from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4026, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,6,13]))
pari: [g,chi] = znchar(Mod(79,4026))
Basic properties
| Modulus: | \(4026\) | |
| Conductor: | \(671\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{671}(79,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4026.gk
\(\chi_{4026}(79,\cdot)\) \(\chi_{4026}(409,\cdot)\) \(\chi_{4026}(481,\cdot)\) \(\chi_{4026}(787,\cdot)\) \(\chi_{4026}(1063,\cdot)\) \(\chi_{4026}(1447,\cdot)\) \(\chi_{4026}(1471,\cdot)\) \(\chi_{4026}(1987,\cdot)\) \(\chi_{4026}(2125,\cdot)\) \(\chi_{4026}(2389,\cdot)\) \(\chi_{4026}(2593,\cdot)\) \(\chi_{4026}(2701,\cdot)\) \(\chi_{4026}(3109,\cdot)\) \(\chi_{4026}(3361,\cdot)\) \(\chi_{4026}(3385,\cdot)\) \(\chi_{4026}(3967,\cdot)\)
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
\((1343,1465,3235)\) → \((1,e\left(\frac{1}{10}\right),e\left(\frac{13}{60}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4026 }(79, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) |
sage: chi.jacobi_sum(n)