from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9680, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,22,33,38]))
pari: [g,chi] = znchar(Mod(6313,9680))
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 \(\chi_{4840}(3893,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9680.ee
\(\chi_{9680}(153,\cdot)\) \(\chi_{9680}(857,\cdot)\) \(\chi_{9680}(1033,\cdot)\) \(\chi_{9680}(1737,\cdot)\) \(\chi_{9680}(1913,\cdot)\) \(\chi_{9680}(2617,\cdot)\) \(\chi_{9680}(2793,\cdot)\) \(\chi_{9680}(3497,\cdot)\) \(\chi_{9680}(3673,\cdot)\) \(\chi_{9680}(4377,\cdot)\) \(\chi_{9680}(4553,\cdot)\) \(\chi_{9680}(5257,\cdot)\) \(\chi_{9680}(5433,\cdot)\) \(\chi_{9680}(6137,\cdot)\) \(\chi_{9680}(6313,\cdot)\) \(\chi_{9680}(7193,\cdot)\) \(\chi_{9680}(7897,\cdot)\) \(\chi_{9680}(8073,\cdot)\) \(\chi_{9680}(8777,\cdot)\) \(\chi_{9680}(9657,\cdot)\)
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: | 44.44.212907048724487789150739362720525788138562346514417698933058002742934795048229975182497632223232000000000000000000000000000000000.1 |
Values on generators
\((3631,2421,1937,4721)\) → \((1,-1,-i,e\left(\frac{19}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 9680 }(6313, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{35}{44}\right)\) | \(-1\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(i\) | \(e\left(\frac{15}{22}\right)\) |
sage: chi.jacobi_sum(n)