from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6930, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,15,50,48]))
pari: [g,chi] = znchar(Mod(47,6930))
Basic properties
Modulus: | \(6930\) | |
Conductor: | \(3465\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3465}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6930.ix
\(\chi_{6930}(47,\cdot)\) \(\chi_{6930}(1193,\cdot)\) \(\chi_{6930}(1307,\cdot)\) \(\chi_{6930}(1433,\cdot)\) \(\chi_{6930}(1697,\cdot)\) \(\chi_{6930}(2567,\cdot)\) \(\chi_{6930}(2693,\cdot)\) \(\chi_{6930}(2957,\cdot)\) \(\chi_{6930}(3083,\cdot)\) \(\chi_{6930}(3953,\cdot)\) \(\chi_{6930}(4217,\cdot)\) \(\chi_{6930}(4343,\cdot)\) \(\chi_{6930}(5087,\cdot)\) \(\chi_{6930}(5603,\cdot)\) \(\chi_{6930}(6473,\cdot)\) \(\chi_{6930}(6737,\cdot)\)
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
\((1541,1387,2971,2521)\) → \((e\left(\frac{1}{6}\right),i,e\left(\frac{5}{6}\right),e\left(\frac{4}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
\( \chi_{ 6930 }(47, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(i\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{59}{60}\right)\) |
sage: chi.jacobi_sum(n)