from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6930, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,45,40,18]))
pari: [g,chi] = znchar(Mod(5233,6930))
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 \(\chi_{3465}(1768,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6930.iy
\(\chi_{6930}(193,\cdot)\) \(\chi_{6930}(457,\cdot)\) \(\chi_{6930}(1327,\cdot)\) \(\chi_{6930}(1843,\cdot)\) \(\chi_{6930}(2587,\cdot)\) \(\chi_{6930}(2713,\cdot)\) \(\chi_{6930}(2977,\cdot)\) \(\chi_{6930}(3847,\cdot)\) \(\chi_{6930}(3973,\cdot)\) \(\chi_{6930}(4237,\cdot)\) \(\chi_{6930}(4363,\cdot)\) \(\chi_{6930}(5233,\cdot)\) \(\chi_{6930}(5497,\cdot)\) \(\chi_{6930}(5623,\cdot)\) \(\chi_{6930}(5737,\cdot)\) \(\chi_{6930}(6883,\cdot)\)
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
\((1541,1387,2971,2521)\) → \((e\left(\frac{1}{3}\right),-i,e\left(\frac{2}{3}\right),e\left(\frac{3}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
\( \chi_{ 6930 }(5233, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(i\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{49}{60}\right)\) |
sage: chi.jacobi_sum(n)