from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8040, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,33,0,53]))
pari: [g,chi] = znchar(Mod(41,8040))
Basic properties
| Modulus: | \(8040\) | |
| Conductor: | \(201\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{201}(41,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8040.fy
\(\chi_{8040}(41,\cdot)\) \(\chi_{8040}(281,\cdot)\) \(\chi_{8040}(1001,\cdot)\) \(\chi_{8040}(1481,\cdot)\) \(\chi_{8040}(1721,\cdot)\) \(\chi_{8040}(1841,\cdot)\) \(\chi_{8040}(1961,\cdot)\) \(\chi_{8040}(2201,\cdot)\) \(\chi_{8040}(3161,\cdot)\) \(\chi_{8040}(3401,\cdot)\) \(\chi_{8040}(4001,\cdot)\) \(\chi_{8040}(4121,\cdot)\) \(\chi_{8040}(4241,\cdot)\) \(\chi_{8040}(4721,\cdot)\) \(\chi_{8040}(5321,\cdot)\) \(\chi_{8040}(6041,\cdot)\) \(\chi_{8040}(6281,\cdot)\) \(\chi_{8040}(6761,\cdot)\) \(\chi_{8040}(7481,\cdot)\) \(\chi_{8040}(7841,\cdot)\)
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
\((6031,4021,2681,3217,5161)\) → \((1,1,-1,1,e\left(\frac{53}{66}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8040 }(41, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{33}\right)\) |
sage: chi.jacobi_sum(n)