from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4026, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,24,25]))
pari: [g,chi] = znchar(Mod(467,4026))
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 \(\chi_{2013}(467,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4026.gf
\(\chi_{4026}(467,\cdot)\) \(\chi_{4026}(509,\cdot)\) \(\chi_{4026}(581,\cdot)\) \(\chi_{4026}(1127,\cdot)\) \(\chi_{4026}(1241,\cdot)\) \(\chi_{4026}(1313,\cdot)\) \(\chi_{4026}(1565,\cdot)\) \(\chi_{4026}(1973,\cdot)\) \(\chi_{4026}(2225,\cdot)\) \(\chi_{4026}(2297,\cdot)\) \(\chi_{4026}(2777,\cdot)\) \(\chi_{4026}(2957,\cdot)\) \(\chi_{4026}(3029,\cdot)\) \(\chi_{4026}(3437,\cdot)\) \(\chi_{4026}(3689,\cdot)\) \(\chi_{4026}(3875,\cdot)\)
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
\((1343,1465,3235)\) → \((-1,e\left(\frac{2}{5}\right),e\left(\frac{5}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 4026 }(467, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(i\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) |
sage: chi.jacobi_sum(n)