from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7200, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,15,0,16]))
pari: [g,chi] = znchar(Mod(361,7200))
Basic properties
| Modulus: | \(7200\) | |
| 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 \(\chi_{400}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7200.ek
\(\chi_{7200}(361,\cdot)\) \(\chi_{7200}(1081,\cdot)\) \(\chi_{7200}(2521,\cdot)\) \(\chi_{7200}(3241,\cdot)\) \(\chi_{7200}(3961,\cdot)\) \(\chi_{7200}(4681,\cdot)\) \(\chi_{7200}(6121,\cdot)\) \(\chi_{7200}(6841,\cdot)\)
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
\((6751,901,6401,577)\) → \((1,-i,1,e\left(\frac{4}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7200 }(361, a) \) | \(1\) | \(1\) | \(-1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) |
sage: chi.jacobi_sum(n)