from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4026, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,42,49]))
pari: [g,chi] = znchar(Mod(7,4026))
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 \(\chi_{671}(7,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4026.fr
\(\chi_{4026}(7,\cdot)\) \(\chi_{4026}(193,\cdot)\) \(\chi_{4026}(547,\cdot)\) \(\chi_{4026}(871,\cdot)\) \(\chi_{4026}(1129,\cdot)\) \(\chi_{4026}(1405,\cdot)\) \(\chi_{4026}(1531,\cdot)\) \(\chi_{4026}(1909,\cdot)\) \(\chi_{4026}(1921,\cdot)\) \(\chi_{4026}(2239,\cdot)\) \(\chi_{4026}(2527,\cdot)\) \(\chi_{4026}(2983,\cdot)\) \(\chi_{4026}(3043,\cdot)\) \(\chi_{4026}(3451,\cdot)\) \(\chi_{4026}(3643,\cdot)\) \(\chi_{4026}(3955,\cdot)\)
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
\((1343,1465,3235)\) → \((1,e\left(\frac{7}{10}\right),e\left(\frac{49}{60}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 4026 }(7, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{41}{60}\right)\) |
sage: chi.jacobi_sum(n)