from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3450, base_ring=CyclotomicField(10))
M = H._module
chi = DirichletCharacter(H, M([5,3,5]))
pari: [g,chi] = znchar(Mod(689,3450))
Basic properties
| Modulus: | \(3450\) | |
| Conductor: | \(1725\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(10\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1725}(689,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3450.r
\(\chi_{3450}(689,\cdot)\) \(\chi_{3450}(1379,\cdot)\) \(\chi_{3450}(2069,\cdot)\) \(\chi_{3450}(2759,\cdot)\)
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: | \(\Q(\zeta_{5})\) |
| Fixed field: | 10.10.1193261222076416015625.1 |
Values on generators
\((1151,277,1201)\) → \((-1,e\left(\frac{3}{10}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3450 }(689, a) \) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)