from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4235, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,22,24]))
pari: [g,chi] = znchar(Mod(573,4235))
Basic properties
Modulus: | \(4235\) | |
Conductor: | \(4235\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4235.ce
\(\chi_{4235}(188,\cdot)\) \(\chi_{4235}(342,\cdot)\) \(\chi_{4235}(573,\cdot)\) \(\chi_{4235}(958,\cdot)\) \(\chi_{4235}(1112,\cdot)\) \(\chi_{4235}(1343,\cdot)\) \(\chi_{4235}(1497,\cdot)\) \(\chi_{4235}(1728,\cdot)\) \(\chi_{4235}(1882,\cdot)\) \(\chi_{4235}(2113,\cdot)\) \(\chi_{4235}(2267,\cdot)\) \(\chi_{4235}(2498,\cdot)\) \(\chi_{4235}(2652,\cdot)\) \(\chi_{4235}(2883,\cdot)\) \(\chi_{4235}(3037,\cdot)\) \(\chi_{4235}(3422,\cdot)\) \(\chi_{4235}(3653,\cdot)\) \(\chi_{4235}(3807,\cdot)\) \(\chi_{4235}(4038,\cdot)\) \(\chi_{4235}(4192,\cdot)\)
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
\((2542,1816,2906)\) → \((-i,-1,e\left(\frac{6}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
\( \chi_{ 4235 }(573, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{44}\right)\) | \(-i\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(-1\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) |
sage: chi.jacobi_sum(n)