from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6040, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,45,34]))
pari: [g,chi] = znchar(Mod(113,6040))
Basic properties
| Modulus: | \(6040\) | |
| Conductor: | \(755\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{755}(113,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6040.eh
\(\chi_{6040}(113,\cdot)\) \(\chi_{6040}(217,\cdot)\) \(\chi_{6040}(377,\cdot)\) \(\chi_{6040}(753,\cdot)\) \(\chi_{6040}(1657,\cdot)\) \(\chi_{6040}(2137,\cdot)\) \(\chi_{6040}(2633,\cdot)\) \(\chi_{6040}(2793,\cdot)\) \(\chi_{6040}(3217,\cdot)\) \(\chi_{6040}(3457,\cdot)\) \(\chi_{6040}(3737,\cdot)\) \(\chi_{6040}(4073,\cdot)\) \(\chi_{6040}(4377,\cdot)\) \(\chi_{6040}(4553,\cdot)\) \(\chi_{6040}(5633,\cdot)\) \(\chi_{6040}(5873,\cdot)\)
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
\((1511,3021,2417,761)\) → \((1,1,-i,e\left(\frac{17}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 6040 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(-1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{9}{20}\right)\) |
sage: chi.jacobi_sum(n)