from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,12,5]))
pari: [g,chi] = znchar(Mod(41,2600))
Basic properties
Modulus: | \(2600\) | |
Conductor: | \(325\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{325}(41,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2600.ga
\(\chi_{2600}(41,\cdot)\) \(\chi_{2600}(241,\cdot)\) \(\chi_{2600}(561,\cdot)\) \(\chi_{2600}(721,\cdot)\) \(\chi_{2600}(761,\cdot)\) \(\chi_{2600}(921,\cdot)\) \(\chi_{2600}(1081,\cdot)\) \(\chi_{2600}(1241,\cdot)\) \(\chi_{2600}(1281,\cdot)\) \(\chi_{2600}(1441,\cdot)\) \(\chi_{2600}(1761,\cdot)\) \(\chi_{2600}(1961,\cdot)\) \(\chi_{2600}(2121,\cdot)\) \(\chi_{2600}(2281,\cdot)\) \(\chi_{2600}(2321,\cdot)\) \(\chi_{2600}(2481,\cdot)\)
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
\((1951,1301,1977,1601)\) → \((1,1,e\left(\frac{1}{5}\right),e\left(\frac{1}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 2600 }(41, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{15}\right)\) |
sage: chi.jacobi_sum(n)