from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1620, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,37,0]))
pari: [g,chi] = znchar(Mod(1001,1620))
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 \(\chi_{81}(29,\cdot)\) | 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
\((811,1541,1297)\) → \((1,e\left(\frac{37}{54}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 1620 }(1001, a) \) | \(-1\) | \(1\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{17}{54}\right)\) |
sage: chi.jacobi_sum(n)