from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4026, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,24,23]))
pari: [g,chi] = znchar(Mod(71,4026))
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 \(\chi_{2013}(71,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4026.gi
\(\chi_{4026}(71,\cdot)\) \(\chi_{4026}(383,\cdot)\) \(\chi_{4026}(575,\cdot)\) \(\chi_{4026}(983,\cdot)\) \(\chi_{4026}(1043,\cdot)\) \(\chi_{4026}(1499,\cdot)\) \(\chi_{4026}(1787,\cdot)\) \(\chi_{4026}(2105,\cdot)\) \(\chi_{4026}(2117,\cdot)\) \(\chi_{4026}(2495,\cdot)\) \(\chi_{4026}(2621,\cdot)\) \(\chi_{4026}(2897,\cdot)\) \(\chi_{4026}(3155,\cdot)\) \(\chi_{4026}(3479,\cdot)\) \(\chi_{4026}(3833,\cdot)\) \(\chi_{4026}(4019,\cdot)\)
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
\((1343,1465,3235)\) → \((-1,e\left(\frac{2}{5}\right),e\left(\frac{23}{60}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4026 }(71, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) |
sage: chi.jacobi_sum(n)