from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1620, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,52,0]))
pari: [g,chi] = znchar(Mod(61,1620))
Basic properties
| Modulus: | \(1620\) | |
| Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(27\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{81}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1620.bg
\(\chi_{1620}(61,\cdot)\) \(\chi_{1620}(121,\cdot)\) \(\chi_{1620}(241,\cdot)\) \(\chi_{1620}(301,\cdot)\) \(\chi_{1620}(421,\cdot)\) \(\chi_{1620}(481,\cdot)\) \(\chi_{1620}(601,\cdot)\) \(\chi_{1620}(661,\cdot)\) \(\chi_{1620}(781,\cdot)\) \(\chi_{1620}(841,\cdot)\) \(\chi_{1620}(961,\cdot)\) \(\chi_{1620}(1021,\cdot)\) \(\chi_{1620}(1141,\cdot)\) \(\chi_{1620}(1201,\cdot)\) \(\chi_{1620}(1321,\cdot)\) \(\chi_{1620}(1381,\cdot)\) \(\chi_{1620}(1501,\cdot)\) \(\chi_{1620}(1561,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 27 polynomial |
Values on generators
\((811,1541,1297)\) → \((1,e\left(\frac{26}{27}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1620 }(61, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{27}\right)\) |
sage: chi.jacobi_sum(n)