from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2268, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,31,18]))
pari: [g,chi] = znchar(Mod(65,2268))
Basic properties
Modulus: | \(2268\) | |
Conductor: | \(567\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{567}(65,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2268.db
\(\chi_{2268}(65,\cdot)\) \(\chi_{2268}(221,\cdot)\) \(\chi_{2268}(317,\cdot)\) \(\chi_{2268}(473,\cdot)\) \(\chi_{2268}(569,\cdot)\) \(\chi_{2268}(725,\cdot)\) \(\chi_{2268}(821,\cdot)\) \(\chi_{2268}(977,\cdot)\) \(\chi_{2268}(1073,\cdot)\) \(\chi_{2268}(1229,\cdot)\) \(\chi_{2268}(1325,\cdot)\) \(\chi_{2268}(1481,\cdot)\) \(\chi_{2268}(1577,\cdot)\) \(\chi_{2268}(1733,\cdot)\) \(\chi_{2268}(1829,\cdot)\) \(\chi_{2268}(1985,\cdot)\) \(\chi_{2268}(2081,\cdot)\) \(\chi_{2268}(2237,\cdot)\)
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
\((1135,1541,325)\) → \((1,e\left(\frac{31}{54}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 2268 }(65, a) \) | \(-1\) | \(1\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) |
sage: chi.jacobi_sum(n)