from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2093, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,12]))
pari: [g,chi] = znchar(Mod(8,2093))
Basic properties
Modulus: | \(2093\) | |
Conductor: | \(299\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{299}(8,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2093.cv
\(\chi_{2093}(8,\cdot)\) \(\chi_{2093}(190,\cdot)\) \(\chi_{2093}(239,\cdot)\) \(\chi_{2093}(330,\cdot)\) \(\chi_{2093}(372,\cdot)\) \(\chi_{2093}(463,\cdot)\) \(\chi_{2093}(512,\cdot)\) \(\chi_{2093}(554,\cdot)\) \(\chi_{2093}(694,\cdot)\) \(\chi_{2093}(785,\cdot)\) \(\chi_{2093}(876,\cdot)\) \(\chi_{2093}(1191,\cdot)\) \(\chi_{2093}(1373,\cdot)\) \(\chi_{2093}(1513,\cdot)\) \(\chi_{2093}(1646,\cdot)\) \(\chi_{2093}(1695,\cdot)\) \(\chi_{2093}(1737,\cdot)\) \(\chi_{2093}(1968,\cdot)\) \(\chi_{2093}(2010,\cdot)\) \(\chi_{2093}(2059,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((1795,1289,1730)\) → \((1,i,e\left(\frac{3}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
\( \chi_{ 2093 }(8, a) \) | \(-1\) | \(1\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) |
sage: chi.jacobi_sum(n)