from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1620, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,25,0]))
pari: [g,chi] = znchar(Mod(101,1620))
Basic properties
| Modulus: | \(1620\) | |
| Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{81}(20,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1620.bq
\(\chi_{1620}(41,\cdot)\) \(\chi_{1620}(101,\cdot)\) \(\chi_{1620}(221,\cdot)\) \(\chi_{1620}(281,\cdot)\) \(\chi_{1620}(401,\cdot)\) \(\chi_{1620}(461,\cdot)\) \(\chi_{1620}(581,\cdot)\) \(\chi_{1620}(641,\cdot)\) \(\chi_{1620}(761,\cdot)\) \(\chi_{1620}(821,\cdot)\) \(\chi_{1620}(941,\cdot)\) \(\chi_{1620}(1001,\cdot)\) \(\chi_{1620}(1121,\cdot)\) \(\chi_{1620}(1181,\cdot)\) \(\chi_{1620}(1301,\cdot)\) \(\chi_{1620}(1361,\cdot)\) \(\chi_{1620}(1481,\cdot)\) \(\chi_{1620}(1541,\cdot)\)
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 54 polynomial |
Values on generators
\((811,1541,1297)\) → \((1,e\left(\frac{25}{54}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1620 }(101, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{29}{54}\right)\) |
sage: chi.jacobi_sum(n)