from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9680, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,0,16]))
pari: [g,chi] = znchar(Mod(221,9680))
Basic properties
Modulus: | \(9680\) | |
Conductor: | \(1936\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1936}(221,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9680.ej
\(\chi_{9680}(221,\cdot)\) \(\chi_{9680}(661,\cdot)\) \(\chi_{9680}(1101,\cdot)\) \(\chi_{9680}(1541,\cdot)\) \(\chi_{9680}(1981,\cdot)\) \(\chi_{9680}(2861,\cdot)\) \(\chi_{9680}(3301,\cdot)\) \(\chi_{9680}(3741,\cdot)\) \(\chi_{9680}(4181,\cdot)\) \(\chi_{9680}(4621,\cdot)\) \(\chi_{9680}(5061,\cdot)\) \(\chi_{9680}(5501,\cdot)\) \(\chi_{9680}(5941,\cdot)\) \(\chi_{9680}(6381,\cdot)\) \(\chi_{9680}(6821,\cdot)\) \(\chi_{9680}(7701,\cdot)\) \(\chi_{9680}(8141,\cdot)\) \(\chi_{9680}(8581,\cdot)\) \(\chi_{9680}(9021,\cdot)\) \(\chi_{9680}(9461,\cdot)\)
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
\((3631,2421,1937,4721)\) → \((1,-i,1,e\left(\frac{4}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 9680 }(221, a) \) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{1}{22}\right)\) | \(-1\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(-i\) | \(e\left(\frac{19}{44}\right)\) |
sage: chi.jacobi_sum(n)