from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4030, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,55,22]))
pari: [g,chi] = znchar(Mod(1749,4030))
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 \(\chi_{2015}(1749,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4030.ho
\(\chi_{4030}(579,\cdot)\) \(\chi_{4030}(1159,\cdot)\) \(\chi_{4030}(1189,\cdot)\) \(\chi_{4030}(1419,\cdot)\) \(\chi_{4030}(1749,\cdot)\) \(\chi_{4030}(1779,\cdot)\) \(\chi_{4030}(1939,\cdot)\) \(\chi_{4030}(2039,\cdot)\) \(\chi_{4030}(2099,\cdot)\) \(\chi_{4030}(2559,\cdot)\) \(\chi_{4030}(2749,\cdot)\) \(\chi_{4030}(2979,\cdot)\) \(\chi_{4030}(3049,\cdot)\) \(\chi_{4030}(3599,\cdot)\) \(\chi_{4030}(3919,\cdot)\) \(\chi_{4030}(3959,\cdot)\)
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
\((807,1861,2731)\) → \((-1,e\left(\frac{11}{12}\right),e\left(\frac{11}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 4030 }(1749, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{29}{30}\right)\) |
sage: chi.jacobi_sum(n)