from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6045, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,15,25,36]))
pari: [g,chi] = znchar(Mod(97,6045))
Basic properties
| Modulus: | \(6045\) | |
| Conductor: | \(2015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2015}(97,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6045.oy
\(\chi_{6045}(97,\cdot)\) \(\chi_{6045}(163,\cdot)\) \(\chi_{6045}(388,\cdot)\) \(\chi_{6045}(748,\cdot)\) \(\chi_{6045}(1558,\cdot)\) \(\chi_{6045}(2017,\cdot)\) \(\chi_{6045}(2143,\cdot)\) \(\chi_{6045}(2992,\cdot)\) \(\chi_{6045}(3412,\cdot)\) \(\chi_{6045}(4063,\cdot)\) \(\chi_{6045}(4162,\cdot)\) \(\chi_{6045}(4387,\cdot)\) \(\chi_{6045}(4747,\cdot)\) \(\chi_{6045}(5038,\cdot)\) \(\chi_{6045}(5458,\cdot)\) \(\chi_{6045}(5557,\cdot)\)
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
\((4031,4837,1861,2731)\) → \((1,i,e\left(\frac{5}{12}\right),e\left(\frac{3}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 6045 }(97, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) |
sage: chi.jacobi_sum(n)