from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4026, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,36,59]))
pari: [g,chi] = znchar(Mod(31,4026))
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 \(\chi_{671}(31,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4026.fw
\(\chi_{4026}(31,\cdot)\) \(\chi_{4026}(181,\cdot)\) \(\chi_{4026}(421,\cdot)\) \(\chi_{4026}(823,\cdot)\) \(\chi_{4026}(1027,\cdot)\) \(\chi_{4026}(1039,\cdot)\) \(\chi_{4026}(1081,\cdot)\) \(\chi_{4026}(1291,\cdot)\) \(\chi_{4026}(1945,\cdot)\) \(\chi_{4026}(2161,\cdot)\) \(\chi_{4026}(2335,\cdot)\) \(\chi_{4026}(2935,\cdot)\) \(\chi_{4026}(2995,\cdot)\) \(\chi_{4026}(3007,\cdot)\) \(\chi_{4026}(3085,\cdot)\) \(\chi_{4026}(3337,\cdot)\)
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
\((1343,1465,3235)\) → \((1,e\left(\frac{3}{5}\right),e\left(\frac{59}{60}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 4026 }(31, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)