot)\) \(\chi_{8040}(4201,\cdot)\) \(\chi_{8040}(4321,\cdot)\) \(\chi_{8040}(4441,\cdot)\) \(\chi_{8040}(5041,\cdot)\) \(\chi_{8040}(5281,\cdot)\) \(\chi_{8040}(6241,\cdot)\) \(\chi_{8040}(6481,\cdot)\) \(\chi_{8040}(6601,\cdot)\) \(\chi_{8040}(6721,\cdot)\) \(\chi_{8040}(6961,\cdot)\) \(\chi_{8040}(7441,\cdot)\)
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 33 polynomial |
Values on generators
\((6031,4021,2681,3217,5161)\) → \((1,1,1,1,e\left(\frac{26}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8040 }(121, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{25}{33}\right)\) |
sage: chi.jacobi_sum(n)