from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1224, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,0,40,33]))
pari: [g,chi] = znchar(Mod(41,1224))
Basic properties
| Modulus: | \(1224\) | |
| Conductor: | \(153\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{153}(41,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1224.cw
\(\chi_{1224}(41,\cdot)\) \(\chi_{1224}(65,\cdot)\) \(\chi_{1224}(113,\cdot)\) \(\chi_{1224}(209,\cdot)\) \(\chi_{1224}(329,\cdot)\) \(\chi_{1224}(401,\cdot)\) \(\chi_{1224}(473,\cdot)\) \(\chi_{1224}(617,\cdot)\) \(\chi_{1224}(641,\cdot)\) \(\chi_{1224}(785,\cdot)\) \(\chi_{1224}(857,\cdot)\) \(\chi_{1224}(929,\cdot)\) \(\chi_{1224}(1049,\cdot)\) \(\chi_{1224}(1145,\cdot)\) \(\chi_{1224}(1193,\cdot)\) \(\chi_{1224}(1217,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((919,613,137,649)\) → \((1,1,e\left(\frac{5}{6}\right),e\left(\frac{11}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1224 }(41, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)