from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4730, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,6,20]))
pari: [g,chi] = znchar(Mod(3863,4730))
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 \(\chi_{2365}(1498,\cdot)\) | 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
\((947,431,1981)\) → \((-i,e\left(\frac{1}{10}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 4730 }(3863, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(-1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{13}{15}\right)\) |
sage: chi.jacobi_sum(n)