from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9680, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,22,33,20]))
pari: [g,chi] = znchar(Mod(3543,9680))
Basic properties
| Modulus: | \(9680\) | |
| Conductor: | \(4840\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{4840}(1123,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9680.eg
\(\chi_{9680}(23,\cdot)\) \(\chi_{9680}(903,\cdot)\) \(\chi_{9680}(1607,\cdot)\) \(\chi_{9680}(1783,\cdot)\) \(\chi_{9680}(2487,\cdot)\) \(\chi_{9680}(3367,\cdot)\) \(\chi_{9680}(3543,\cdot)\) \(\chi_{9680}(4247,\cdot)\) \(\chi_{9680}(4423,\cdot)\) \(\chi_{9680}(5127,\cdot)\) \(\chi_{9680}(5303,\cdot)\) \(\chi_{9680}(6007,\cdot)\) \(\chi_{9680}(6183,\cdot)\) \(\chi_{9680}(6887,\cdot)\) \(\chi_{9680}(7063,\cdot)\) \(\chi_{9680}(7767,\cdot)\) \(\chi_{9680}(7943,\cdot)\) \(\chi_{9680}(8647,\cdot)\) \(\chi_{9680}(8823,\cdot)\) \(\chi_{9680}(9527,\cdot)\)
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
\((3631,2421,1937,4721)\) → \((-1,-1,-i,e\left(\frac{5}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 9680 }(3543, a) \) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{19}{44}\right)\) | \(-1\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(-i\) | \(e\left(\frac{8}{11}\right)\) |
sage: chi.jacobi_sum(n)