from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9680, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,22,40]))
pari: [g,chi] = znchar(Mod(309,9680))
Basic properties
| Modulus: | \(9680\) | |
| Conductor: | \(9680\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | 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 9680.dy
\(\chi_{9680}(309,\cdot)\) \(\chi_{9680}(749,\cdot)\) \(\chi_{9680}(1189,\cdot)\) \(\chi_{9680}(1629,\cdot)\) \(\chi_{9680}(2069,\cdot)\) \(\chi_{9680}(2509,\cdot)\) \(\chi_{9680}(2949,\cdot)\) \(\chi_{9680}(3829,\cdot)\) \(\chi_{9680}(4269,\cdot)\) \(\chi_{9680}(4709,\cdot)\) \(\chi_{9680}(5149,\cdot)\) \(\chi_{9680}(5589,\cdot)\) \(\chi_{9680}(6029,\cdot)\) \(\chi_{9680}(6469,\cdot)\) \(\chi_{9680}(6909,\cdot)\) \(\chi_{9680}(7349,\cdot)\) \(\chi_{9680}(7789,\cdot)\) \(\chi_{9680}(8669,\cdot)\) \(\chi_{9680}(9109,\cdot)\) \(\chi_{9680}(9549,\cdot)\)
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
\((3631,2421,1937,4721)\) → \((1,i,-1,e\left(\frac{10}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 9680 }(309, a) \) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{4}{11}\right)\) | \(-1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(-i\) | \(e\left(\frac{9}{44}\right)\) |
sage: chi.jacobi_sum(n)