from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4730, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([15,42,40]))
pari: [g,chi] = znchar(Mod(3747,4730))
Basic properties
| Modulus: | \(4730\) | |
| Conductor: | \(2365\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2365}(1382,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4730.cp
\(\chi_{4730}(337,\cdot)\) \(\chi_{4730}(393,\cdot)\) \(\chi_{4730}(767,\cdot)\) \(\chi_{4730}(853,\cdot)\) \(\chi_{4730}(1283,\cdot)\) \(\chi_{4730}(1597,\cdot)\) \(\chi_{4730}(1713,\cdot)\) \(\chi_{4730}(2543,\cdot)\) \(\chi_{4730}(2917,\cdot)\) \(\chi_{4730}(3317,\cdot)\) \(\chi_{4730}(3747,\cdot)\) \(\chi_{4730}(3863,\cdot)\) \(\chi_{4730}(4177,\cdot)\) \(\chi_{4730}(4263,\cdot)\) \(\chi_{4730}(4637,\cdot)\) \(\chi_{4730}(4693,\cdot)\)
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
\((947,431,1981)\) → \((i,e\left(\frac{7}{10}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 4730 }(3747, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(-1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) |
sage: chi.jacobi_sum(n)