from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4030, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,30,38]))
pari: [g,chi] = znchar(Mod(2833,4030))
Basic properties
| Modulus: | \(4030\) | |
| 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 \(\chi_{2015}(818,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4030.gv
\(\chi_{4030}(207,\cdot)\) \(\chi_{4030}(363,\cdot)\) \(\chi_{4030}(623,\cdot)\) \(\chi_{4030}(1013,\cdot)\) \(\chi_{4030}(1377,\cdot)\) \(\chi_{4030}(2027,\cdot)\) \(\chi_{4030}(2183,\cdot)\) \(\chi_{4030}(2287,\cdot)\) \(\chi_{4030}(2677,\cdot)\) \(\chi_{4030}(2807,\cdot)\) \(\chi_{4030}(2833,\cdot)\) \(\chi_{4030}(3093,\cdot)\) \(\chi_{4030}(3483,\cdot)\) \(\chi_{4030}(3587,\cdot)\) \(\chi_{4030}(3613,\cdot)\) \(\chi_{4030}(3847,\cdot)\)
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
\((807,1861,2731)\) → \((-i,-1,e\left(\frac{19}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 4030 }(2833, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) |
sage: chi.jacobi_sum(n)