from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1620, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,37,27]))
pari: [g,chi] = znchar(Mod(29,1620))
Basic properties
Modulus: | \(1620\) | |
Conductor: | \(405\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{405}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1620.bl
\(\chi_{1620}(29,\cdot)\) \(\chi_{1620}(149,\cdot)\) \(\chi_{1620}(209,\cdot)\) \(\chi_{1620}(329,\cdot)\) \(\chi_{1620}(389,\cdot)\) \(\chi_{1620}(509,\cdot)\) \(\chi_{1620}(569,\cdot)\) \(\chi_{1620}(689,\cdot)\) \(\chi_{1620}(749,\cdot)\) \(\chi_{1620}(869,\cdot)\) \(\chi_{1620}(929,\cdot)\) \(\chi_{1620}(1049,\cdot)\) \(\chi_{1620}(1109,\cdot)\) \(\chi_{1620}(1229,\cdot)\) \(\chi_{1620}(1289,\cdot)\) \(\chi_{1620}(1409,\cdot)\) \(\chi_{1620}(1469,\cdot)\) \(\chi_{1620}(1589,\cdot)\)
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
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 1620 }(29, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{17}{54}\right)\) |
sage: chi.jacobi_sum(n)