from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6040, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,0,25,49]))
pari: [g,chi] = znchar(Mod(79,6040))
Basic properties
Modulus: | \(6040\) | |
Conductor: | \(3020\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3020}(79,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6040.dz
\(\chi_{6040}(79,\cdot)\) \(\chi_{6040}(359,\cdot)\) \(\chi_{6040}(479,\cdot)\) \(\chi_{6040}(959,\cdot)\) \(\chi_{6040}(1199,\cdot)\) \(\chi_{6040}(1839,\cdot)\) \(\chi_{6040}(1879,\cdot)\) \(\chi_{6040}(1919,\cdot)\) \(\chi_{6040}(2759,\cdot)\) \(\chi_{6040}(3199,\cdot)\) \(\chi_{6040}(3799,\cdot)\) \(\chi_{6040}(3999,\cdot)\) \(\chi_{6040}(4199,\cdot)\) \(\chi_{6040}(4359,\cdot)\) \(\chi_{6040}(4439,\cdot)\) \(\chi_{6040}(5199,\cdot)\) \(\chi_{6040}(5439,\cdot)\) \(\chi_{6040}(5519,\cdot)\) \(\chi_{6040}(5839,\cdot)\) \(\chi_{6040}(5959,\cdot)\)
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
\((1511,3021,2417,761)\) → \((-1,1,-1,e\left(\frac{49}{50}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 6040 }(79, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{50}\right)\) |
sage: chi.jacobi_sum(n)