from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4410, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,7,12]))
pari: [g,chi] = znchar(Mod(127,4410))
Basic properties
| Modulus: | \(4410\) | |
| Conductor: | \(245\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{245}(127,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4410.cu
\(\chi_{4410}(127,\cdot)\) \(\chi_{4410}(253,\cdot)\) \(\chi_{4410}(757,\cdot)\) \(\chi_{4410}(1387,\cdot)\) \(\chi_{4410}(1513,\cdot)\) \(\chi_{4410}(2017,\cdot)\) \(\chi_{4410}(2143,\cdot)\) \(\chi_{4410}(2773,\cdot)\) \(\chi_{4410}(3277,\cdot)\) \(\chi_{4410}(3403,\cdot)\) \(\chi_{4410}(3907,\cdot)\) \(\chi_{4410}(4033,\cdot)\)
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: | Number field defined by a degree 28 polynomial |
Values on generators
\((3431,2647,1081)\) → \((1,i,e\left(\frac{3}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4410 }(127, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(-1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{28}\right)\) |
sage: chi.jacobi_sum(n)