from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4140, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,55,0,6]))
pari: [g,chi] = znchar(Mod(761,4140))
Basic properties
| Modulus: | \(4140\) | |
| Conductor: | \(207\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{207}(140,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4140.dg
\(\chi_{4140}(41,\cdot)\) \(\chi_{4140}(101,\cdot)\) \(\chi_{4140}(581,\cdot)\) \(\chi_{4140}(761,\cdot)\) \(\chi_{4140}(821,\cdot)\) \(\chi_{4140}(1001,\cdot)\) \(\chi_{4140}(1181,\cdot)\) \(\chi_{4140}(1301,\cdot)\) \(\chi_{4140}(1361,\cdot)\) \(\chi_{4140}(1481,\cdot)\) \(\chi_{4140}(2201,\cdot)\) \(\chi_{4140}(2381,\cdot)\) \(\chi_{4140}(2441,\cdot)\) \(\chi_{4140}(2561,\cdot)\) \(\chi_{4140}(2741,\cdot)\) \(\chi_{4140}(2801,\cdot)\) \(\chi_{4140}(3341,\cdot)\) \(\chi_{4140}(3521,\cdot)\) \(\chi_{4140}(3821,\cdot)\) \(\chi_{4140}(4061,\cdot)\)
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
\((2071,461,1657,3961)\) → \((1,e\left(\frac{5}{6}\right),1,e\left(\frac{1}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4140 }(761, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) |
sage: chi.jacobi_sum(n)