from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,5,0,9]))
pari: [g,chi] = znchar(Mod(37,3600))
Basic properties
| Modulus: | \(3600\) | |
| Conductor: | \(400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{400}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3600.eg
\(\chi_{3600}(37,\cdot)\) \(\chi_{3600}(253,\cdot)\) \(\chi_{3600}(973,\cdot)\) \(\chi_{3600}(1477,\cdot)\) \(\chi_{3600}(2197,\cdot)\) \(\chi_{3600}(2413,\cdot)\) \(\chi_{3600}(2917,\cdot)\) \(\chi_{3600}(3133,\cdot)\)
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: | 20.0.104857600000000000000000000000000000000000.2 |
Values on generators
\((3151,901,2801,577)\) → \((1,i,1,e\left(\frac{9}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3600 }(37, a) \) | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)