from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4026, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,36,49]))
pari: [g,chi] = znchar(Mod(251,4026))
Basic properties
| Modulus: | \(4026\) | |
| Conductor: | \(2013\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2013}(251,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4026.gh
\(\chi_{4026}(251,\cdot)\) \(\chi_{4026}(311,\cdot)\) \(\chi_{4026}(323,\cdot)\) \(\chi_{4026}(401,\cdot)\) \(\chi_{4026}(653,\cdot)\) \(\chi_{4026}(1373,\cdot)\) \(\chi_{4026}(1523,\cdot)\) \(\chi_{4026}(1763,\cdot)\) \(\chi_{4026}(2165,\cdot)\) \(\chi_{4026}(2369,\cdot)\) \(\chi_{4026}(2381,\cdot)\) \(\chi_{4026}(2423,\cdot)\) \(\chi_{4026}(2633,\cdot)\) \(\chi_{4026}(3287,\cdot)\) \(\chi_{4026}(3503,\cdot)\) \(\chi_{4026}(3677,\cdot)\)
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
\((1343,1465,3235)\) → \((-1,e\left(\frac{3}{5}\right),e\left(\frac{49}{60}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4026 }(251, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)