from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4140, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,55,33,42]))
pari: [g,chi] = znchar(Mod(59,4140))
Basic properties
| Modulus: | \(4140\) | |
| Conductor: | \(4140\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4140.dh
\(\chi_{4140}(59,\cdot)\) \(\chi_{4140}(119,\cdot)\) \(\chi_{4140}(239,\cdot)\) \(\chi_{4140}(959,\cdot)\) \(\chi_{4140}(1139,\cdot)\) \(\chi_{4140}(1199,\cdot)\) \(\chi_{4140}(1319,\cdot)\) \(\chi_{4140}(1499,\cdot)\) \(\chi_{4140}(1559,\cdot)\) \(\chi_{4140}(2099,\cdot)\) \(\chi_{4140}(2279,\cdot)\) \(\chi_{4140}(2579,\cdot)\) \(\chi_{4140}(2819,\cdot)\) \(\chi_{4140}(2939,\cdot)\) \(\chi_{4140}(2999,\cdot)\) \(\chi_{4140}(3479,\cdot)\) \(\chi_{4140}(3659,\cdot)\) \(\chi_{4140}(3719,\cdot)\) \(\chi_{4140}(3899,\cdot)\) \(\chi_{4140}(4079,\cdot)\)
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
\((2071,461,1657,3961)\) → \((-1,e\left(\frac{5}{6}\right),-1,e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4140 }(59, a) \) | \(1\) | \(1\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) |
sage: chi.jacobi_sum(n)