from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9200, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,0,9,0]))
pari: [g,chi] = znchar(Mod(737,9200))
Basic properties
| Modulus: | \(9200\) | |
| Conductor: | \(25\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{25}(12,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9200.co
\(\chi_{9200}(737,\cdot)\) \(\chi_{9200}(1473,\cdot)\) \(\chi_{9200}(2577,\cdot)\) \(\chi_{9200}(3313,\cdot)\) \(\chi_{9200}(4417,\cdot)\) \(\chi_{9200}(5153,\cdot)\) \(\chi_{9200}(8097,\cdot)\) \(\chi_{9200}(8833,\cdot)\)
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: | Number field defined by a degree 20 polynomial |
Values on generators
\((1151,6901,2577,1201)\) → \((1,1,e\left(\frac{9}{20}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 9200 }(737, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(i\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) |
sage: chi.jacobi_sum(n)