from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9800, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,14,7,6]))
pari: [g,chi] = znchar(Mod(1357,9800))
Basic properties
| Modulus: | \(9800\) | |
| Conductor: | \(1960\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1960}(1357,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9800.dq
\(\chi_{9800}(1357,\cdot)\) \(\chi_{9800}(1693,\cdot)\) \(\chi_{9800}(2757,\cdot)\) \(\chi_{9800}(3093,\cdot)\) \(\chi_{9800}(4157,\cdot)\) \(\chi_{9800}(4493,\cdot)\) \(\chi_{9800}(5557,\cdot)\) \(\chi_{9800}(5893,\cdot)\) \(\chi_{9800}(7293,\cdot)\) \(\chi_{9800}(8357,\cdot)\) \(\chi_{9800}(8693,\cdot)\) \(\chi_{9800}(9757,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | 28.28.3771654561118105678109014156786321272080955342848000000000000000000000.1 |
Values on generators
\((7351,4901,1177,5001)\) → \((1,-1,i,e\left(\frac{3}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 9800 }(1357, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(-1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)