from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6930, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,15,20,12]))
pari: [g,chi] = znchar(Mod(37,6930))
Basic properties
Modulus: | \(6930\) | |
Conductor: | \(385\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{385}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6930.ip
\(\chi_{6930}(37,\cdot)\) \(\chi_{6930}(163,\cdot)\) \(\chi_{6930}(487,\cdot)\) \(\chi_{6930}(1423,\cdot)\) \(\chi_{6930}(1747,\cdot)\) \(\chi_{6930}(1873,\cdot)\) \(\chi_{6930}(2557,\cdot)\) \(\chi_{6930}(3007,\cdot)\) \(\chi_{6930}(3133,\cdot)\) \(\chi_{6930}(3943,\cdot)\) \(\chi_{6930}(4393,\cdot)\) \(\chi_{6930}(4447,\cdot)\) \(\chi_{6930}(5527,\cdot)\) \(\chi_{6930}(5707,\cdot)\) \(\chi_{6930}(5833,\cdot)\) \(\chi_{6930}(6913,\cdot)\)
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
\((1541,1387,2971,2521)\) → \((1,i,e\left(\frac{1}{3}\right),e\left(\frac{1}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
\( \chi_{ 6930 }(37, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(-i\) | \(e\left(\frac{31}{60}\right)\) |
sage: chi.jacobi_sum(n)