from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1800, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,10,10,19]))
pari: [g,chi] = znchar(Mod(1763,1800))
Basic properties
| Modulus: | \(1800\) | |
| Conductor: | \(600\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{600}(563,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1800.cu
\(\chi_{1800}(323,\cdot)\) \(\chi_{1800}(467,\cdot)\) \(\chi_{1800}(683,\cdot)\) \(\chi_{1800}(827,\cdot)\) \(\chi_{1800}(1187,\cdot)\) \(\chi_{1800}(1403,\cdot)\) \(\chi_{1800}(1547,\cdot)\) \(\chi_{1800}(1763,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | 20.0.184528125000000000000000000000000000000.1 |
Values on generators
\((1351,901,1001,577)\) → \((-1,-1,-1,e\left(\frac{19}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1800 }(1763, a) \) | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)