from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,36,35]))
pari: [g,chi] = znchar(Mod(271,2600))
Basic properties
| Modulus: | \(2600\) | |
| Conductor: | \(1300\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1300}(271,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2600.fh
\(\chi_{2600}(71,\cdot)\) \(\chi_{2600}(111,\cdot)\) \(\chi_{2600}(271,\cdot)\) \(\chi_{2600}(431,\cdot)\) \(\chi_{2600}(591,\cdot)\) \(\chi_{2600}(631,\cdot)\) \(\chi_{2600}(791,\cdot)\) \(\chi_{2600}(1111,\cdot)\) \(\chi_{2600}(1311,\cdot)\) \(\chi_{2600}(1471,\cdot)\) \(\chi_{2600}(1631,\cdot)\) \(\chi_{2600}(1671,\cdot)\) \(\chi_{2600}(1831,\cdot)\) \(\chi_{2600}(1991,\cdot)\) \(\chi_{2600}(2191,\cdot)\) \(\chi_{2600}(2511,\cdot)\)
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
\((1951,1301,1977,1601)\) → \((-1,1,e\left(\frac{3}{5}\right),e\left(\frac{7}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 2600 }(271, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) |
sage: chi.jacobi_sum(n)