from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6069, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,40,45]))
pari: [g,chi] = znchar(Mod(40,6069))
Basic properties
| Modulus: | \(6069\) | |
| Conductor: | \(119\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{119}(40,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6069.bt
\(\chi_{6069}(40,\cdot)\) \(\chi_{6069}(1081,\cdot)\) \(\chi_{6069}(1405,\cdot)\) \(\chi_{6069}(1510,\cdot)\) \(\chi_{6069}(1669,\cdot)\) \(\chi_{6069}(1774,\cdot)\) \(\chi_{6069}(2098,\cdot)\) \(\chi_{6069}(3139,\cdot)\) \(\chi_{6069}(3244,\cdot)\) \(\chi_{6069}(3337,\cdot)\) \(\chi_{6069}(3832,\cdot)\) \(\chi_{6069}(4177,\cdot)\) \(\chi_{6069}(5071,\cdot)\) \(\chi_{6069}(5416,\cdot)\) \(\chi_{6069}(5911,\cdot)\) \(\chi_{6069}(6004,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((2024,4336,3760)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{15}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
| \( \chi_{ 6069 }(40, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(i\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{7}{16}\right)\) |
sage: chi.jacobi_sum(n)