from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7200, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,40,33]))
pari: [g,chi] = znchar(Mod(223,7200))
Basic properties
| Modulus: | \(7200\) | |
| Conductor: | \(900\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{900}(223,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7200.hg
\(\chi_{7200}(223,\cdot)\) \(\chi_{7200}(1087,\cdot)\) \(\chi_{7200}(1183,\cdot)\) \(\chi_{7200}(1663,\cdot)\) \(\chi_{7200}(2047,\cdot)\) \(\chi_{7200}(2527,\cdot)\) \(\chi_{7200}(2623,\cdot)\) \(\chi_{7200}(3103,\cdot)\) \(\chi_{7200}(3487,\cdot)\) \(\chi_{7200}(3967,\cdot)\) \(\chi_{7200}(4063,\cdot)\) \(\chi_{7200}(4927,\cdot)\) \(\chi_{7200}(5503,\cdot)\) \(\chi_{7200}(5983,\cdot)\) \(\chi_{7200}(6367,\cdot)\) \(\chi_{7200}(6847,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((6751,901,6401,577)\) → \((-1,1,e\left(\frac{2}{3}\right),e\left(\frac{11}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7200 }(223, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{8}{15}\right)\) |
sage: chi.jacobi_sum(n)