from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7200, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,50,6]))
pari: [g,chi] = znchar(Mod(7079,7200))
Basic properties
| Modulus: | \(7200\) | |
| Conductor: | \(3600\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3600}(779,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7200.hx
\(\chi_{7200}(119,\cdot)\) \(\chi_{7200}(839,\cdot)\) \(\chi_{7200}(1319,\cdot)\) \(\chi_{7200}(1559,\cdot)\) \(\chi_{7200}(2039,\cdot)\) \(\chi_{7200}(2279,\cdot)\) \(\chi_{7200}(2759,\cdot)\) \(\chi_{7200}(3479,\cdot)\) \(\chi_{7200}(3719,\cdot)\) \(\chi_{7200}(4439,\cdot)\) \(\chi_{7200}(4919,\cdot)\) \(\chi_{7200}(5159,\cdot)\) \(\chi_{7200}(5639,\cdot)\) \(\chi_{7200}(5879,\cdot)\) \(\chi_{7200}(6359,\cdot)\) \(\chi_{7200}(7079,\cdot)\)
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
\((6751,901,6401,577)\) → \((-1,i,e\left(\frac{5}{6}\right),e\left(\frac{1}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7200 }(7079, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) |
sage: chi.jacobi_sum(n)