from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2400, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,15,10,3]))
pari: [g,chi] = znchar(Mod(233,2400))
Basic properties
| Modulus: | \(2400\) | |
| Conductor: | \(1200\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1200}(1133,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2400.dg
\(\chi_{2400}(233,\cdot)\) \(\chi_{2400}(377,\cdot)\) \(\chi_{2400}(713,\cdot)\) \(\chi_{2400}(1337,\cdot)\) \(\chi_{2400}(1673,\cdot)\) \(\chi_{2400}(1817,\cdot)\) \(\chi_{2400}(2153,\cdot)\) \(\chi_{2400}(2297,\cdot)\)
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: | 20.20.6191736422400000000000000000000000000000000000.2 |
Values on generators
\((1951,901,1601,577)\) → \((1,-i,-1,e\left(\frac{3}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2400 }(233, a) \) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) |
sage: chi.jacobi_sum(n)