from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8993, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([11,28]))
pari: [g,chi] = znchar(Mod(795,8993))
Basic properties
| Modulus: | \(8993\) | |
| Conductor: | \(391\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{391}(13,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8993.q
\(\chi_{8993}(795,\cdot)\) \(\chi_{8993}(863,\cdot)\) \(\chi_{8993}(1016,\cdot)\) \(\chi_{8993}(1228,\cdot)\) \(\chi_{8993}(1313,\cdot)\) \(\chi_{8993}(1764,\cdot)\) \(\chi_{8993}(2053,\cdot)\) \(\chi_{8993}(2911,\cdot)\) \(\chi_{8993}(2979,\cdot)\) \(\chi_{8993}(3132,\cdot)\) \(\chi_{8993}(3821,\cdot)\) \(\chi_{8993}(3880,\cdot)\) \(\chi_{8993}(4169,\cdot)\) \(\chi_{8993}(5937,\cdot)\) \(\chi_{8993}(6218,\cdot)\) \(\chi_{8993}(6320,\cdot)\) \(\chi_{8993}(8105,\cdot)\) \(\chi_{8993}(8190,\cdot)\) \(\chi_{8993}(8334,\cdot)\) \(\chi_{8993}(8436,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((530,7940)\) → \((i,e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8993 }(795, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) |
sage: chi.jacobi_sum(n)