from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6040, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,25,25,38]))
pari: [g,chi] = znchar(Mod(29,6040))
Basic properties
| Modulus: | \(6040\) | |
| Conductor: | \(6040\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6040.dx
\(\chi_{6040}(29,\cdot)\) \(\chi_{6040}(229,\cdot)\) \(\chi_{6040}(429,\cdot)\) \(\chi_{6040}(1029,\cdot)\) \(\chi_{6040}(1469,\cdot)\) \(\chi_{6040}(2309,\cdot)\) \(\chi_{6040}(2349,\cdot)\) \(\chi_{6040}(2389,\cdot)\) \(\chi_{6040}(3029,\cdot)\) \(\chi_{6040}(3269,\cdot)\) \(\chi_{6040}(3749,\cdot)\) \(\chi_{6040}(3869,\cdot)\) \(\chi_{6040}(4149,\cdot)\) \(\chi_{6040}(4309,\cdot)\) \(\chi_{6040}(4429,\cdot)\) \(\chi_{6040}(4749,\cdot)\) \(\chi_{6040}(4829,\cdot)\) \(\chi_{6040}(5069,\cdot)\) \(\chi_{6040}(5829,\cdot)\) \(\chi_{6040}(5909,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
| Fixed field: | Number field defined by a degree 50 polynomial |
Values on generators
\((1511,3021,2417,761)\) → \((1,-1,-1,e\left(\frac{19}{25}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 6040 }(29, a) \) | \(1\) | \(1\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{25}\right)\) |
sage: chi.jacobi_sum(n)